40 DM12 Abstracts

Home , Hoffman graph, List of graphs, Stable marriage problem

40 DM12 Abstracts

IP0 [email protected] Dnes Knig Prize Lecture: Talk Title TBD Abstract not available at time of publication. IP4 Adding and Counting Zeev Dvir Princeton University In mathematics, the stuff of partitions seems like mere [email protected] child’s play. The speaker will explain how the simple task of adding and counting has fascinated many of the world’s leading mathematicians: Euler, Ramanujan, IP1 Hardy, Rademacher, Dyson, to name a few. And as is typ- Cell Complexes in Combinatorics ical in number theory, many of the most fundamental (and simple to state) questions have remained open. In 2010, Cell complexes of various kinds have been invented in the speaker, with the support of the American Institute for topology to help analyze manifolds and other spaces. By Mathematics and the National Science Foundation, assem- introducing a combinatorial structure they make algo- bled an international team of researchers to attack some rithms for computing topological invariants possible. Sim- of these problems. Come hear Professor Ono speak about plicial complexes are well-known examples. In the other their findings: new theories which solve some of the famous direction, several structures studied in combinatorics nat- old questions. urally suggest associated cell complexes. Can the link to topology provided by these cell complexes be of use for Ken Ono dealing with purely combinatorial questions, or are they Emory University just idle curiosities? The answer is definitely ”yes” in the Department of Mathematics and Computer Science simplicial case, as testified by several successes of what has [email protected] come to be called ”topological combinatorics”. But what about more general cell complexes? In the talk this question will be discussed. Examples, old and new, arising in IP5 graph, group and number theory will be reviewed. Coloring 3-colorable Graphs; Graph Theory Fi- nally Strikes Back! Anders Björner Royal Institute of Technology (KTH) We consider the problem of coloring a 3-colorable graph in [email protected] polynomial time using as few colors as possible. Starting with Wigderson in 1982 (O(n1/2) colors), Blum in 1990 came with the first polynomial improvements (O(n3/8) IP2 colors). His improvement is based on graph theoretical On Sidorenko’s Conjecture approach. Karger, Motwani, Sudan in 1994 is the first to use semi-definite programming (SDP) to give improve- The Erdos-Simonovits-Sidorenko conjecture is well-known ment, and then Karger and Blum in 1997 combines Blum’s in combinatorics but it has equivalent formulations in anal- method with the SDP improvement to show that O(n0.2142 ) ysis and probability theory. The shortest formulation is an colors suffices. Since then, the only improvements in semi- integral inequality related to Mayer integrals in statistical definite programming have been made (Arora, Chlamtac, mechanics and Feynman integrals in quantum field theory. and Charikar in 2006 (O(n0.2111 ) colors), and Chlamtac We present new progress in the area. Part of the talk is in 2007 (O(n0.2072 ) colors)). We present the first im- based on joint results with J.L. Xiang Li. In particular we provement on the graph theory side since Blum in 1990. present a type of calculus (based on logarithmic functions) With a purely graph theoretical approach, we get down to which can be used to prove inequalities between subgraph O(n4/11) colors (over Blum’s O(n3/8) colors). Combining densities. it with SDP, we get down to O(n0.2038 ) colors. Joint work Balázs Szegedy with Mikkel Thorup (ATT Research) University of Toronto Ken-ichi Kawarabayashi Department of Mathematics National Institute of Informatics, Japan [email protected] k [email protected]

IP3 IP6 Forcing Large Transitive Subtournamets Talk Title TBA - Singh The Erdos Hajnal Conjecture states roughly that a graph Abstract not available at time of publication. with some induced subgraph excluded has a large clique or a large stable set. A similar statement can be formu- Mona Singh lated for tournaments (a tournament is an orientation of a Princeton University complete graph), replacing cliques and stable sets by tran- [email protected] sitive subtournaments; and the two conjectures turn out to be equivalent. This talk will survey a number of recent results related to the latter conjecture. In particular, we IP7 will discuss a new inﬁnite class of tournaments excluding The Hub Labeling Algorithm which forces large transitive subtournaments; to the best of our knowledge this is the ﬁrst such class not obtained This is a survey of Hub Labeling results for general and by the so-called substitution operation. road networks. Given a weighted graph, a distance oracle takes as an input a pair of vertices and returns the dis- Maria Chudnovsky tance between them. The labeling approach to distance Columbia DM12 Abstracts 41

oracle design is to precompute a label for every vertex so [email protected] that distances can be computed from the corresponding labels. This approach has been introduced by [Gavoille et Muhammad Javaid al. ’01], who also introduced the Hub Labeling algorithm FAST-National University of Computer and Emerging (HL). HL has been further studied by [Cohen et al. ’02]. Sciences We study HL in the context of graphs with small high- Lahore Campus, Lahore, Pakistan way dimension (e.g., road networks). We show that under [email protected] this assumption HL labels are small and the queries are sublinear. We also give an approximation algorithm for computing small HL labels that uses the fact that short- CP1 est path set systems have small VC-dimension. Although Distingushing with Nordhaus Gaddum graphs polynomial-time, precomputation given by theory is too slow for continental-size road networks. However, heuris- Albertson and Collins introduced the distinguishing num- tics guided by the theory are fast, and compute very small ber in 1996 and Collins and Trenk introduced the distin- labels. This leads to the fastest currently known practi- guishing chromatic number of a graph which requires that cal distance oracles for road networks. The simplicity of the coloring be proper as well as distinguishing in 2006. HL queries allows their implementation inside of a rela- We revisit the Nordhaus-Gaddum inequalities which give tional database (e.g., in SQL), and query efficiency assures bounds on the sum and the product of χ(G)andχ(G)for real-time response. Furthermore, including HL data in the any graph G, and give analogues of these inequalities for database allows efficient implementation of more sophis- the distinguishing chromatic number. We provide a new ticated location-based queries. This approach brings the characterization of those graphs that achieve equality for power of location-based services to SQL programmers the upper bound in the sum Nordhaus-Gaddum inequality, which leads to a polynomial-time recognition algorithm for Andrew Goldberg this class and efficient computation of their chromatic num- Microsoft Research Silicon Valley bers. [email protected] Karen Collins Wesleyan University IP8 [email protected] Algorithms, Graph Theory, and the Solution of Laplacian Linear Equations Ann N. Trenk Wellesley College We survey several fascinating concepts and algorithms in [email protected] graph theory that arise in the design of fast algorithms for solving linear equations in the Laplacian matrices of graphs. We will begin by explaning why linear equations CP1 in these matrices are so interesting. The problem of solving On Super (a,d)-Edge Antimagic Total Labeling of linear equations in these matrices motivates a new notion Subdivided Caterpillar of what it means for one graph to approximate another. This leads to a problem of graph sparsification–the ap- Let G =(V,E) be a graph with v = |V (G)| vertices and proximation of a graph by a sparser graph. Our algorithms e = |E(G)| edges. An(a, d)-edge antimagic total labeling for solving Laplacian linear equations will exploit surpris- is a bijection “f” from V (G)UE(G)to the set of consec- ingly strong approximations of graphs by sparse graphs, utive integers 1, 2, .., v + e, such that the weights of the and even by trees. We will survey the roles that spectral edge W = {w(xy):xy in E(G)} form an arithmetic pro- graph theory, random matrix theory, graph sparsification, gression with the initial term “a” and common difference low-stretch spanning trees and local clustering algorithms “d” where w(xy)=f(x)+f(y)+f(xy). “W” is called play in the design of fast algorithms for solving Laplacian the set of edge-weights of the graph G. Additionally, if linear equations. f(V )=1, 2, ..., |V (G)| then f is called super (a, d)-edge antimagic total labeling and G is called super(a, d)-edge Daniel Spielman antimagic total. In this paper we formulate, super (a, d)- Yale University edge antimagic total labeling on subdivided caterpillar for [email protected] different integral values of d. Muhammad Javaid CP1 FAST-National University of Computer and Emerging On Antimagic Vertex Labeling Of Hypergraphs Sciences Lahore Campus, Lahore, Pakistan Let H =(V,E) be a graph with vertex set V (H)andedge [email protected] set E(H). Moreover suppose that v = |V (H)|,e = |E(H)| and N denotes the set of non negative integers. An antimagic (magic) vertex labeling is a bijection “f” from Akhlaq Bhatti V (H) to the set of consecutive integers 1, 2,...,v if the FAST-NUCES LAHORE, PAKISTAN induced edge labeling “g” from E(H) to the set of non neg- [email protected] ative integers N defined by g(e)=? f(v) for all v in E(G)is injective (constant) function. A hypergraph H is called an- CP1 timagic (magic) iff there exist an antimagic (magic) vertex labeling of H. In this paper we formulate antimagic vertex On Two Conjectures About Graceful Digraphs labeling on star and disjoint union of star hypergraph. AdigraphD with p vertices and q arcs is labeled by assign- { } Akhlaq Bhatti ing a distinct integer value g(v)from 0, 1,...,q to each FAST-NUCES LAHORE, PAKISTAN vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v)whereg(u, v)=(g(v)−g(u))(mod q +1). 42 DM12 Abstracts

If the arc values are all distinct then the labeling is called [email protected] a graceful labeling of digraph. In this talk we present the proofs of two conjectures namely, Bloom and Hsu’s conjec- Matthieu Dufour ture (1985) on graceful unicyclic wheels and Du and Sun’s −→ University of Quebec at Montreal conjecture (1994) on the gracefulness of the digraph n.Cm. [email protected]

Shivaraj Kumar CP2 NITK Surathkal Structure of Weighted Graphs with Forbidden Sub- shivaraj [email protected] division and Graph Sharing Games

Suresh Hegde We show that every weighted graph excluding subdivision NITK Surathkal, India of a fixed (small) graph contains one of the following struc- [email protected] tures: • a connected subgraph separating the graph into heavy balanced components, CP2 Saving Sets of Vertices in the Firefighter Problem • a heavy set of vertices connected by paths as in a cycle. The Firefighter Problem is a discrete-time model of the This characterization yields a strategy for the first player to spread of a fire using a simple graph. At each time step a gather a positive fraction of the total weight in a variant player has an opportunity to defend some vertex (or set of of graph sharing game played on graphs with forbidden vertices) and then the fire spreads from all burning vertices subdivision. to all of their undefended neighbours. Here we examine the decision problem that asks if a specified set of vertices can Piotr Micek be saved from burning. Jagellionian University [email protected] Christopher Duffy, Gary MacGillivrary University of Victoria duff[email protected], [email protected] Adam Gagol MariaCurie-SklodowskaUniversity [email protected] CP2 A Game on Zero Forcing Sets Bartosz Walczak Jagiellonian University We consider the two player game on a graph where players [email protected] alternate turns, with one player selecting a vertex to color black on each turn and the other player selecting a vertex to color white on each turn. The black player wins if after CP3 all of the vertices have been colored, the black vertices form 2-Switches and Isomorphism Classes a zero forcing set. This game will be discussed for several classes of graphs. A 2-switch is an operation that changes adjacencies in a graph while preserving vertex degrees. The resulting graph Kara Greenfield, Alyssa Gottshall may or may not have the same isomorphism class; we show Worcester Polytechnic Institute that if a 2-switch changes the isomorphism class of a graph, kgreenfi[email protected], [email protected] then it takes place in one of four configurations. We also present a sufficient condition for 2-switches to change graph Steve Butler isomorphism classes and examine consequences for graphs UC Los Angeles determined up to isomorphism by their degree sequences. USA [email protected] Michael Barrus Black Hills State University Department of Mathematics Young Michael [email protected] Iowa State University [email protected] CP3 CP2 The Number of Spanning Trees in Self-Similar Graphs. A Generalization of the Nim and Wythoff Games G \ ≤ The number of spanning trees of a graph is an important We define a two-player game ( , )(withk n)asfol- invariant related to its topological and dynamic proper- lows: each player chooses up to k stacks from a total of ties (reliability, communication, synchronization,etc.) Its n stacks and removes an equal number of tokens from the calculation remains a challenge, particularly for large net- selected stacks. The first player who cannot move loses. G ∞ \ G ∈ ∈ works. We describe a method to find an exact analytical ( , ) is ordinary Nim and ( , ) is Wythoff. We will expression for the number of spanning trees (and tree en- present a conjecture on the structure of the losing posi- tropy) for several relevant graph families, including Hanoi tions of this class of games and prove a special case of the graphs, Apollonian networks and some maximal outerpla- conjecture. nar graphs. The method is based on the self-similarity of Silvia Heubach the graphs considered. California State University Los Angeles Francesc Comellas DM12 Abstracts 43

Universitat Politecnica de Catalunya well-covered. [email protected] David Tankus Zhongzhi Zhang Ariel University Center of Samaria, Israel Fudan University, China [email protected] [email protected] Vadim E. Levit Ariel University Center of Samaria and Holon Inst. of CP3 Tech. On the 2-Edge Clique Cover Numbers of Graphs Dept. of Computer Science & Mathematics [email protected] Let F be a multifamily of subsets of the vertex set of a graph G. The multifamily F is called a 2-edge clique cover of G if for any two distinct I, J ∈F, I ∩ J is a clique of G, CP4 and the family {I ∩ J | I,J ∈F, I = J} is an edge clique Construction of nonuniform p-wavelet packets cover of G. The minimum size of a 2-edge clique cover of G is called the 2-edge clique cover number of G. In general, A generalization of Mallat’s classical multiresolution anal- it is hard to compute a 2-edge clique cover of a graph. In ysis for which the translation set is a discrete set which is this talk, we study relationships between the 2-edge clique not a group, was deﬁned by Gabardo and Nashed. In this cover number and the edge clique cover number of a graph. talk, we introduce nonuniform multiresolution p-analysis 2 + Then we focus on the 2-edge clique cover numbers of a path on L (R ), analogy of Gabardo-Nashed deﬁnition and then and a cycle, and then give a bound for 2-edge clique cover we will construct the associated p-wavelet packets for such 2 + number of a star K1,n by using a block design. an multiresolution p-analysis on L (R ). Jung Yeun Lee Jahangir Cheshmavar National Institute of Mathematical Science Payame Noor University (PNU) South Korea [email protected] [email protected] CP4 Suh-Ryung Kim Seoul National University Application of New Variational Method Using [email protected] Hamiltonian for Nonlinear Oscillators with Discon- tinuities

Boram Park In this paper, we used Hamiltonian for nonlinear oscillators DIMACS with discontinuities. The maximal relative error for the Rutgers University frequency obtained by new variational method compared [email protected] with the exact solution indicates the remarkable precision of this method. Some examples are given to illustrate the Yoshio Sano effectiveness and convenience of the method National Institute of Informatics [email protected] Waseem A. Khan CIIT, ISLAMABAD, PAKISTAN [email protected] CP3 Dual-Chordal and Strongly Dual-Chordal Graphs CP4 If a subgraph is defined to be strength-k whenever it is in ≥ A Comparison of Conventional and Fuzzy Methods k maxcliques, then a graph is (strongly) chordal if and only in Performance Appraisal Process if (for all k ≥ 1) every cycle of (strength-k) edges either has a (strength-k) chord or is a (strength-k) triangle. A The purpose of this paper is to address the performance ap- carefully crafted notion of dual strength (using cycle/cutset praisal process via conventional and Fuzzy satisfied meth- duality) allows parallel characterizations of (strongly) dual- ods.Conventional methods for solving performance ap- chordal graphs and a listing of all 3-connected strongly praisal decision making problem are by using statistical dual-chordal graphs. methods(chi-square, correlation etc)on the measurement of performance based on specific appraisal criteria.This re- Terry McKee search shows that the Fuzzy satisfied method is an easy to Wright State University understand and realistic technique as it enables us to un- Department of Mathematics & Statistics derstand the relative strengths and weaknesses of employ- [email protected] ees,thus providing useful information for improving com- petitive advantage.

CP3 Idowu A. Osinuga Well-Covered Graphs Without C4,C5,C6 Department of Mathematics, College of Natural Sciences, Federal University of Agriculture, Abeokuta, Nigeria Let G =(V,E) be a graph and w : V −→ R be a weight [email protected] function. Then G is w-well-covered if all its maximal independent sets are of the same weight. The set of weight Adegboyega Adekoya functions w such that G is w-well-covered is a vector space. Department of Mathematical Sciences, Olabisi Onabanjo Given an input graph G without cycles of length 4, 5, and Univ 6 (not necessarily induced), we characterize polynomially Ago-Iwoye, Nigeria the vector space of weight functions w for which G is w- 44 DM12 Abstracts

[email protected] 2n−2 + 1 has order dimension at least n.

Jonathan E. Beagley CP4 George Mason University On the Structure of Some Groups Containing [email protected] L2(13)wrl2(17) In this paper, we will show the structure of some groups CP5 containing the wreath product L2(13)wrL2(17).Some sym- Universal Cycles for Weak Orders and a Problem metric and alternating groups are constructed in view of of Knuth such wreath product. Some other related cases are also { } included. A weak order on [n]= 1, 2, ..., n is a string w1w2...wn such that {w1,w2, ..., wn} =[h], for some height h ≤ n. Basmah Shafee Knuth asked: For which n does there exist a Gray code for Umm Al qura University all weak orders on [n] using operations of the form wi ↔ dr.basmah [email protected] wi+1 and wi ← wi+1? Weprovesuchexistenceforaclassof weak orders with the additional operation wi ↔ wi+2 and discuss open problems for fixed height weak orders that are CP4 related to solving the original problem. Furthermore, we Fixed Points Of a Fuzzy Semantics Function present constructions of universal cycles for weak orders, as well as for weak orders of fixed height or weight. The meaning of a program is given by specifying the function (from input to output) that corresponds to the pro- Victoria Horan, Glenn Hurlbert gram. The denotational semantic definition, thus maps Arizona State University syntactical things into functions. A fuzzy relational se- [email protected], [email protected] mantics is a mapping of programs to relations. We consider that the input-output semantics of a program is given by a fuzzy relation on its set of states. In a nondeterministic CP5 context, this relation is calculated by considering the worst Zeta Polynomials for Shellable Posets and Their behavior of the program (demonic fuzzy relational seman- Applications tics). In this paper, we concentrate on while loops. We will present some interesting results about the fixed points of The zeta polynomial Z(P, n) for a poset P is the number of − the while fuzzy semantics function; f(X)=Q∪f uzPf uzX multichains in P of length n 2, and a shellable poset P is where PQ =, by taking P := tB and Q := t, one gets the a graded poset whose maximal chains explain the structure demonic semantics we have assigned to while loops in pre- of chains in P . Using shellability, we establish theorems on vious papers. We will show that the least angelic fixed computations of zeta polynomials for shellable posets. Ap- point is equal to the greatest demonic fixed point of the plying these theorem, we also derive new resluts on multi- fuzzy semantics function. dimensional partitions, multidimensional Young tableaux, lattice paths, and multiset permutations. Fairouz Tchier King saud University Joon Yop Lee [email protected] ASARC, KAIST fl[email protected]

CP4 Binary Sequences, Fractals, and Primes CP5 The Largest Size Versus the Largest Weight of The concept of a fractal transformation is discussed. A Families of Sets Without a Poset P 1-dimensional example leads to connections between the topics in the title. We are interested in how large a family of subsets of [n]:={1, 2 ...,n} not containing a given poset P as a Andrew Vince subposet can be. Such a family is called a P -free fam- Department of Mathematics ily and the largest size of a P -free family is denoted University of Florida by La(n, P ). Griggs and Lu conjectured that π(P ):= n avince@uﬂ.edu limn→∞ La(n, P )/ n exists and is an integer for any ﬁ- 2 F nite P . The Lubell function of a family of subsets of Michael Barnsley \ −∞ [n]ish¯ (F):= which can be viewed as the Australian National University n E∈F |E| [email protected] weighted sum of sets in F. It gives an upper bound of |F|, which is crucial in solving the forbidden subposet problem above. We will study the limit of λn(P ):=maxF h¯n(F) CP5 over all P -free families F as n increases and demonstrate a Order Dimension and Coloring of Planar Point Sets class of posets satisfying π(P ) = limn λn(P ). We will also show that for many posets limn λn(P )existsandcanbe The Erd˝os-Szekeres Conjecture of planar point sets in gen- evaluated, although π(P ) is unknown. eral position will be discussed using convex geometries or anti-matroids. We study this problem with two graphs, Jerry Griggs one based on the lattice of closed sets and the other from University of South Carolina, Columbia the copoints of the convex geometry. We use these graphs [email protected] to show that any point set in general position with at least Wei-Tian Li Department of Mathematics DM12 Abstracts 45

Univeristy of South Carolina CP6 [email protected] On The Structure Of The γ-Graph Of A Tree The γ-graph of a graph G has vertex set comprised of all CP5 minimum dominating sets of G. There are two ways to Searching Algorithms in Partially Ordered Set define adjacency in the γ-graph: where dominating sets are adjacent if they agree in all but one vertex, or where Using conceptual graphs and graph homomorphism it is dominating sets are adjacent if they agree in all but one possible to build a basic query-answering mechanism based vertex and the differing vertices are adjacent. Here we on semantic search. Because graph homomorphism is answer open questions on maximum degree, diameter, and NP-Complete, the main requirement for conceptual graph order of γ-graphs for trees. database building and managing algorithms is to reduce the homomorphism checks. Graph homomorphism defines Michelle Edwards, Gary Macgillivray a partial order over conceptual graphs. Searching is a basic Mathematics and Statistics operation for building and managing operations as order- University of Victoria ing, updating, and retrieval. We propose two new searching [email protected], [email protected] algorithms in a poset.

Abdurashid R. Mamadolimov CP6 MIMOS, Malaysia Some Unsolved Problems on Cycles [email protected] Hajos’ conjecture that every simple even graph on n vertices can be decomposed into at most (n − 1)/2cycles. CP5 Let f(n) be the maximum number of edges in a graph on The Bkr Inequality on Finite Distributive Lattices n vertices in which no two cycles have the same length. P. Erdos raised the problem of determining f(n). P. Er- The van den Berg-Kesten-Reimer inequality, or BKR in- dos conjectured that there exists a positive constant c such equality, is a well-known result in combinatorial probabil- 1+1/k that ex(n, C2k) ≥ cn . This talk summarizes some re- ity with applications in percolation. The BKR inequality sults on these problems and the conjectures that relate to applies to product spaces X equipped with product prob- these. It seems to me that Hajós conjecture is false. ability measures. For events A and B, the box product of A and B is the set of tuples x in X that lie in A and B Chunhui Lai disjointly, i.e. x lies in A due to the values of one set of x’s Zhangzhou Normal University coordinates and in B due to a disjoint set of coordinates. [email protected] The BKR inequality states that for any two events A and B of any finite product probability space the measure of the Mingjing Liu box product of A and B is less than or equal to the product Department of Mathematics and Information Science of the measures of A and B. We extend the box product Zhangzhou Normal University and the BKR inequality to finite distributive lattices with [email protected] log modular measures, analogues of product measures. A product space can be made into a distributive lattice by linearly ordering each member of the product. The new box CP6 product of A and B will then contain the old box product. On Signed Star Domination and Domatic Numbers Furthermore the containment is often strict, improving the of Complete Multipartite Graphs BKR inequality. The signed star domination and domatic numbers of a Clifford D. Smyth graph G, γSS(G)anddSS(G), were first introduced in University or North Carolina Greensboro [B. Xu, On edge domination numbers of graphs, Discrete [email protected] Mathematics, 294 (2005), 311–316.] and [M. Atapour, S. M. Sheikholeslami, A. N. Ghameshlou and L. Volk- mann, Signed star domatic number of a graph, Discrete CP6 Applied Mathematics, 158 (2010), 213–218.]. We give new On the Locating-Chromatic Number of a Corona constructions to complete the proofs of γSS(Km,n)and Product of Two Graphs dSS(Km,n). We also obtain γSS(Km(n))anddSS(Km(n)). The problem of determining the locating-chromatic num- Wu-Hsiung Lin ber for a general graph is an NP-hard problem. Therefore, Departement of Applied Mathematics there is no efficient algorithm to determine this number National Chiao Tung University of arbitrary graph. Some heuristics have been developed [email protected] and some studies were done to find the locating-chromatic number for certain graphs. In this paper, we investigate ChiuyuanChen,WellY.Chiu,ALLENY.Wang the locating-chromatic number for a corona product G H. Departement of Applied Mathematics, Some upper bound will be derived. We also determine the National Chiao Tung University locating-chromatic number of certain graphs. [email protected], [email protected], dog- Edy T. Baskoro, Ira Purwasih [email protected] Institut Teknologi Bandung Indonesia CP6 [email protected], ira apni [email protected] Decompositions of Complete Multipartite Graphs 46 DM12 Abstracts

into Gregarious Long Cycles Good Edge-Labelling

The notion of gregarious cycles in complete multipartite A good edge-labelling of a simple graph is a labelling of its graphs was introduced by Billington and Hoffman in 2003 edges with real numbers such that, for any ordered pair of and was modified by Billington, Smith, and Hoffman in vertices (u, v), there is at most one non-decreasing path 2007. In this talk, we propose more general definition of from u to v. Say a graph is good if it admits a good “gregarious” cycles which is a common generalization of edge-labelling, and is bad otherwise. Our main result is both the definitions. With our definition, we can consider that any good n-vertex graph whose maximum degree is the problem to characterize complete multipartite graphs within a constant factor of its average degree (in particu- which are decomposable into gregarious long cycles, and lar, any good regular graph) has at most n1+o(1) edges. As we give some results on this problem. a corollary, we show that there are bad graphs with arbitrarily large girth, answering a question of Bode, Farzad Jeongmi Park, Jung Rae Cho and Theis. We also prove that for any Δ, there is a g such Department of Mathematics that any graph with maximum degree at most Δ and girth Pusan National University at least g is good. [email protected], [email protected] Abbas Mehrabian Yoshio Sano Department of Combinatorics and Optimization National Institute of Informatics University of Waterloo [email protected] [email protected]

CP6 CP7 A New Upper Bound for the Broadcast Domina- Bounds on Shannon Capacity and Ramsey Num- tion Number of a Graph bers from Product of Graphs

The broadcast domination number of a graph G,de- We study Shannon capacity of channels in the context of noted γb(G), is the minimum cost of a dominating Ramsey numbers. We overview some of the results on chan- broadcast on G. We present an improvement to the nel capacity, and how Ramsey-type constructions may en- well-known Bollobás-Cockayne bound from γb(G) ≤ hance them. A new lower bound for a special type of multi- { }≤ − ≤ 3 min rad(G),γ(G) 2ir(G) 1, to γb(G) 2 ir(G). We color Ramsey numbers is presented, which implies that the also show the existence of a graph G with γb(G)− ir(G)=k supremum of the Shannon capacity over all graphs with in- for any positive integer k. dependence 2 cannot be achieved by any finite graph power. This generalizes to graphs with any bounded independence Laura E. Teshima, Christina Mynhardt number. University of Victoria [email protected], c.m. mynhardt Xiaodong Xu Guangxi Academy of Sciences [email protected] CP7 The Edge Density of Critical Digraphs Stanislaw P. Radziszowski Department of Computer Science Let χ(G) denote the chromatic number of a graph G.We Rochester Institute of Technology say that G is k-critical if χ(G)=k and χ(H) 1 − 1 |D|2 hypergraph with girth at least 6 then -critical digraphs for which 2 2k−1 .One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair − of integers k, l ≥ 3. This extends a result by Bokal et. rn 1 Δ(H) ≥ c , al. who used a probabilistic approach to demonstrate the ln n existence of one such digraph.

Richard Hoshino, Ken-Ichi Kawarabayashi where c>0 is an absolute constant. National Institute of Informatics [email protected], k [email protected] Dmitry A. Shabanov Moscow State University [email protected] CP7 On the Density of Nearly Regular Graphs with a Andrei Kapavskii Moscow Institute of Physics and Technology [email protected] DM12 Abstracts 47

CP7 CP8 Embedding Spanning Bipartite Graphs of Small The Minimum Rank of Universal Adjacency Ma- Bandwidth trices Böttcher, Schacht and Taraz gave a condition on the mini- For a simple graph G on n vertices, a matrix of the form mum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded U(α, β, γ, δ)=αA + βI + γJ + δD degree and of bandwidth o(n), thereby proving a conjec- where A is the 0,1-adjacency matrix of G, J is the all ones ture of Bollobás and Komlós. We strengthen this result in matrix of size n, I is the identity matrix of size n,andD the case when H is bipartite by relaxing the condition on is the diagonal matrix with the degrees of the vertices in G to a condition on the degree sequence of G. the main diagonal, and α =0 ,β,γ,δ are scalars, is called a Andrew Treglown,FiachraKnox universal adjacency matrix of G. An analogous parameter University of Birmingham to the minimum rank of a given graph G is the minimum [email protected]ff.cuni.cz, [email protected] rank over all matrices in the set of universal adjacency matrices of G. This parameter is called the minimum universal rank of G, and is denoted by mur(G). Graphs with CP8 mur(G) equal to zero and one are characterized. The min- Forbidden Submatrices imum universal rank of some families of graphs such as complete graphs, complete bipartite graphs, paths and cy- We consider an extremal hypergraph problem, where the cles are presented. A formula for the minimum universal order of the elements and of the hyperedges matter. In rank of a regular graph is given. Finally, it is shown that matrix terms, let F be a given (0,1)-matrix. Among all m- mur(G) is not monotone on induced subgraphs. rowed (0,1)-matrices with no repeated columns and no sub- This is joint work with Discrete Mathematics Research matrix F , what is the maximum number of columns? We Group at the University of Regina. call this fs(m, F ). It was conjectured by Anstee, Frankl, Shahla Nasserasr, Bahman Ahmadi, Fatemeh Furedi, Pach that if F has k rows then fs(m, F )isO(mk). Alinaghipour We give results for 2-rowed F . University of Regina Richard P. Anstee [email protected], [email protected], Mathematics Department [email protected] University of British Columbia [email protected] Shaun M. Fallat University of Regina Ruiyuan Chen Canada UBC [email protected] Vancouver, Canada [email protected] Yi-Zheng Fan Anhui University Attila Sali [email protected] Renyi Institute [email protected] Karen Meagher University of Regina Saskatchewan, Canada CP8 [email protected] Intertwining Connectivities in Matroids Repre- sentable over a Finite Field CP8 Let N1 and N2 be matroids. An intertwine of N1 and N2 The Supertail of a Subspace Partition is a matroid that has both an N1-andN2-minor, and is minor-minimal with this property. Bonin proved that there Let V = V (n, q) be a vector space of dimension n over the exist matroids N1 and N2 with infinitely many intertwines. finite field with q elements, and let d1

Bert Gerards Esmeralda L. Nastase CWI Amsterdam Xavier University [email protected] [email protected]

Stefan van Zwam Olof Heden Princeton University KTH [email protected] [email protected]

Julianne Lehmann Bremen University [email protected] ¡jule.germany@googlema 48 DM12 Abstracts

Papa Sissokho CP9 Illinois State University An Approximation Algorithm for the Multilevel [email protected] Bottleneck Assignment Problem We deal with the multilevel bottleneck assignment prob- CP8 lem. The multilevel bottleneck assignment problem is NP- Rank-Width and Well-Quasi-Ordering of Skew- complete, and some approximation algorithms have been Symmetric Or Symmetric Matrices proposed. We propose an approximation algorithm that uses the element’s rank in each set sorted in the ascend- We prove that every infinite sequence of skew-symmetric ing order. We show the effectiveness of our algorithm by or symmetric matrices M1, M2, ... over a fixed finite field numerical experiments for the case that elements’ values must have a pair Mi, Mj (i

Sang-Il Oum CP9 KAIST Greedy Is Good to Approximate Minimum Rain- [email protected] bow Subgraphs We consider the Minimum Rainbow Subgraph problem CP9 (MRS): Given a graph G, whose edges are coloured with Applications of the Traveling Salesman Problem p colours. Find a subgraph F ⊆ G of minimum order and and Perfect B-Matching for Finding Genomic Me- with p edges such that each colour occurs exactly once. In dians this talk we will show that the Greedy algorithm for the Δ ln Δ+1 MRS problem has an approximation ratio of 2 + 2 Genomic medians model an ancestor of several species. for graphs with maximum degree Δ. If the average degree There has been extensive research on ﬁnding medians. We d of a minimum rainbow subgraph is known, then the ap- study the Breakpoint Median Problem (BMP) and present proximation ratio is d + ln d +1 . a novel and successful approach in which the NP-hard prob- 2 2 lem of BMP for linear multichromosomal genomes is trans- Ingo Schiermeyer formed into a variation of the Traveling Salesman Problem. Technische Universitaet Bergakademie Freiberg We also use prefect b-matchings to show, for the ﬁrst time, Institut fuer Diskrete Mathematik und Algebra that BMP for circular or mixed unsigned genomes is solv- [email protected] able in polynomial time.

Maryam Haghighi, Sylvia Boyd CP10 University of Ottawa Constructing Self-Dual Graph Embeddings in Sur- [email protected], [email protected] faces

CP9 A cellular automorphism of a surface S is a 2-cell graph embedding G in S together with an automorphism of the A Branch, Bound, and Remember Algorithm for cellular structure given by G. We detail methods which the Simple Assembly Line Balancing Problem allow one, given a catalog of all cellular automorphisms A new exact algorithm is presented for the assembly line in any particular surface S (such as we have produced for balancing problem, which finds and verifies the optimal five surfaces), to construct all self-dual graph embeddings solution for every problem in the benchmarks of Hoffmann, in S. Our methods account for all free and all non-free Talbot, and Scholl one-half second per problem, on average, automorphisms, of all possible orders. including one problem that has remained open for over ten Lowell Abrams years. The previous best known algorithm solved 257 of the Department of Mathematics 269 benchmarks. The new algorithm is based on a branch The George Washington University and bound method using memory to eliminate redundant [email protected] subproblems. Sheldon H. Jacobson Daniel Slilaty University of Illinois Department of Mathematics and Statistics Dept of Computer Science Wright State University [email protected] [email protected]

Edward Sewell CP10 Southern Illinois University Edwardsville -Metric Dimension of Graphs [email protected] k Let G(V,E) be a connected graph. A subset S of V is said DM12 Abstracts 49

to be k-resolving set of G, if for every pair of distinct ver- some cellular network graphs and their dual graphs. tices u, v ∈ V − S, there exists a vertex w ∈ S such that |d(u, w) − d(v, w)|≥k,forsomek ∈ Z+.Amongallk- Shreedhar K resolving sets of G, the set having minimum cardinality is KVG College of Engineering, Sullia called a k-metric basis of G and its cardinality is called the shreedhar.k@rediffmail.com k-metric dimension of G, denoted by βk(G). In this paper, we characterize the graphs with prescribed k-metric dimen- Sooryanarayana B sions and also extend some of the earlier known results on Dr. Ambedkar Institute of Technology, Bangalore metric dimension of graphs. dr [email protected] Sooryanarayana B Dr. Ambedkar Institute of Technology, Bangalore CP10 dr [email protected] Latin Squares and Competition Numbers In this talk we show how latin squares are used to find CP10 bounds on the competition numbers of complete multipar- Almost Empty Monochromatic Triangles in Planar tite graphs. Let G be a graph and let Ik denote the graph Point Sets on k isolated vertices. The minimum k such that G ∪ Ik is the competition graph of an acyclic digraph is called the For positive integers c, s ≥ 1, let M (c, s) be the least in- competition number of G. We show that if n ≥ 5 is odd, 4 2 2 teger such that any set of at least M (c, s)pointsinthe then k(Kn) ∈{n − 4n +7,n − 4n +8}. For the general plane, no three on a line and colored with c colors, contains case, we show that if n is a prime integer and m ≤ n,then m 2 a monochromatic triangle with at most s interior points. k(Kn ) ≤ n − 2n +3. The case s = 0, which corresponds to empty monochromatic triangles, has been studied extensively over the last Jaromy S. Kuhl few years. In particular, it is known that M (1, 0) = 3, University of West Florida M (2, 0) = 9 and M (c, 0) = ∞,forc ≥ 3. In this paper [email protected] we extend these results when c ≥ 4ands ≥ 1. We prove that the least integer λ(c) such that M (c, λ(c)) < ∞ satisfies: CP10 c − 1 Metric Dimension of Amalgamation of Graphs p ≤ λ(c) ≤ c − 2, 2 A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the ver- where c ≥ 2andcp is the largest prime number less − ≤ tices in S. The metric dimension of G is the minimum than or equal to c. We also show that M (c, c 2) { } (c +1)2,forc ≥ 2. Finally, we determine the exact values cardinality of resolving set of G.Let G1,G2,...,Gn be a finite collection of graphs and each Gi has a fixed vertex of M (c, s) for several small values of c and s, and discuss { } related problems for general convex polygons. v0i called a terminal. The amalgamation Amal Gi; v0i is formed by taking all the Gi’s and identifying their termi- { } Bhaswar B. Bhattacharya nals.HerewestudythemetricdimensionofAmal Gi; v0i { } Department of Statistics for G1,G2,...,Gn a finite collection of arbitrary graphs. Stanford University We give lower and upper bounds for the dimension, show [email protected] that the bounds are tight, and construct infinitely many graphs for each possible value of dimensions.

Deepan Basu Rinovia Simanjuntak, SALADDIN Uttunggadewa, Suhadi Indian Statistical Institute, Kolkata Wido Saputro [email protected] Institut Teknologi Bandung [email protected], s [email protected], Sandip Das [email protected] ISI, Calcutta, India [email protected] CP11 The Periodicity of Winning/Losing States in Sub- CP10 traction Games On The Metric Dimension Of Generalized Wheel And Other Graphs A subtraction game is a game involving a pile of tokens and a ﬁnite set S of positive integers. Two players move { } For an ordered set W = w1,w2, ..., wk of vertices and alternately, each removing some m tokens provided that m avertexv in a connected graph G, the representation is an element in S. The player who makes the last move of v with respect to W is the ordered k-tuple r(v | wins. We introduce an observation on the periodicity of W )=(d(v,w1),d(v, w2), ..., d(v, wk)), where d(x, y)rep- the winning/losing positions of subtraction games which resents the distance between the vertices x and y.Theset have a connection to the periodicity of the nim-sequences. W is called a resolving set for G if every two vertices of G have distinct representations. A resolving set containing a minimum number of vertices is called a basis for G.The Nhan Bao Ho elements of a metric basis are called land marks and the La Trobe University metric dimension of G, denoted by β(G), is the number of [email protected] vertices in a basis of G. A generalized wheel graph is Wmn = Cm + Kn.Inthis Paper we determine the metric dimensions of Generalized CP11 Wheel graphs. We also obtain the metric dimension of Towards a De Bruijn-Erdos Theorem in the L1- 50 DM12 Abstracts

Metric tive Formula” in this talk. The main results are: (1) a more rigorous formulation (via a discretization of the derivative) In 1948, de Bruijn and Erd˝os proved that every set of n to a parameter at the antiderivative formula, and (2) an ex- points in the plane is either collinear, or it induces at least tension of the antiderivative formula to more general types n lines. The notion of a line can be extended in a nat- of domains, via a (novel) discrete integration method over ural way to an arbitrary metric space as follows. If δ is curves in the plane. a metric, we say that the point x is between the points a and b if δ(a, b)=δ(a, x)+δ(x, b). The line induced by the Amir Shachar points a and b consists of the points a, b,andallpoints Mathematical Institute, Hebrew University, Jerusalem, x such that one of a, b, x is between the other two. With Israel the usual Euclidean, or L2, metric, this is equivalent to the [email protected] usual definition of a line. Chen and Chvátal conjectured in 2006 that a statement analogous to the de Bruijn–Erd˝os theorem holds in any finite metric space. Despite several CP11 partial results, this question is still open. We prove that Graphs with Minimum Identifying Code and Min- with the L1 (also called Manhattan) metric, every set of n imum Size points in the plane is either collinear, or it induces at least Ω(n6/7−o(1)) lines. In a graph G,letB(u)bethesetofu with all its neighbors in G. A subset S of vertices is called an identifying Ida Kantor code of G if, for every pair of distinct vertices u and v, Charles University both B(u) ∩ S and B(v) ∩ S are nonempty and distinct. [email protected]ff.cuni.cz A minimum identifying code of a graph G is an identifying code of G with minimum cardinality and M(G)is Balázs Patkós the cardinality of a minimum identifying code for G.A Alfr minimum identifying code graph G of order n is a graph ’{e}dR with M(G)= log2(n +1) having the minimum number ’{e}nyi Institute of Mathematics, Budapest of edges. Moncel constructed minimum identifying code m − ≥ [email protected] graphs of order 2 1form 2 and left the same problem, for arbitrary order, to be open. In this talk, we pur- posed the construction of connected minimum identifying CP11 code graphs to solve this problem. Radio Number of kth Power of a Path Li-Da Tong Let G be a connected graph. For any two vertices u and v, National Sun Yat-sen University, Taiwan let d(u, v) denotes the distance between u and v in G.The [email protected] maximum distance between any pair of vertices is called the diameter of G and is denoted by diam(G). A radio CP12 labeling of a connected graph G is an assignment of distinct positive integers to the vertices of G,withx ∈ V (G) labeled Eulerian Circuits with No Monochromatic Transi- f(x), such that |f(u)−f(v)|+d(u, v) ≥ 1+diam(G) for all tions. u, v ∈ V,u = v. The radio number rn(f) of a radio labeling Let G be an eulerian digraph with a fixed edge coloring f of G is the maximum label assigned by f to a vertex of (not necessarily a proper edge coloring). A compatible cir- G. The radio number rn(G)ofG is the min{rn(f)} over cuit T is an eulerian circuit of G such that no consecutive all radio labeling f of G.Thekth power of a graph G, k edges in the circuit have the same color. We provide suf- denoted by G , is the graph on the vertices of G with two ficient conditions for the existence of a compatible circuit, k ≤ vertices u and v are adjacent in G , whenever d(u, v) k strengthening results of Fleischner and Isaak. We also show in G. In this paper, we completely determine the radio several examples of digraphs that do not have compatible k ≥ number of the graph Pn for positive integers n, k with k circuits. 2. James Carraher P. Devadas Rao University of Nebraska Srinivas Institute of Technology [email protected] [email protected] Stephen Hartke Sooryanarayana B Department of Mathematics Dr. Ambedkar Institute of Technology, Bangalore University of Nebraska - Lincoln dr [email protected] [email protected]

Chandru Hegde PES Institute of Technology, Bangalore CP12 [email protected] Labeled Embeddings of Graphs

We present a recent variant of the graph embedding prob- CP11 lem on labeled graphs. Given a graph G =(V,E), a k- Discrete Operators Applied to Further Generalize labeled embedding of G is a labeling f : V →{1..k} such the Integral Image Algorithm that there exists an edge-disjoint placement of two copies of G into the complete graph K|V | preserving f.Theob- The ”Integral Image” algorithm was extended to continu- jective consists in maximizing the number of labels k.We ous and more general domains by Wang, Doretto et al in give upper bounds in the general case, exact values for 2007. Their theorem will be abbreviated the ”Antideriva- some families of graphs, and extensions for the placement DM12 Abstracts 51

of more than two copies of G. Woongbae Park, Suh-Ryung Kim Seoul National University Eric Duchene, Hamamache Kheddouci [email protected], [email protected] Lyon 1 University [email protected], [email protected] CP12 Flow-Continuous Mappings – Influence of the Richard Nowakowski Group Department of Mathematics and Statistics Dalhousie University Mapping f : E(G) → E(H)isM-flow-continuous if for [email protected] every flow φ on H using values in M the composition φ ◦ f is a flow on G. This notion was introduced by Jaeger to Amine Tahraoui approach flow-related conjectures (like CDC). We discuss Lyon 1 University the influence of the group on flow-continuous mappings. [email protected] On one hand, groups for which this mapping exists exhibit nice structure. On the other hand, in the important case of snarks we can restrict to Z2, Z3 or Z. CP12 Robert Samal Dynamic Coloring and List Dynamic Coloring of Charles University Planar Graphs [email protected]ff.cuni.cz

A dynamic chromatic number χd(G) of a graph G is the least number k such that G has a proper k-coloring of the Jaroslav Nesetril vertex set V (G) so that for each vertex of degree at least 2, department of Applied Mathematics its neighbors receive at least two distinct colors. We show Charles University that χd(G) ≤ 4 for every planar graph except C5.Thelist [email protected]ff.cuni.cz dynamic chromatic number chd(G)ofG is the least number k such that for any assignment of k-element lists to the vertices of G, there is a dynamic coloring of G where CP12 the color on each vertex is chosen from its list. Based on Total Weight Choosability of Cartesian Product of Thomassen’s result that every planar graph is 5-choosable, Graphs an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not. AgraphG =(V,E) is called (k, k )-choosable if the following is true: For any total list assignment L which assigns We answer this question by showing that chd(G) ≤ 5for every planar graph. to each vertex x asetL(x)ofk real numbers, and assigns to each edge e asetL(e)ofk real numbers, there is a mapping f : V ∪ E → R such that f(y) ∈ L(y)for Seog-Jin Kim Konkuk University any y ∈ V ∪ E and for any two adjacent vertices x, x , [email protected] e∈E(x) f(e)+f(x) = e∈E(x) f(e)+f(x ). In this talk, we will show that if G is the Cartesian product of an even Sang June Lee number of even cycles, or the Cartesian product of an odd Emory University, USA number of even cycles and at least one of the cycles has [email protected] length 4n for some positive integer n,thenG is (1, 3)- choosable. Won-Jin Park Jiaojiao Wu,Tsai-LienWong Seoul National University, Korea Department of Applied Mathematics [email protected] National Sun Yat-sen University [email protected], [email protected] CP12 The Competition Graphs of Powers of Digraphs Xuding Zhu Zhejiang Normal University, China The competition graph C(D) of a digraph D has the same [email protected] vertex set as D and has an edge between vertices u and v if and only if there exists a common prey of u and v in D. Given a digraph D, the convergence of the sequence CP13 m ∞ Counting Minimal Sturmian Words {C(D )}m=1 has been studied by several researches that are related to the notion of competition index. In this paper, given a digraph D, we focus on the limit graph of the The subword complexity pw(n)ofawordw is the number m ∞ of distinct subwords of length n of w.Wecallawordw sequence {C(D )}m=1 if it converges. We find the limit m ∞ Sturmian of order N,ifpw(n)=n +1 for n =1,...,N. graph of {C(D )}m=1 when D hasatmosttwostrongly connected components, and then we give a sufficient and It is minimal if it has minimal length among all Sturmian necessary condition for its limit graph consisting of only words of order N. Equivalently, it is minimal if it has complete connected components. length 2N. Recently, Blanchet-Sadri and Lensmire considered such words and provided an algorithm for generating Boram Park all of them using techniques from graph theory. In this pa- DIMACS per, we exploit their approach in order to count the number Rutgers University of minimal Sturmian words of order N and we show that [email protected] this number is connected to Euler’s totient function from number theory. We also present some other results that come from applying this graph theoretical framework to 52 DM12 Abstracts

subword complexity. and length of the words.

Francine Blanchet-Sadri Francine Blanchet-Sadri University of North Carolina University of North Carolina [email protected] [email protected]

Sean Simmons Laure Flapan MIT Yale University [email protected] laure.ﬂ[email protected]

Bob Chen Steven Ji University of California at San Diego MIT [email protected] [email protected]

Elizabeth Reiland CP13 Harvey Mudd College Recurrence in Inﬁnite Partial Words [email protected]

The recurrence function Rw(n) of an infinite word w was introduced by Morse and Hedlund in relation to symbolic Stephen Watkins dynamics. It is the size of the smallest window such that, Vanderbilt University wherever its position on w, all length n subwords of w [email protected] will appear at least once inside that window. The recurrence quotient ρ(w)ofw, defined as limsup of the quotient CP13 of R (n)andn, is useful for studying the growth rate of w Solving Wilf’s Sigma-Tau Problem Rw(n). It is known that if w is periodic, then ρ(w)=1, ≥ while if w is not, then ρ(w) 3. A long standing conjec- We honor Herb Wilf (1931-2012) by solving Exercise 6 ture from Rauzy√ states that the latter can be improved to ≥ ∼ from his textbook Combinatorial Algorithms with Albert ρ(w) (2.5+.5 5) 3.618, this bound being true for each Nijenhuis (1975). The symmetric group S is generated by Sturmian word and being reached by the Fibonacci word. n σ =(12 ··· n)andτ = (1 2), and Wilf asks if Sn can be In this paper, we study in particular the spectrum of values sequenced ”so that each is obtained from its predecessor” taken by the recurrence quotients of infinite partial words, by σ or τ. We settle Wilf’s open problem with a cyclic which are sequences that may have some undefined posi- sequence when n is odd, and a non-cyclic sequence when n tions. In this case, we determine exactly the spectrum of is even. values, which turns out to be 1, every real number greater than or equal to 2, and ∞. More precisely, if an infinite par- Aaron M. Williams tial word w is “ultimately factor periodic”, then ρ(w)=1, University of Victoria while if w is not, then ρ(w) ≥ 2, and we give constructions Department of Computer Science of infinite partial words achieving each value. [email protected] Francine Blanchet-Sadri University of North Carolina CP14 [email protected] Monotone Path Systems in Regions with Holes Bob Chen A monotone path system (MPS) is a finite set of pairwise University of California at San Diego disjoint paths (polygonal arcs) in the plane such that every [email protected] horizontal line intersects each of the paths in at most one point. Consider a simple polygon in the xy-plane which Sinziana Munteanu bounds the polygonal region D.LetT and B be two finite, Princeton University disjoint, equicardinal sets of points of D. Cameron and [email protected] Sachs gave a polytime algorithm for finding a MPS joining T with B. We consider polygonal regions with holes.

CP13 Kathie Cameron Unavoidable Sets Wilfred Laurer University, Canada [email protected] Partial words are sequences over a finite alphabet that may contain some undefined positions called holes. We first Katie Tsuji consider unavoidable sets of partial words of equal length. Wilfrid Laurier University We compute the minimum number of holes in sets satisfy- [email protected] ing some conditions (summed over all partial words in the sets). We also construct sets that achieve this minimum. This is a step towards the difficult problem of fully charac- CP14 terizing all unavoidable sets of partial words. We are next Lattice-point Generating Functions for Free Sums concerned with the complexity of deciding the avoidability of Convex Sets of sets of partial words over an arbitrary alphabet. Towards this, we investigate the minimum size of unavoidable sets The join of two convex sets in Euclidean space is the convex of partial words with a fixed number of holes. Addition- hull of their union. This join is called a free sum under cer- ally, we analyze the complexity of variations on the decision tain conditions on the convex sets. We present a multivari- problem when placing restrictions on the number of holes ate generating function that gives information about the lattice points in the free sum. This work is motivated by DM12 Abstracts 53

(and recovers) a product formula of Braun for the Ehrhart behind studying this family, and some applications of our series of the free sum of certain lattice polytopes. description. This is a joint work with Luis Goddyn.

Pallavi Jayawant Maryam Verdian Rizi Bates College Simon Fraser University [email protected] [email protected]

Matthias Beck San Francisco State University CP15 [email protected] n-Tournaments That Have n-Integer Signatures A compact representation for many digraphs is defined in Tyrrell B. Mcallister a ”labeled graph” context. Each positive integer of an n- University of Wyoming element subset S (called the signature of the digraph) of [email protected] Zm\{0}, is assigned to a vertex of a digraph with n vertices. An arc from a vertex x to another vertex y exists − CP14 if and only if (y x) mod m is an element of the signature. A digraph defined by such a signature is called a A New Lower Bound Based on Gromovs Method mod difference digraph. In this presentation, we discuss of Selecting Heavily Covered Points the n-Tournaments Tn having uniquely defined n-Integer signatures. Classical theorem of Bárány states that there exists cd > 0 such that for every n-set P of points in Rd in general posi- d n Vasudeva Acharya tion, there exists a point of R contained in at least cd d+1 Srinivas Institute of Technology, Mangalore (d + 1)-simplices with vertices at the points of P .Gro- [email protected] mov improved the known lower bound on cd by topological means. Using methods from extremal combinatorics, we Suresh M. Hegde improve one of the quantities appearing in Gromov’s ap- National Institute of Technology Karnataka proach and thereby provide a new stronger lower bound on Surathkal cd for arbitrary d. [email protected] Lukas Mach Computer Science Institute of Charles University (IUUK) CP15 Lesser Town Square, no. 25, Prague New Characterizations of Proper Interval Bigraphs [email protected] The class of proper interval bigraphs was introduced and Daniel Král characterized by Sen and Sanyal in terms of monotone con- Computer Science Institute of Charles University (IUUK) secutive arrangement of adjacency matrix.In this paper [email protected]ff.cuni.cz we define astral triple of edges in a bigraph and characterize proper interval bigraph in terms of this and other Jean-Sébastien Sereni notions.Tucker characterized proper circular arc graphs in CNRS (LIAFA) terms of circularly compatible 1’s of adjacency matrices.We [email protected]ff.cuni.cz have shown an inter-relation between monotone consecutive arrangement and circularly compatible 1’s.

CP14 Ashok K. Das On Structures of Geometrically Realizable Trian- Dept. of Pure Mathematics, University of Calcutta. India gulations on the Möbius Band [email protected] Let G be a graph embedded in to a surface F 2. A geometric realization of G is an embedding of F 2 into a Euclidian 3- CP15 space with no self-intersection such that each face of G is a House of Graphs: a Database of Interesting Graphs flat polygon. In 1983, Brehm found a Möbius triangulation (i.e., a triangulation on the on the Möbius band) which In this talk we will present a new searchable database of has no geometric realization. In this talk, we characterize graphs. Next to complete lists of some graph classes (such geometrically realizable Möbius triangulations. as fullerenes or snarks), also a list of graphs that already turned out to be interesting and relevant in the study of Shoichi Tsuchiya graph theoretic problems is offered. We will demonstrate Tokyo University of Science how users can perform queries on this database and how [email protected] they can add new interesting graphs to it. House of Graphs is accessible at http://hog.grinvin.org

CP14 Jan Goedgebeur Grid Representation for the Triangulations of the Ghent University Torus [email protected]

William Thurston describes all triangulations of the sphere Gunnar Brinkmann such that each vertex has degree six or less. In this talk, University of Ghent, we present a grid representation for the triangulations of Belgium the torus with at most two odd vertices where each vertex [email protected] has degree ﬁve or more. We also discuss the motivation 54 DM12 Abstracts

Kris Coolsaet CP16 Ghent University Cayley Factorization and the Area Method [email protected] Given a bracket polynomial B, i.e. a polynomial in formal Hadrien Melot determinants. Assume B is homogenous and has integer Universite de Mons coeﬃcients. Let e(B,C) be the evaluation of B for a point [email protected] conﬁguration C. The Cayley factorization problem asks for a synthetic construction for testing e(B,C) = 0. We give an explicit algorithm that generates a bracket monomial CP15 M and a construction which tests e(M · B,C)=0.This Immersions of Complete Graphs in Graphs with construction is more geometrically motivated and compact Minimum Degree n − 1 than the one of Sturmfels and Whiteley. M has lower degree. In 2003, F. Abu-Khzam and M. Langston conjectured that an n-chromatic graph contains an immersion of a complete Susanne Apel graph on n vertices. We will look at an attempt to prove Technische Universitaet Muenchen this by looking at a stronger statement involving minimum [email protected] degree. We will consider examples that show a graph with minimum degree n − 1 need not contain an immersion of a complete graph on n vertices for n greater than or equal CP16 to 8. Variations on the Erdos Discrepancy Problem Megan Heenehan, Karen Collins The Erdos Discrepancy Problem asks if there exists a se- { } {− } Wesleyan University quence ti i>0 over 1, 1 and a constant c such that | k | ∈ [email protected], [email protected] i=1 tid

cursion parameter c1,c2, ···,cl. We also derive exact values of and tighter bounds on M and M∗ for some parameters. See In this presentation, we show how enumerations of the http://arxiv.org/abs/0805.4059 for the full manuscript. backwards iterates of certain piecewise defined functions lead to general second order linear recurrence relations. In Guangyue Han particular, for each integer k ≥ 2, we provide a piecewise Department of Mathematics function fk that acts on k equal subdivisions of the unit University of Hong Kong interval [0, 1], and the recurrence relation that results from [email protected] counting the solutions of fk(x)=ψ for some ψ in (0, 1). Christopher Raridan, Elliot Krop CP17 Clayton State University Another Proof for Lovász’s Cathedral Theorem [email protected], [email protected] A graph with perfect matchings is satuated if addition of any complement edges creates new perfect match- CP17 ings. Cathedral theorem, which is originally investi- Universal H-Colourable Graphs gated by Lovász (1986) and later given another proof by Szigeti (1993), is a constructive charactrization of sat- Rado constructed a (simple) denumerable graph R with the urated graphs and appears in proving the theorem on positive integers as vertex set with the following edges: For toatally-k-covered graphs. Here, we gave the third proof given m and n with m

CP17 A Generalization of Menger’s Theorem CP17 Saturation Numbers for Families of Graph Subdi- Consider an acyclic directed network G with sources visions S1,S2, ···,Sl and distinct sinks R1,R2, ···,Rl.Fori = 1, 2, ···,l,letc denote the min-cut between S and R . For a family F of graphs, a graph G is F-saturated if G i i i F Then, by Menger’s theorem, there exists a group of ci contains no member of but the addition of any edge in edge-disjoint paths from S to R , which will be referred G results in one. The minimum number of edges in an i i F F to as a group of Menger’s paths from Si to Ri in this -saturated graph of order n is denoted sat(n, ). For a paper. Although within the same group they are edge- graph H, S(H) denotes the family of graphs resulting from disjoint, the Menger’s paths from different groups may replacing the edges of H with internally disjoint paths. We havetomergewitheachother.Inthispaper,weprove determine sat(n, S(C )) for small t, and provide a bound that by choosing Menger’s paths appropriately, the num- when t is arbitrarily large. We also examine sat(n, S(K )) ber of mergings among different groups of Menger’s paths in light of a result of Wagner’s on minors. is always bounded by a constant, which is independent of the size and the topology of G. The tightest such con- Craig Tennenhouse stant for the all the above-mentioned networks is denoted University of New England [email protected] by M(∞, ∈, ···, ∈)whenallSi’s are distinct, and by ∗ M (∞, ∈, ···, ∈)whenallSi’s are in fact identical. It turns out that M and M∗ are closely related to the net- Michael Ferrara work encoding complexity for a variety of networks. Com- University of Colorado Denver putation of these two functions, however, appears to be [email protected] rather difficult; so far there are no explicit formulas for ∗ M(∞, ∈, ···, ∈)andM (∞, ∈, ···, ∈) for a generic Michael Jacobson 56 DM12 Abstracts

University of Colorado, Denver Tomasz Krawczyk, Grzegorz Matecki = Jagiellonian University [email protected] [email protected], [email protected]

Kevin Milans University of South Carolina MS1 [email protected] Forbidden Induced Posets in the Boolean Lattice The induced Turán number La∗(n, P) is the maximum size Paul Wenger of a family of elements in the n-dimensional Boolean lattice University of Colorado Denver that does not contain P as an induced subposet. Not much [email protected] is known about La∗(n, P) when the Hasse diagram of P contains cycles. We present bounds on La∗(n, P) for some MS1 such posets. This is joint work with Linyuan Lu. Variations on the Majorization Order Linyuan Lu, Kevin Milans University of South Carolina The key role played by the majorization (aka dominance) [email protected], [email protected] order on integer partitions as a tool in both pure and applied mathematics has been widely recognized and docu- mented. Recently, variants of the classical majorization or- MS1 der have appeared in the study of posets defined by certain Title Not Available at Time of Publication symmetric function inequalities. We will describe these variants, focusing on differences and similarities between Abstract not available at time of publication these posets and the classical majorization order. William T. Trotter Curtis Greene School of Mathematics Haverford College Georgia Institute of Technology [email protected] [email protected]

MS1 MS2 Linear Extension Diameter and Reversal Ratio Lie Theory for Hyperplane Arrangements

Deﬁne the distance between two linear extensions L1 and There is a well-known isomorphism of representations be- L2 of a poset P as the number of incomparable pairs x, y tween the homology of the partition lattice and the mul- with xEconomics cializing to the case of braid arrangements. This is joint [email protected] work in progress with Swapneel Mahajan.

Graham Brightwell Marcelo Aguiar Department of Mathematics Texas A&M London School of Economics [email protected] [email protected] Swapneel Mahajan IIT Mumbai MS1 [email protected] Forbidden Structures for Eﬃcient First-Fit Chain Partitioning MS2 Using natural numbers for identifying chains, First-Fit al- Tesler Matrices, Parking Functions, and Diagonal gorithm processes elements of a poset P in some order and Harmonics for each element it assigns the smallest natural number such that all elements with the same number form a chain. We show how the Hilbert series of the space of Diagonal Let P be a class of posets with bounded-width, closed un- Harmonics can be expressed in terms of combinatorial ob- der taking induced subposets. We prove that there is a jects called Tesler matrices. This leads to some open pos- constant c such that all posets from P are partitioned by itivity questions involving what we call “colored parking First-Fitintoatmostcchainsifandonlyifthereisawidth functions”. This is joint work with D. Armstrong, A. Gar- 2posetnotinP. sia, B. Rhoades, and B. Sagan.

Bartlomiej E. Bosek Jim Haglund Jagellion University, Poland University of Pennsylvania [email protected] Department of Mathematics DM12 Abstracts 57

[email protected] Demazure characters.

Peter Tingley MS2 MIT Combinatorial Ergodicity in Products of Chains [email protected]

Many cyclic actions τ on a finite set S of combinatorial objects, along with many natural statistics φ on S,ex- MS3 hibit“combinatorial ergodicity’: the average of φ over each Conjectures Equivalent to the Borodin-Kostochka τ-orbit in S is the same as the average of φ over the whole Conjecture that a Priori Seem Weaker set S. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Borodin and Kostochka conjectured that every graph G Stump, and Thomas proved his conjecture in 2011. We de- with maximum degree Δ ≥ 9satisfiesχ ≤ max{ω,Δ − 1}. scribe a theoretical framework for results of this kind, and We carry out an in-depth study of minimum counterexam- discuss old and new results for products of chains. ples to the Borodin-Kostochka conjecture. Our main tool is the classification of graph joins A ∗B with |A|≥2, |B|≥2 Tom Roby that are f-choosable, where f(v):=d(v) − 1foreachver- University of Connecticut/MIT tex v. Since such a join cannot be an induced subgraph [email protected] of a vertex critical graph with χ =Δ,wehaveawealth of structural information about minimum counterexamples James Propp to the Borodin-Kostochka conjecture. Our main result is UMassLowell to prove that certain conjectures that a priori seem weaker [email protected] than the Borodin-Kostochka Conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with χ ≥ Δ=9containsK3 ∗K6 as a subgraph. MS2 Classical and Quasi Symmetric Hall-Littlewood Daniel Cranston Polynomials and Transition Matrices Virginia Commonwealth University An interesting property of Hall-Littlewood polynomials is Department of Mathematics and Applied Mathematics that they interpolate between the Schur and monomial [email protected] symmetric functions at t = 0 and 1, respectively. Hivert has constructed a family of quasisymmetric functions with MS3 a parameter t, which interpolate in a similar way between the fundamental and monomial quasisymmetric functions. List-coloring on Surfaces with Some Small Lists In this talk we present a combinatorial interpretation for C. Thomassen’s celebrated 5-list-coloring theorem for pla- the expansion of Hall-Littlewood polynomials in terms of nar graphs proves more. Among its consequences are the Hivert functions, as well as other transition matrices be- results that 1) a planar graph with one vertex with a 1-list tween the latter and other families of quasisymmetric func- and the rest with 5-lists and 2) a planar graph with 3-lists tions. on the vertices of one face and 5-lists elsewhere can be list- Nicholas Loehr colored. We study generalizations of these two variations Virginia Tech for graphs on nonplanar surfaces. We obtain initial results [email protected] for graphs on the projective plane and torus and begin a study for surfaces with larger Euler genus. Luis Serrano Alice M. Dean LACIM Skidmore College [email protected] Mathematics and Computer Science Department [email protected] Gregory Warrington University of Vermont Joan P. Hutchinson [email protected] Macalester College Department of Mathematics [email protected] MS2 Demazure Crystals, Kirillov-Reshetikhin Crystals, and the Energy Function MS3 Fractional Colorings of Cubic Graphs I will discuss a paper with Anne Schilling which surveys and expands upon some relationships between Demazure Fractional coloring is a relaxation of classical graph col- crystals of non-exceptional affine Kac-Moody algebras and oring. Each independent set is assigned weight and each Kirillov-Reshetikhin (KR) crystals. In particular, certain vertex is required to be in independent sets of weight at Demazure crystals are isomorphic as classical crystals to least one. The goal is to minimuze the total weight of in- tensor products of KR crystals, and we show that this dependent sets. In the talk, we present two recent results: isomorphism intertwines the natural affine grading on the The fractional chromatic number of any triangle-free cu- Demazure crystals with a combinatorially defined energy bic graph is at most 32/11 and the fractional chromatic function. This leads to a formula for the Demazure char- number of any cubic graph with sufficiently large girth is acter in terms of the energy function, and has applications smaller than 23/10. to symmetric function theory since certain specializations of Macdonald polynomials are equal to specializations of David Ferguson London School of Economics 58 DM12 Abstracts

[email protected] Douglas B. West University of Illinois, Urbana Tomas Kaiser Department of Mathematics University of West Bohemia [email protected] [email protected] MS4 Frantisek Kardos INRIA Sophia-Antipolis Title Not Available at Time of Publication [email protected] Abstract not available at time of publication. Daniel Kral Hamed Hatami Charles University Department of Mathematics [email protected]ff.cuni.cz Princeton Univ. [email protected] Jan Volec IUUK Charles University [email protected]ff.cuni.cz MS4 On a Problem of Erd˝os and Rothschild on Edges in Triangles MS3 4-critical Graphs on Surfaces without Contractible Erd˝os and Rothschild asked to estimate the maximum Cycles of Length at Most 4 number h(n, c) such that every n-vertex graph with at least cn2 edges, each of which is contained in at least one trian- We show that every 4-critical graph G embedded in a fixed gle, must contain an edge that is in at least h(n, c) triangles. surface Σ so that every contractible cycle has length at In 1987, Erd˝os asked whether for every c>0, there is >0 O(1/ log log n) least 5 can be expressed as G = G ∪ G1 ∪ G2 ∪ ...∪ Gk, such that h(n, c) >n.Weproveh(n, c)=n where |V (G )| and k are bounded by a constant (depending for every fixed c<1/4, the first improvement over an old linearly on the genus of Σ) and G1, ..., Gk are graphs (of construction of Alon and Trotter. unbounded size) whose structure we describe exactly. The proof is computer-assisted— we use computer to enumerate Jacob Fox all plane 4-critical graphs of girth 5 with a precolored cycle MIT of length at most 16, that are used in the basic case of the [email protected] inductive proof of the statement. Po-Shen Loh Zdenek Dvorak Department of Mathematical Sciences Department of Applied Mathematics Carnegie Mellon University Charles University, Prague [email protected] [email protected]ff.cuni.cz

Bernard Lidicky MS4 University of Illinois, Urbana, IL Turan’s Brickyard Problem and Flag Algebras [email protected] Recently, Razborov developed a ”ﬂag algebra” calculus which captures in pure form many of the techniques used MS3 in extremal combinatorics. This calculus has been success- Extending Graph Choosability Results to fully used in the last few years to improve the best known Paintability bounds on many of the long-standing open problems in extremal graph theory. In this talk we describe an appli- Introduced independently by Schauz and by Zhu, the cation to the Turan’s brickyard problem: the problem of Marker/Remover game is an on-line version of list col- determining the crossing number of the complete bipartite oring. The resulting graph parameter, paintability,isat graph Km,n. Beas on joint work with Yori Zwols. least the list chromatic number (the choosability). We strengthen several choosability results to paintability. We Sergey Norin study paintability of joins with complete graphs and de- McGill University termine bounds on the paintability of complete bipartite [email protected] graphs. We characterize 3-paint-critical graphs and show that claw-free perfect graphs with ω(G) ≤ 3 have paintabil- MS4 ity equal to chromatic number. Finally, we introduce and k study sum-paintability, the analogue of sum-choosability. Tur´an Densities of Hypergraphs Related to Kk+1

(k) James Carraher Let Bi be the k-uniform hypergraph on the vertex set University of Nebraska S ∪ T with |S| = i and |T | = k − 1 with the edges consist- [email protected] ing of all k-sets containing S or T . We derive upper and (k) lower bounds for the Turán density of Bi that are close to Sarah Loeb, Thomas Mahoney, Gregory Puleo, Mu-Tsun each other as k →∞. We also obtain asymptotically tight Tsai bounds for the Turán density of several other infinity fam- University of Illinois ilies of hypergraphs. The construction that supports the [email protected], [email protected], lower bounds is derived from elementary number theory [email protected], [email protected] by probabilistic arguments. The upper bounds are derived DM12 Abstracts 59

from the results of de Caen, Sidorenko, and Keevash. some results towards its solution.

Yi Zhao Andrea Burgess Georgia State University Ryerson University [email protected] [email protected]

J´ozsef Balogh Patrick Niesink University of Illinois . [email protected] [email protected]

Tom Bohman Mateja Sajna Carnegie Mellon University of Ottawa [email protected] [email protected]

Béla Bollobás Trinity College, Cambridge MS5 University of Memphis Defining Sets in Combinatorial Arrays [email protected] A defining set in a combinatorial array is a partially filled- in array with a unique completion. We compare what is MS4 known about the defining sets of (0, 1)-matrices and Latin Extremal results in sparse pseudorandom graphs squares. While these arrays on the surface seem quite different, for certain cases the minimum possible size for a Szemerédi’s regularity lemma is a fundamental tool in ex- defining set appears to be the same. tremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Ko- Nicholas Cavenagh hayakawa and Rödl proved an analogue of Szemerédi’s reg- University of Waikato ularity lemma for sparse graphs as part of a general pro- [email protected] gram toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity MS5 lemma use an associated counting lemma. In order to prove analogues of these results for sparse graphs, it re- Graph Decompositions and Convexity mained a well-known open problem to prove a counting One of the most interesting conjectures in the area of graph lemma in sparse graphs. The main advance of this paper decompositions is due to Nash-Williams. It states that lies in a new counting lemma, proved following the func- the edge set of any sufficiently large graph with all even tional approach of Gowers, which complements the sparse degrees, 3m edges, v vertices and minimum degree of at regularity lemma of Kohayakawa and Rödl, allowing us to least 3/4v can be partitioned into triangles. We discuss a count small graphs in regular subgraphs of a sufficiently new approach for exploring this problem using the inclusion pseudorandom graph. We use this to prove analogues of matrix of the Johnson scheme and the facets of the convex several well-known combinatorial theorems, including the cone it generates. graph removal lemma, the Erd˝os-Stone-Simonovits theorem, and Ramsey’s theorem, for subgraphs of sparse pseu- Kseniya Garaschuk dorandom graphs. These results extend and improve upon University of Victoria a substantial body of previous work. [email protected] David Conlon University of Oxford Peter Dukes [email protected] Assistant Professor [email protected] Jacob Fox MIT MS5 [email protected] Extending the Bruck-Ryser-Chowla Theorem to Coverings Yufei Zhao Massachusetts Institute of Technology The celebrated Bruck-Ryser-Chowla theorem rules out the [email protected] existence of various balanced incomplete block designs, including projective planes of infinitely many orders. In this talk I will discuss how (parts of) the Bruck-Ryser-Chowla MS5 theorem can be extended to establish the nonexistence of On the Directed Oberwolfach Problem various covering designs and, consequently, the nonexistence of certain decompositions of a complete graph into − The Oberwolfach problem asks whether the Kn (or Kn I copies of a smaller complete graph minus an edge. if n is even) admits a 2-factorization in which each factor is isomorphic to a specified 2-factor F . We consider a di- Daniel Horsley rected variant of this problem, in which we are now required Memorial University, Newfoundland, Canada ∗ to factorize the complete symmetric digraph Kn.Inpar- [email protected] ticular, we consider the question of determining necessary and sufficient conditions for the existence of a resolvable ∗ Darryn Bryant, Melinda Buchanan, Barbara Maenhaut, decomposition of Kn into directed m-cycles, and present Victor Scharaschkin University of Queensland 60 DM12 Abstracts

[email protected], [email protected], MS6 [email protected], [email protected] Understanding SHAPE-directed RNA Secondary Structure Prediction MS5 We investigate the interplay between experimental data Trinal Decompositions of Steiner Triple Systems and thermodynamic model via stochastic simulations. RNA prediction under the NNTM is a discrete optimiza- Let STS(n) be a Steiner triple system of order n.Atri- tion problem whose accuracy can be improved by the in- angle T is a set of three pairwise intersecting triples of an corporation of auxiliary information. However, our results STS(n) whose intersection is empty. Z. Füredi posed a demonstrate that improvements in SHAPE-directed pre- question whether the set of triples of any STS(n)canbe dictions are non-uniform and correlated with original MFE decomposed into triangles. This question remains largely accuracies. This analysis suggests that further advances unanswered, although examples of such decomposition are in computational molecular biology are needed to reliably known for every admissible order n ≡ 1or9(mod 18). A predict RNA secondary structures. triangle T = {{a, b, c}, {c, d, e}, {e, f, a}} in an STS(n)is sometimes called a hexagon triple because the outer edges Zsuzsanna Sukosd ab, bc, cd, de, ef, fa form a hexagon. Depending on a Nanoscience Center graph-theoretic or geometric representation, respectively, Aarhus University that triangle determines naturally two more triples: the [email protected] inner triple {a, c, e} and the midpoint triple {b, d, f}.In either case, the number of inner triples (called type 1) or Christine E. Heitsch that of midpoint triples (type 2) equals one third of the to- School of Mathematics tal number of triples in an STS(n). A problem which will Georgia Tech be discussed concerns the existence of three distinct decom- [email protected] positions of an STS(n) into triangles such that the union of three collections of type 1 triples (type 2, respectively) from these three decompositions form a set of triples of a MS6 Steiner triple system of the same order n. Such decomposi- Detecting SNP-Induced Structural Changes in tions are called trinal decompositions of type 1 and type 2, RNA: Application to Disease Studies respectively. Solutions to the existence question for trinal decompositions of type 1 and type 2 will be presented. Single Nucleotide Polymorphisms (SNPs) are often linked to critical phenotypes such as diseases. However, the spe- Mariusz Meszka cific molecular mechanisms by which a causal SNP acts is AGH University of Science and Technology usually not obvious. Changes in RNA secondary structure [email protected] emerge as a possible explanation. We introduce remuRNA, an efficient method to compute the distance between the Charles C. Lindner native and mutant RNA structure Boltzmann ensembles. Auburn University We applied remuRNA to determine which of the disease- [email protected] associated non-coding SNPs are potentially related to RNA structural changes. Alexander Rosa Department of Mathematics Raheleh Salari McMaster University Stanford University [email protected] [email protected]

Teresa Przytycka MS6 NCBI Structural Requirements for RNA Elements [email protected]

RNA plays a major role in many regulatory circuits of cells, and new and often suprising function are discovered con- MS6 tinously, The structure of RNA elements determines the Efficient Algorithms to Explore the RNA Muta- associated function for many ncRNAs. Clustering by sim- tional Landscape ilarity on the level of sequence and structure is now an accepted approach for annotating ncRNAs and for deter- Understanding the relationship between RNA sequences mining new ncRNA classes. The problem, however, is the and structures is essential to decipher evolutionary pro- large number of RNAs to be clustered (e.g., up to 450.000 cesses, predict deleterious mutations and design synthetic predicted ncRNAs in human), and the complexity of the molecules. We introduce RNAmutants, the first algorithm similarity test. We have introduced a new alignment-free for exploring complete RNA sequence-structures maps in method for clustering that avoids the complex all-againts- polynomial time and space. Using statistical mechanics all pairwise comparisons, and is still able to cluster accord- and importance sampling techniques, we estimate the ther- ing to sequential and structural properties. Furthermore, modynamical pressure applied on sequences and explore re- we will discuss application problems like the bacterial adap- gions of the mutational landscape preserving the nucleotide tive immunesystem called CRISPR that is based on RNA. composition. We show that C+G-contents influence the evolutionary accessible structural ensembles.

Rolf Backofen Jerome Waldispuhl University of Freiburg McGill University [email protected] [email protected]

Bonnie Berger DM12 Abstracts 61

Applied Mathematics, MIT size minor that is not a projection of a graphic matroid. [email protected] Bertrand Guenin Srinivas Devadas University of Waterloo CSAIL, MIT [email protected] [email protected] Irene Pivotto Peter Clote Simon Fraser University, BC Biology, Boston College [email protected] [email protected] Paul Wollan Yann Ponty Department of Mathematics LIX, Ecole Polytechnique University of Hamburg [email protected] [email protected]

MS7 MS7 Path Graphs, PR-trees, and Split Decomposition Fixed Weight De Bruijn Graphs In this work we preset new characterizations of (directed) Let S be the set of all binary strings of length n with weight (number of 1s) w and w-1. The de Bruijn graph for S is path graphs*. First, we describe a new data structure (PR- − trees) which captures the set of path-tree models of a path adirectedgraphG(S) whose nodes are the length n 1 graph (generalizing PQ-trees). This allows us to character- prefixes and suffixes of the strings in S. There is an arc labeled x ∈{0, 1} from α = a1 ···an−1 to β = a2 ···an−1x ize and recognize (directed) path graphs via split decompo- ∈ sition. The recognition algorithm runs in O(|V| ∗ |E|)for if αx S. In addition to considering constructions of fixed both directed path graphs and path graphs. * The inter- weight de Bruijn cycles, we consider the problem of deter- section graphs of (directed) paths in (directed) trees. mining the diameter of the de Bruijn graph G(S).

Steven Chaplick Joe Sawada Department of Computer Science, University of Guelph University of Toronto Dept. of Computer Science [email protected] [email protected]

MS7 MS7 Algorithms for Unipolar and Generalized Split Induced and Distance-k Matchings and Some Re- Graphs lated Min-Max Relations Adistance-k matching in a graph is a matching such that G=(V1 ∪ V2,E)isunipolar when V1 is a clique and V2 induces the disjoint union of cliques. A generalized split the distance between any two edges of the matching is at graph is either unipolar or its complement is unipolar. We least k. Computing the size of a largest induced matching present an O(nm )-time recognition algorithm for unipo- (distance-2 matching) in a graph is NP-hard. We consider lar graphs, where m is the number of edges in a minimal finding a largest distance-k matching in certain classes of triangulation of the given graph. We efficiently solve four graphs. In some cases, along comes a relation between classic optimization problems on unipolar and generalized the size of a largest distance-k matching and the size of a split graphs and prove that perfect code is NP-complete smallest appropriate cover. for unipolar graphs. R Sritharan,ArthurBusch Elaine M. Eschen The University of Dayton Lane Department of CSEE [email protected], [email protected] West Virginia University [email protected] Feodor F. Dragan Kent State University Xiaoqiang Wang Dept. of Computer Science West Virginia University [email protected] [email protected] Chandra Krishnamurthy The University of Dayton MS7 [email protected] Recognizing Even-Cycle Matroids A matroid M is an even-cycle matroid if the cycles of M MS8 correspond to the even cycles of a signed graph. The prob- Q2-Free Families in the Boolean Lattice lem of recognizing whether a binary matroid (represented by a 0,1 matrix) is an even-cycle matroid is open. Progress For a family of subsets of {1,...,n}, ordered by inclusion on this problem has been hampered by the fact that even and a partially-ordered set P , we say that the family is P - cycle matroids can have an arbitrary number of pairwise free if it does not contain a subposet isomorphic to P .We inequivalent representations (two signed graph are equiva- are interested in finding ex(n, P ), the largest size of a P - lent if they are related by a sequence of Whitney-flips and free family of subsets of [n]. It is conjectured that, for any n signature exchanges). We discuss how to proceed for the fixed P , this quantity is (k+o(1)) 2 for some fixed integer recognition problem for the case where we are given a fixed 62 DM12 Abstracts

k, depending only on P . Recently, Boris Bukh has verified tichains in a (k + k)-free poset of width w is (k − 1)w−1. the conjecture for P which are in a “tree shape’. There This is easily shown to be true for w ≤ 2. We give the are some other small posets P for which the conjecture proof for w = 3 and present several different constructions has been verified. The smallest for which it is unknown is attaining the conjectured extremal value for w ≥ 3. De- Q2, the Boolean lattice on two elements. We will discuss spite this support we still feel to be far away from the proof improvements of the bounds of Griggs, Li, and Lu. This for all w. is joint work with Ryan Martin, Iowa State University and Michael Young, Iowa State University. MichalLasoń, Piotr Micek Jagiellonian University Lucas J. Kramer, Ryan R. Martin, Michael Young [email protected], [email protected] Iowa State University [email protected], [email protected], Noah Streib [email protected] Georgia Institute of Technology [email protected] MS8 William T. Trotter First-Fit Coloring of Ladder-Free Posets School of Mathematics Bosek and Krawczyk provided a subexponential bound for Georgia Institute of Technology the on-line chain partitioning of posets of width w and [email protected] observed the problem could be reduced to First-Fit chain partitioning of 2w2-ladder-free posets of width w.Wepro- Bartosz Walczak vide a subexponential upper bound (in terms of w with m Jagiellonian University fixed) for the performance of First-Fit chain partitioning [email protected] on m-ladder-free posets. With the Bosek-Krawczyk observation, this yields an slightly improved upper bound for the general problem. MS8 The Dimension of Posets with Planar Cover Matt E. Smith Graphs Arizona State University [email protected] We study the conditions that bound the dimension of posets with planar cover graphs. We show that if P is H. A. Kierstead a poset with a planar comparability graph, then the di- Mathematics Department mension of P is at most four. We also show that if P has Arizona State University an outerplanar cover graph, then the dimension of P is at [email protected] most four. Finally, if P has an outerplanar cover graph and the height of P is two, then the dimension of P is at most three. These three inequalities are all best possible. MS8 Stefan Felsner Dimension and Height for Posets with Planar Technical University of Berlin Cover Graphs Institute for Mathematics Planar posets can have arbitrarily large dimension. How- [email protected] ever, we show that the dimension of a planar poset is bounded as a function of its height. More precisely, we William T. Trotter show that for each integer h ≥ 2, there exists a least pos- School of Mathematics itive integer ch so that if P isaposetwithaplanarcover Georgia Institute of Technology graph and height h, then the dimension of P is at most [email protected] ch. Trivially, c1 = 2. Felsner, Li and Trotter showed that c2 = 4, but their proof techniques do not seem to apply Veit Wiechert when h ≥ 3. In this paper, we establish the existence of TU - Berlin ch, although we suspect that the upper bound provided by [email protected] our proof is far from best possible. From below, a construction of Kelly is modified to show that ch is at least h +2. MS9 Tropisms, Surfaces and the Puiseux Series Noah Streib Georgia Institute of Technology We present a polyhedral method to develop Puiseux series [email protected] expansions for surfaces, defined by a system of polynomials. Our starting point is the construction of cones of nor- William T. Trotter mal vectors to the Newton polytopes, associated with the School of Mathematics system of polynomials. Generating vectors of the normal Georgia Institute of Technology cone, which lead to an initial form system with regular, [email protected] isolated solutions are considered pretropisms. When the generating vectors of the normal cone lead to an exact, finite representation of the surface or to its Puiseux series MS8 expansion, then they are referred to as tropisms. The Width of the Family of Maximum Antichains Danko Adrovic,JanVerschelde Felsner, Krawczyk and Micek conjectured in 2010 that the Department of Mathematics, Statistics and Computer maximum possible width of the family of maximum an- Science DM12 Abstracts 63

University of Illinois at Chicago [email protected] [email protected], [email protected]

MS10 MS9 Globally Fair Stable Matchings Divisors on Tropical Varieties Stable marriage instances generally have many stable I will present notions of divisors, linear equivalence, and matchings. In spite of this, Gale and Shapley’s famous rank on tropical models. A tropical model is a simplicial algorithm for computing stable matchings can only out- complex together with some numerical data telling how the put two kinds: the man-optimal/woman-pessimal or the affine linear structures relate. Although the motivations woman-optimal/man-pessimal stable matchings. This has come from algebraic varieties, I will discuss the combina- motivated the study of fair stable matchings. This talk is torial aspects. about stable matchings whose fairness is derived from the fact that they are ”good” representatives of the distributive Dustin Cartwight lattice of stable matchings of the instance. Department of Mathematics Yale University Christine T. Cheng [email protected] University of Wisconsin-Milwaukee [email protected] MS9 Orbits of Projective Point Configurations MS10 Stable Matching as a Heuristic for Cost-Based Given an r × n matrix, thought of as representing n points Matching Problems in projective (r−1)-space, we consider the closure of the orbit of all projectively equivalent matrices. I will discuss the Minimum-cost assignment problems arise in many situa- equations cutting out this variety, its finely graded Hilbert tions in practice. The optimal solution of such problems series, and their relation to the matroid of the point con- usually requires network flow techniques, which can be figuration. computationally taxing for very large problem instances. Here, we discuss potential heuristics for minimum-cost as- Alex Fink signment problems based on stable matchings, which can North Carolina State University generally be computed much more efficiently. A number arfi[email protected] of approaches will be discussed, along with the results of empirical testing to assess their viability in practice. Andrew Berget University of California, Davis Brian C. Dean, John Dabney [email protected] Clemson University [email protected], [email protected]

MS9 Log-concavity of Characteristic Polynomials and MS10 Tropical Intersection Theory Some Open Problems in Matchings with Prefer- ences In a recent joint work with June Huh, we proved the log concavity of the characteristic polynomial of a realizable Matching problems involving preferences have been stud- matroid by relating its coeﬃcients to intersection numbers ied for many years, and have a range of real-world appli- on an algebraic variety and applying an algebraic geomet- cations. The stable marriage problem is perhaps the most ric inequality. In this talk, we outline that proof which famous such problem, but many variants of it have been involves algebraic geometric positivity. studied, involving extensions and variations of the basic model. Despite the extensive literature in this area, many Eric Katz challenging (and fascinating) open problems remain. I will University of Texas Austin survey some of these open problems that have connections [email protected] with discrete mathematics, algorithms and complexity.

David F. Manlove MS9 University of Glasgow Tropical Torelli Space and Tropical Period Map- [email protected] ping I will discuss the Teichmuller type approach to the mod- MS10 uli of tropical curves. We also define the tropical period A Unified Approach to Equivalence Results in Ob- mapping and investigate the injectivity of this map on the ject Allocation Torelli space (i.e. the generalized Torelli problem). We will describe the image of the period map (i.e. the Schot- We consider probabilistic mechanisms that allocate indi- tky locus). To better understand the Torelli problem and visible objects to agents by hierarchical exchange using the the Schottky locus, we consider the (marked) moduli of top-trading cycles algorithm. The main result is a general “metric” regular matroids. Various compactifications of technique for proving that seemingly different probabilistic the moduli of tropical curves will be obtained from this mechanisms are in fact equivalent. This approach simpli- point of view. fies and unifies several equivalence results in the literature. The same technique is used to generalize these results to Farbod Shokrieh mechanisms in which the priority structure for each ob- Georgia Institute of Technology ject is given by a tree (instead of a linear ordering of the 64 DM12 Abstracts

agents). MS11 An Erdos-Stone Theorem for Hypergraphs Jay Sethuraman Columbia University This talk is about complete k-partite subgraphs in hyper- [email protected] graphs with suﬃciently many k-cliques. These results generalize the Erd˝os-Stone theorem for 2-graphs. Thiam Lee Columbia Vladimir Nikiforov [email protected] Department of Mathematical Sciences, University of Memphis Memphis, TN 38152-3240 , United States MS11 [email protected] Homeomorphically Irreducible Spanning Trees Let G be a graph. A spanning tree of G is called a homeo- MS11 morphically irreducible spanning tree (HIST) if it does not On Independent Sets in Steiner Systems contain vertices of degree 2. In 1979, Albertson, Berman, Hutchinson, and Thomassen asked the following two ques- For a hypergraph H,letα(H) denote the largest set of tions: vertices containing no edge of H. Inthistalkwegivenear- optimal bounds on the size of a largest independent set in 1. Does every triangulation of a surface contain a HIST Steiner systems on n points. We conjecture that the mini- except the triangle? mum value of α(H√) over all Steiner systems H on n vertices 2. Does every graph with every edge on two triangles is asymptotic to 3n log n as n →∞.Themethodsarea contain a HIST? combination of spectral techniques and probabilistic com- We have conﬁrmed both questions positively. The outlines binatorics, which are useful for various other problems on of the proofs will be given in this talk. independent sets in hypergraphs.

Guantao Chen Jacques Verstraete Department of Mathematics and Statistics University of California-San Diego Georgia State University [email protected] [email protected] MS12 Songling Shan Graph Homomorphisms: Mixing and Homotopies Georgia State University [email protected] Given a k-colouring of a graph, consider the process of recolouring a single vertex to obtain a new k-colouring. Provided k is sufficiently large, one can generate all k- MS11 colourings of the graph. We study circular colourings giv- Decompositions of (Hyper)Graphs into Cliques or ing bounds on how large k/q must be to ensure all circular Bicliques colourings can be generated by recolouing. We connect our work to discrete homotopies of graph homomorphisms and This talk will be a brief survey of recent results regard- pre-colourings of graphs. ing edge-decompositions of (hyper)graphs into cliques or bicliques. These include various extensions of De Bruijn- Richard Brewster Erdos Theorem and Graham-Pollak Theorem. Despite re- Department of Mathematics and Statistics cent progress, many problems remain open. Thompson Rivers University [email protected] Sebastian Cioaba University of Delaware at Newark [email protected] Jon Noel McGill University [email protected] MS11 Multipartite Version of the Alon-Yuster Theorem MS12 In this talk, we prove the asymptotic multipartite version Defective Colorings and Colorings that Avoid of the Alon-Yuster theorem. That is, if k ≥ 3 is an integer, Large Monochromatic Components H is a k-colorable graph and γ>0 is fixed, then for sufficiently large n and for every balanced k-partite graph G We consider two types of colorings. Those that avoid monochromatic graphs with large degree and those that on kn vertices with each of its corresponding k bipartite 2 avoid monochromatic graphs with large components. We k−1 subgraphs having minimum degree at least k n + ηn,the discuss relations between these two parameters and make graph G has a subgraph consisting of n/|V (H)| vertex- several remarks on issues of computational complexity. disjoint copies of H. John Gimbel Ryan R. Martin University of Alaska Fairbanks Iowa State University [email protected] [email protected]

Jozef Skokan MS12 London School of Economics Kempe-equivalence Classes for 3-edge-colored Cu- [email protected] DM12 Abstracts 65

bic Graphs Gary Macgillivray Mathematics and Statistics An edge-Kempe change switches the colors in a maximal University of Victoria two-colored chain in a proper edge coloring of a graph. Two [email protected] proper edge colorings of a graph are called Kempe equivalent if you can get from one to the other by a sequence of edge-Kempe changes. This work examines Kempe equiv- MS13 alence in 3-edge colorable cubic graphs. Not all proper TBD on Genome Assembly Algorithms edge colorings of a 3-edge colorable cubic graph are Kempe equivalent. We give results about the number of non- Abstract not available at time of publication. equivalent proper edge colorings there can be for certain classes of cubic graphs. Max Alekseyev University of South Carolina Ruth Haas [email protected] Smith College [email protected] MS13 Sarah-Marie Belcastro De Bruijn Graph Based Genome Assembly for Sin- Smith College gle Cells Hampshire College Summer Studies in Mathematics [email protected] Characterization of environmental bacteria, the vast majority of which elude cultivation, is necessary for many applications including discovery of rare novel species. The MS12 ability to generate high-quality draft genome assemblies Distinguishing Edge Colourings of Graphs that support annotation of the majority of genes will drive advances in characterizing such uncultured organisms. Re- An edge colouring of a graph G is distinguishing provided cent advances in DNA amplification technology have en- that the identity is the only automorphism of G that pre- abled whole genome sequencing directly from individual serves the edge colours. The minimum number of colours cells without requiring growth in culture. These meth- required to produce such a colouring is the edge distin- ods amplify femtograms of DNA extracted from a single guishing chromatic number of G. In this talk I will describe bacterial cell into micrograms of DNA needed for current results that we have obtained concerning the edge distin- sequencing platforms. Based on these advances, we devel- guishing chromatic number and the more general edge dis- oped a specialized software tool for assembling sequencing tinguishing number of a graph. reads from single cells and applied it to assembly of two known genomes, E. coli and S. aureus, and an unknown Karen Seyffarth marine genome, SAR324 Deltaproteobacterium. These Mathematics and Statistics draft de novo single cell assemblies, with no efforts to close University of Calgary gaps and resolve repeats, identify more than 90% of genes. kseyff[email protected] DNA amplification using Multiple Displacement Amplifi- cation (MDA), however, creates serious coverage bias and Richard Brewster chimeric fragments. Our algorithm successfully deals with Department of Mathematics and Statistics these errors. Our assembler is based on the celebrated de Thompson Rivers University Bruijn graph in which nodes are (k −1)-mers and edges are [email protected] k-mers. We present the general de Bruijn graph sequence assembly algorithm as well as our modifications to it. Stacey Lamont Hamidreza Chitsaz Department of Mathematics and Statistics Wayne State University University of Calgary [email protected] [email protected]

MS13 MS12 Quantifying Uniformity of Mapped Reads Obstructions to Homomorphisms Involving the Graft Extension We describe a tool for quantifying the uniformity of mapped reads in high-throughput sequencing experiments. Let H1 be a digraph that has an X-enumeration, say { } Our statistic directly measures the uniformity of both read h1,h2,...,hn .LetH2 be a digraph such that H2- position and fragment length, and we explain how to com- colouring is polynomial. Form a new digraph H by deleting pute a p-value that can be used to quantify biases arising the vertex hn from H1 and replacing it by the digraph H2: ∈ from experimental protocols and mapping procedures. Our every vertex hi V (H1)thatisadjacentto(from)hn is method is useful for comparing different protocols in exper- now adjacent to (from) every vertex in H2. The digraph iments such as RNA-Seq. H is denoted by graft(H1,H2). Given a digraph H, a dual (or obstruction) of H is a digraph F such that G → H if Valerie Hower, Richard Starfield, Adam Roberts and only if F → G. In this talk we aim to identify the University of California, Berkeley obstructions of H inthecasewhereH =graft(H1,H2). [email protected], rstarfi[email protected], [email protected] Jacobus Swarts Mathematics Vancouver Island University Lior Pachter [email protected] University of California, Berkeley Dept. of Mathematics 66 DM12 Abstracts

[email protected] Department of Mathematics Oberlin College [email protected] MS13 Combinatorial Designs for Sequencing Pooled Sam- ples MS14 Graph Pebbling: Past, Present, Future Rare variants account for missing heritability that cannot be explained by common variants. DNA Pooling is a cost In this talk we aim to give a gentle introduction to the sub- effective method to sequence a large number of samples and ject of graph pebbling, including its history from combina- to discover rare variants. In this talk, I will present two new torial number theory, its growth as a purely graph theoretic combinatorial designs for overlapped pooling strategies and topic, and its development into an area of combinatorial show their effectiveness in discovering rare variants and in optimization. We shall describe some of its variations as identifying variant carriers. well, such as optimal pebbling, cover pebbling, fractional pebbling, pebbling thresholds, and rubbling, along with its Wenhui Wang, Xiaolin Yin, Matthew Hayes, Yoon Soo connections to computer science. Lastly, we will present Pyon, Jing Li a few of the major techniques in the field and offer some Case Western Reserve University tantalizing open problems for further research. [email protected], [email protected], [email protected], [email protected], [email protected] Glenn Hurlbert Arizona State University [email protected] MS13 Spaced Seeds and their Application in Next Gen- eration Sequencing MS14 Graph Rubbling We will briefly overview the spaced seeds technique for improving the speed and sensitivity of homology search. Graph rubbling is a version of graph pebbling where an Then, a variation of spaced seeds is introduced to provide additional move is allowed. The new move adds one peb- full sensitivity reads mapping for next generation sequenc- ble at vertex u in exchange for removing one pebble each ing analysis. from two vertices that are neighbors of u. Rubbling numbers are defined analogously to pebbling numbers. The Bin Ma purpose of this talk is to give a summary of what is known University of Waterloo about rubbling and optimal rubbling and to present a set [email protected] of unanswered questions.

Nandor Sieben MS14 Department of Mathematics The Pebbling Threshold of Graph Sequences Northern Arizona University [email protected] A pebbling threshold for a family of graphs is a function g(n) such that any pebbling configuration of size much larger than g(n) is almost surely solvable and of size much MS14 less than g(n) is almost surely unsolvable. We will dis- Pebbling Graphs of Diameter Three and Four cuss pebbling threshold for an arbitrary graph family and present the results for particular families of graphs and dis- Given a configuration of pebbles on the vertices of a con- cuss open problems. We will also introduce a probabilistic nected graph G, a pebbling move is defined as the removal version of Graham’s pebbling conjecture for the threshold. of two pebbles from some vertex and the placement of one of these on an adjacent vertex. The pebbling number of a Airat Bekmetjev graph G is the smallest integer k such that for each vertex Mathematics Department v and each configuration of k pebbles on G there is a se- Hope College quence of pebbling moves that places at least one pebble [email protected] on v. We improve on a bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most 3n/2+2, and this bound is best possible. MS14 Further, we will describe an asymptotic bound for graphs Complexity of Diameter Two Graph Pebbling of diameter four and an asymptotic bound for pebbling graphs of arbitrary diameter obtained via discharging. We present tight bounds on the number of vertices with two and three pebbles that can exist in an unsolvable configu- Carl Yerger ration on a diameter two graph in terms of the size of the Department of Mathematics graph. We use the construction from this result to prove Davidson College that determining reachability of a vertex is NP-complete, [email protected] even in graphs of diameter two. Charles Cusack, Timothy Lewis, Daniel Simpson Noah Streib Department of Mathematics Georgia Institute of Technology Hope College [email protected] [email protected], [email protected], [email protected] Luke Postle Georgia Tech Samuel Taggart [email protected] DM12 Abstracts 67

MS15 MS15 Phase Transition in Random Graph Processes The Fractal Nature of the Abelian Sandpile through the Lens of PDE and Singularity Analy- sis The Abelian sandpile is a diffusion process for configurations of chips on the integer lattice; a vertex with at least 4 Erd˝os and Rényi show that in the standard random graph chips topples, distributing one chip to each of its neighbors. on n vertices, a phase transition takes place when the num- One of the most striking unexplained features of the sand- ber of edges reaches n/2 and a giant component emerges. pile is that it appears to produce terminal configurations Since this seminal work, various random graph processes converging to a peculiar fractal pattern when begun from have been studied. In this talk we discuss new approaches increasingly large stacks of chips at the origin. We will to study the size and structure of components near the crit- present a mathematical explanation for this phenomenon. ical point of random graph processes: key techniques are a quasi-linear PDE and its singularity analysis. Wesley Pegden Mihyun Kang Courant Institute Technische Universität Graz [email protected] Institut für Optimierung und Diskrete Mathematik [email protected] Charles Smart Massachusetts Institute of Technology [email protected] MS15 Self-similarity of Graphs Lionel Levine An old problem of Jacobson and Schönheim asks for the Cornell University maximum s such that every m-edge graph contains a pair [email protected] of edge-disjoint isomorphic subgraphs with s edges. We prove that every m-edge graph contains a pair of edge- MS15 disjoint isomorphic subgraphs with at least c(m log m)2/3 Achlioptas Processes: Recent Results and New edges. We also construct graphs that show this estimate Problems is correct up to a constant factor. Our results improve bounds of Erd˝os, Pach, and Pyber from 1987. Recent results have shown that the phase transition of a large class of Achlioptas processes looks qualitatively simi- Choongbum Lee lar to that of the Erdos-Renyi random graph process. I will University of California, Los Angeles describe these results and pay particular attention to one [email protected] technique involved in the proofs, combining the differential equation method with Flajolet-Sedgewick singularity Po-Shen Loh analysis. I will then propose some open questions about Department of Mathematical Sciences Achlioptas processes related to both random graphs and Carnegie Mellon University random constraint satisfaction problems. [email protected] Will Perkins Benny Sudakov Georgia Tech Department of Mathematics [email protected] UCLA [email protected] Mihyun Kang Technische Universität Graz Institut für Optimierung und Diskrete Mathematik MS15 [email protected] Hunting the k-SAT Threshold Joel Spencer We prove that the threshold for the existence of solutions − New York University in random k-NAESAT is 2k 1 ln 2 − ( ln 2 + 1 )+ε ,where 2 4 k [email protected] −(1−ok(1))k |εk|≤2 . According to deep but non-rigorous arguments from statistical mechanics, the insufficiency of current methods to prove such results is due to a change MS16 in the geometry of the set of solutions called condensation Rational Noncrossing Partitions that occurs shortly before the actual threshold for the existence of solutions. In the talk I will describe a new method For each positive rational number α/β in lowest terms we that is inspired by the sophisticated but non-rigorous for- define a poset of noncrossing partitions NC(α/β)withthe malism called Survey Propagation. property that Konstantinos Panagiotou 1 2α + β |NC(α/β)| = . Max-Planck-Institute for Informatics 2α + β α [email protected] For the rational number α/β = n/(kn − n +1)weobtain Amin Coja-Oghlan the poset of “k-divisible” noncrossing partitions of the cycle University of Warwick (1, 2,...,kn), in which the size of each block of a partition [email protected] is divisible by k.Fork = 1 we obtain the “good old’ NC(n)=NC(n/1). Drew Armstrong 68 DM12 Abstracts

University of Miami namely a shifted staircase without the box in its upper [email protected] right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus Nathan Williams a box. The proofs involve finding the generating function University of Minnesota of the corresponding plane partitions using interpretations [email protected] and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. MS16 Separation Probabilities for Products of Permuta- Greta Panova tions UCLA [email protected] We study the mixing properties of permutations obtained as product of two uniformly random permutations of fixed types. For instance, we give an exact formula for the prob- MS16 ability that elements 1, 2,...,k are in distinct cycles of the The Möbius Function of Generalized Subword Or- random permutation of {1, 2,...,n} obtained as product der of two uniformly random n-cycles. Let P be a poset and let P ∗ be the set of all finite length Olivier Bernardi words over P . Generalized subword order is the partial order on P ∗ obtained by letting u ≤ w if and only if there MIT [email protected] is a subword u of w having the same length as u such that each element of u is less than or equal to the corresponding

Rosena Du element of u in the partial order on P . Classical subword East China Normal University order arises when P is an antichain, while letting P be a [email protected] chain gives an order on compositions. For any finite poset P , we give a simple formula for the Möbius function of P ∗ in terms of the Möbius function of P . This permits us to Alejandro Morales rederive in an easy and uniform manner previous results of MIT Björner, Sagan and Vatter, and Tomie. We are also able to [email protected] determine the homotopy type of all intervals in P ∗ for any finite P of rank at most 1. This is joint work with Peter Richard Stanley McNamara Massachusetts Institute of Technology [email protected] Bruce Sagan Michigan State University Department of Mathematics MS16 [email protected] Bijections for Lattice Paths Between Two Bound- aries MS17 We prove that on the set of lattice paths with north and Closed 2-cell Embeddings Under Partial Duality east (unit) steps that lie between two boundaries B and T , the statistics ‘number of east steps shared with B’and In 2009 Chmutov introduced a partial duality operation, ‘number of east steps shared with T ’ have a symmetric using only a subset of the edges, for graph embeddings. joint distribution. We give an involution that switches The topological consequences of this operation have not these statistics, and a generalization to paths that con- yet been intensively investigated. One important property tain south steps. We show that a similar result relates to of embeddings is being closed 2-cell, when the boundary of the Tutte polynomial of a matroid. Finally, we extend our every face is a cycle in the graph. We present a necessary main theorem to k-tuples of paths, and we provide con- and sufficient condition for a closed 2-cell embedding to nections to flagged semistandard Young tableaux and to remain closed 2-cell after taking a partial dual. k-triangulations. Mark Ellingham Sergi Elizalde Department of Mathematics Department of Mathematics Vanderbilt University Dartmouth [email protected] [email protected] Xiaoya Zha Martin Rubey Middle Tennessee State University - [email protected] [email protected]

MS17 MS16 Biembedding Designs and Minimum Genus Em- Tableaux and Plane Partitions of Truncated Shapes beddings

We consider a new kind of straight and shifted plane par- In this presentation I will discuss results establishing, for titions/Young tableaux — ones whose diagrams are no n ≡ 3 (mod 36), that Kn has a face 2-colourable, blue and longer of partition shape, but rather Young diagrams with green say, nonorientable embedding in which there are (n− boxes erased from their upper right ends. We ﬁnd formu- 1)/2 blue faces each of which have a Hamilton cycle as their las for the number of standard tableaux in certain cases, facial walk and n(n−1)/6 green faces each of which have a DM12 Abstracts 69

triangle as their facial walk. By adapting the construction the minimum span is at most Δ2. It has now been proven results on minimum genus embeddings of Kn +Km are also for all suﬃciently large Δ, but remains open for small Δ, obtained. even for Δ = 3.

Tom McCourt Jerry Griggs University of Bristol University of South Carolina, Columbia [email protected] [email protected]

MS17 MS18 Genus Distribution of Path-like Graphs: Transfer Backbone Coloring: Tree Backbone in Planar Matrix Method Graphs The Transfer Matrix approach is taken for computing the Backbone coloring is a variation of channel assignment genus distribution of graphs whose structure is repeated in problem. For a graph G and a subgraph H (called back- a way of a long path or a cycle. As an example we provide bone)ofG,abackbone k-coloring of G over H is a proper explicit solutions for genus distributions of the Cartesian vertex coloring of G using colors from {1, 2,...,k},such product of various small graphs with a long path or a cycle. that the colors for any two adjacent vertices in H diﬀer by at least two. The backbone chromatic number of G Bojan Mohar over H, BBC(G, H), is the smallest k of a backbone k- Simon Fraser University coloring admitted by G over H. Broesma et al. showed [email protected] that BBC(G, H) ≤ 2χ(G) − 1holdsforeveryG and H. This implies that if G is planar, then BBC(G, H) ≤ 7for all H. They conjecture that if G is planar and T is a tree, MS17 then BBC(G, H) ≤ 6. We prove this conjecture when T is Polychromatic Coloring of Graphs on Surfaces atreeofdiameteratmost4. Let G beagraphonasurface.Apolychromatick-coloring Victor Campos of G is a vertex-k-coloring of G such that each face of G Unversidade Federal do Ceara receives all the k colors. Recently, Horev et al. proved that [email protected] every cubic bipartite plane graph admits a polychromatic proper 4-coloring, where “4” is best possible, and the cu- Frederic Havet bicity and bipartiteness cannot be omitted in the theorem. INRIA Sophia-Antipolis, France In my talk, we discuss a recent progress on polychromatic [email protected] colorings of graphs on surfaces. Atsuhiro Nakamoto Rudini Sampaio, Ana Shirley Ferreira Silva Yokohama National University Universidade Federal do Ceara Japan [email protected], [email protected] [email protected] MS18 MS17 L(2,1,1)-Labeling Is NP-complete for Trees Obstructions for Embeddings of Graphs in Surfaces An L(p1,p2,p3)-labeling of a graph G with span λ is a Only for two surfaces, the sphere and the projective plane, mapping f that assigns each vertex u of G an integer la- ≤ ≤ | − |≥ the complete list of obstructions (forbidden minors) for em- bel 0 f(u) λ such that f(u) f(v) pi whenever bedding graphs into the surface is known. We present our vertices u and v are of distance i for i ∈{1, 2, 3}.We general results about the obstructions of connectivity 2. As show that testing whether a given graph has an L(2, 1, 1)- a consequence, we obtain all obstructions for the torus of labeling with some given span λ is even -complete for the connectivity 2. class of trees. Bojan Mohar, Petr Skoda Petr A. Golovach Simon Fraser University Durham University [email protected], [email protected] [email protected]

Bernard Lidicky MS18 University of Illinois, Urbana, IL The Δ2 Conjecture for Graph Labellings with Sep- [email protected] aration Conditions Daniel Paulusma In 1988 Roberts described a problem posed to him by Lan- Durham University fear concerning the eﬃcient assignment of channels to a [email protected] network of transmitters in the plane. To understand this problem, Griggs and Yeh introduced the theory of integer vertex λ-labellings of a graph G. To prevent interference, MS18 labels for nearby vertices must be separated by speciﬁed Backbone Colorings of Graphs with Large Girths amounts ki depending on the distance i,1≤ i ≤ p.One seeks the minimum span of such a labelling. The p =2 Backbone coloring is a variation of the channel assignment case with k1 =2andk2 = 1 has attracted the most atten- problem. For a graph G and a subgraph H (called back- tion, particularly the tantalizing conjecture that for such bone)ofG,abackbone k-coloring of G over H is a proper “L(2, 1)-labellings”, if G has maximum degree Δ ≥ 2, then vertex coloring of G using colors from {1, 2,...,k},such 70 DM12 Abstracts

that the the colors of adjacent vertices in H differ by at [email protected] least two. The backbone chromatic number of G over H, BBC(G, H), is the smallest k of a backbone k-coloring of David Pike G over H. Broersma, Fomin, Golovach, and Woeginger Memorial University of Newfoundland showed that BBC(G, H) ≤ 2χ(G) − 1 holds in general. [email protected] Miˇstkuf, Skrekovski,ˇ and Tancer proved that for any n there exists a triangle-free graph G with a spanning tree T such that χ(G)=n and BBC(G, T )=2n − 1. We gen- MS19 eralize this result: For any n there exists a graph G with Schröder Quasigroups and Related Combinatorial arbitrarily large girth and a tree T ⊆ G with χ(G)=n Designs and BBC(G, H)=2n − 1. Moreover, we prove that if T is a tree with degrees bounded by a given constant, and G Schröder quasigroups have been studied quite exten- is a subgraph of the square of T , then BBC(G, T )canbe sively over the years. Apart from corresponding to determined in polynomial time. self-orthogonal Latin squares with the Weisner property, Schröder quasigroups are known to be associated with Yuehua Bu other combinatorial configurations such as orthogonal ar- Zhejiang Normal University, China rays with interesting conjugate invariant properties, a class [email protected] of edge-colored designs with block size 4, and triple tournaments. The purpose of this talk is to survey known ex- Daphne D. Liu istence results for various aspects of these associations and California State University, Los Angeles tomentionsomeopenproblems. Department of Mathematics [email protected] Frank Bennett Mount Saint Vincent University [email protected] Xuding Zhu Zhejiang Normal University, China [email protected] MS19 Decompositions of Complete Graphs into Cycles MS18 and Related Problems Distance Three Labellings of Graphs In 1981, Alspach posed the problem of proving that the obvious necessary conditions for a decomposition of a com- Let h be a positive integer. An L(h, 1, 1)-labelling of a (fi- plete graph into cycles of specified lengths are also suf- nite or infinite) graph is an assignment of nonnegative in- ficient. In this talk I will briefly outline a solution to tegers (labels) to its vertices such that adjacent vertices re- Alspach’s problem and discuss other similar problems. ceive labels with difference at least h, and vertices distance two or three apart receive distinct labels. The span of such Darryn Bryant a labelling is the difference between the maximum and min- University of Queensland imum labels used. Motivated by applications in frequency [email protected] assignment, the L(h, 1, 1)-labelling problem seeks for the minimum span over all L(h, 1, 1)-labellings of a graph together with an optimal L(h, 1, 1)-labelling. In this talk I MS19 will review recent results on the L(h, 1, 1)-labelling prob- Friendship 3-hypergraphs lem. Let (X, B)beasetsysteminwhichB is a set of 3- Sanmaing Zhou subsets of X.Then(X, B)isafriendship 3-hypergraph The University of Melbourne if it satisfies the following property: for distinct elements [email protected] u, v, w ∈ X, there exists a unique fourth element x ∈ X such that {u, v, x}, {u, w, x}, {v, w, x}∈B. If a friendship 3-hypergraph contains a element f ∈ X such that MS19 {f, u,v}∈B for all u, v ∈ X \{f}, then it is called a Cycle Extension Property in BIBD Block- universal friend 3-hypergraph and the element f is called Intersection Graphs a universal friend of the hypergraph. In this presentation, we will show that if (X, B) is a friendship 3-hypergraph AcycleC in a graph G is said to be extendible if there with |X| = n,then|B| ≥ ∈(\−∞)(\−∈)/.Inad- exists a cycle C such that V (C) ⊆ V (C )and|V (C )| = dition, we will also show that if n ≡ 2, 4 (mod 6), then |V (C)|+1. A graph G is said to be cycle extendible if every this bound is met if and only if (X, B) is a universal friend non-Hamiltonian cycle of G is cycle extendible. A balanced 3-hypergraph. incomplete block design BIBD(v, k, λ) consists of a set of blocks, each of which is a k-subset of a point set V of cardi- Ben Li nality v, such that each pair of points occurs in precisely λ University of Manitoba of the blocks of the design. The block-intersection graph of [email protected] adesignD is the graph having the block set of D as its vertex set, and in which two vertices are adjacent if and only if their corresponding blocks have non-empty intersection. MS19 ≥ ≥ We show that for integers k 2andλ 1, the block- Broadcast Systems intersection graph of a BIBD(v, k, λ) is cycle extendable. This is joint work with David Pike. A broadcast system of order n is a decomposition of the complete directed graph Dn into n minimum broadcast Atif Abueida trees, one rooted at each node. I will show that broadcast The University of Dayton systems exist for all orders, and discuss some partial results DM12 Abstracts 71

on uniform broadcast systems where all of the broadcast [email protected] trees are isomorphic.

Matthew Walsh MS20 Indiana-Purdue University, Fort Wayne Speciﬁcation and Optimization of Synthetic Multi- [email protected] cell Behaviors

The process by which a single cell develops into a multi- MS20 celled organism is complicated. A combination of internal Graph Theoretical Design Strategies for DNA Self- logic, control of growth and division, and cell-cell commu- assembly nication must coordinate the behaviors of the growing organism so that it differentiates correctly in time and space, Recent advances in DNA self-assembly have resulted in in spite of environmental perturbations, and intrinsic or nanoscale graphs: cubes, octahedrons, truncated octahe- extrinsic noise. Although several formalisms have been de- dra, and even buckyballs, as well as ultra-fine meshes, lid- fined to describe development to produce CGI plants using ded boxes and 2D figures. These constructs serve emergent L-systems for example the relationship between high level applications in biomolecular computing, nanoelectronics, programs and the possible biochemical implementation of biosensors, drug delivery systems, and organic synthesis. those programs in micro-organisms is tenuous. Said differ- One construction method uses k-armed branched junction ently, it is unknown whether the basic mechanisms avail- molecules, called tiles, whose arms are double strands of able to the synthetic biologist are sufficient to implement DNA with one strand extending beyond the other, forming the pattern formation algorithms defined at higher levels. a ’sticky end’ at the end of the arm that can bond to any Here, we introduce a formal specification and programming other sticky end with complementary Watson-Crick bases. language, called gro, that allows the programmer to write A vertex of degree k in the target graph is formed from a distributed algorithms at a variety of different levels of ab- k-armed tile, and joined sticky ends form the edges. An- straction. For example, the programmer may specify that other construction method ’threads’ a single strand of DNA a cell changes from one state to another, of that cell state through the graphical structure and then uses short ’staple’ is defined at the level of a multi-stable gene network with strands for an origami folding of the DNA into the desired all of the low-copy number noise that would accompany geometric realization of the graph. A third method uses it. The programs can be simulated in an environment that circular single strands of DNA to trace the faces of a topo- models the basic geometry of micro-colony growth and di- logical embedding of the graph. We use graph theory to vision, as one might see under a microscope, and also mod- determine optimal design strategies for biologists produc- els the production, degradation, and diffusion of signaling ing these nanostructures, and conclude with a discussion of molecules. In this talk, we illustrate gro with a variety of how the same mathematics, on the macroscale now, may examples; focus on the example of symmetry-breaking in be adapted to space applications. This is joint work with particular; and discuss a notion abstraction for gro pro- Greta Pangborn, with an undergraduate research compo- grams using Wasserstein pseudometrics. The result is a nent. tool that may someday allow the molecular programmer to specify a developmental process at a high level, and refine Joanna Ellis-Monaghan it, step-by-step, into a low level description of how an im- Saint Michaels College plementation would work, and finally to formally compare Department of Mathematics the specification to an actual experimental implementa- [email protected] tion. Eric Klavins MS20 Department of Electrical Engineering Fluid Models for Self-organized Microtubule Ar- University of Washington rays [email protected] In plant cells, microtubules self-organize into ordered arrays from an initially isotropic system as a result of their MS20 interactions. We develop a mean-field model which is used Inferring Physical Parameters and Assembly Path- to derive sufficient conditions for self-organization by con- ways from Indirect Measures of Viral Self-assembly ducting a stability analysis. Considering parameter regions that satisfy conditions for organization, we develop predic- Virus capsids have inspired many theoretical studies of as- tive methodologies for expected number and average length sembly mechanisms and pathways accessible to a simple set of microtubules over time using fluid models for micro- of self-assembling components. It has generally proven im- tubule dynamics and approximations. Results are tested possible, though, to determine where any particular virus using computer simulations. falls in this universe of possibilities. We have combined discrete assembly models with continuous optimization over Ezgi Eren rate parameters to infer assembly kinetics and pathways Industrial and Systems Engineering from bulk measures of assembly progress. Application to Texas A&M University real viruses suggests a diversity of mechanisms in use in [email protected] nature. Natarajan Gautam Russell Schwartz Texas A&M University Biological Sciences and Computer Science Departments [email protected] Carnegie Mellon University [email protected] Ram Dixit Biology Department Lu Xie Washington University in St. Louis Joint Ph.D. Program in Computational Biology 72 DM12 Abstracts

Carnegie Mellon University (bipartite graphs being a basic example). Edmonds (1965) ﬁ[email protected] characterized the perfect matching polytope PM(G) of a graph G =(V,E) as the set of nonnegative x ∈E sat- Gregory Smith isfying two families of constraints: ‘vertex saturation’ and Department of BIological Sciences ‘blossom’. When the blossom constraints are redundant for Carnegie Mellon University determining PM(G), the graph is non-Edmonds.Afterdis- [email protected] cussing some early results on K-E graphs, we’ll show that these graphs are non-Edmonds. Xian Feng Mark Kayll Department of Biological Sciences University of Montana Carnegie Mellon University [email protected] [email protected]

MS21 MS20 Fractional Independence Number and König- Modelling the Co-operative Roles of Genomic RNA Egerváry Graphs during Virus Assembly The fractional independence number α of a graph is the We demonstrate that the genomes of ssRNA viruses im- f maximum value of w(v ), where w(v ) ∈ [0, 1], for each pact on the assembly of the viral protein containers that i i vertex v of the graph, and w(v )+w(v ) ≤ 1foreach package them. For a test virus, bacteriophage MS2, we i i j pair {v ,v } of adjacent vertices in the graph. This is the show that its genomic sequence has evolved specific pat- i j linear programming relaxation of the integer programming terns that play important functional roles in the assembly formulation for the independence number α of the graph. process. We quantify these via kinetic modelling and a α is an upper bound for α and can be computed efficiently. computation of the assembly pathways and, in combina- f We show that α = α for a graph if, and only if, the tion with graph theory, predict unprecedented details con- f graph is König-Egerváry, and discuss connections to the cerning the organisation of the packaged genome. work of Nemhauser, Trotter, Bourjolly, Pulleyblank, and Reidun Twarock the theory of critical independent sets. Departments of Mathematics and Biology Craig E. Larson University of York Virginia Commonwealth University [email protected] [email protected] Eric Dykeman, Nick Grayson Department of Biology MS21 University of York Independent Sets in almost König-Egerváry [email protected], [email protected] Graphs

Peter Stockley Let α(G) denote the cardinality of a maximum indepen- Astbury Centre for Structural Molecular Biology dent set and μ(G) the size of a maximum matching in University of Leeds G =(V,E). If α(G)+μ(G)=|V |−1, then G is an [email protected] almost König-Egerváry graph.Letcore(G)(corona(G)) be the intersection (union, respectively) of all maximum independent sets, and ker (G) be the intersection of all critical MS21 independent sets. In this talk we prove that if G is an al- Egerváry LPs and Fractional Vertex Packing most König-Egerváry unicyclic graph, then: (i) ker (G)= core(G)and(ii) |corona(G)| + |core(G)| =2α (G)+1. We treat relations of Egerváry (i.e. Hungarian) type algorithms to various early results on fractional vertex packings Vadim E. Levit and coverings by Nemhauser and Trotter, Picard and Quer- Ariel University Center of Samaria and Holon Inst. of anne, Bourjolly and Pulleyblank, Lovász, Korach, Dem- Tech. ing, Sterboul, and Kayll, including that, for any vertex Dept. of Computer Science & Mathematics weighted graph G, every maximal set of vertices with 0s [email protected] and 1s, rather than halves, in an optimum weight fractional vertex packing is the same. It can be extended to an Eugen Mandrescu optimum integer vertex packing of G.Itprovidesaneasy Department of Computer Science way to test whether G is K-E since, with all weights one, Holon Institute of Technology some optimum fractional vertex packing has no fractions if eugen [email protected] and only if G is K-E.

Jack Edmonds MS22 Kitchener Phase Transition in Random Integer Programs [email protected] We consider integer programs on random polytopes in Rn with m facets whose normal vectors are chosen indepen- MS21 dently from any spherically symmetric distribution. We K¨onig-Egerv´ary Graphs: Introduction and a show a phase transition phenomenon: a transition from Warm-up Result integer infeasibility to integer feasibility happens within a constant factor increase in the radius of the largest in- K-E graphs are those for which the maximum size of a scribed ball. Our main tools are: a new connection between matching coincides with the minimum size of a covering DM12 Abstracts 73

integer programming and matrix discrepancy, a bound on and relation to cyclic crystals the discrepancy of random Gaussian matrices and Lovett- Meka’s algorithm for finding low discrepancy solutions. The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This Karthekeyan Chandrasekaran generalizes earlier work of Griggs, Killian and Savage on Georgia Institute of Technology the case of prime order, giving an explicit construction for [email protected] any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of Kashiwara’s sl2 MS22 lowering operator in the theory of crystal bases. This is The Game Chromatic Number of Sparse Random joint work with Anne Schilling. Graphs Patricia L. Hersh Given a graph G and an integer k, two players properly North Carolina State University color the vertices of G using k colors. The first player [email protected] wins iff when the game ends all the vertices are colored. The game chromatic number χg(G) is the minimum k for which the first player has a winning strategy. We present MS23 results regarding the asymptotic behavior of this parameter Volume Bounds for Shadow Covering for random graphs with constant average degree and for random regular graphs. Suppose that K and L are compact convex subsets of Eu- clidean space such that, for every direction u, the orthogo- Simi Haber nal projection of L onto the subspace u⊥ contains a trans- Tel Aviv University late of the corresponding projection of the body K.In [email protected] spite of these covering conditions, it is possible for K to have greater volume than L. The question then turns to V (K) Alan Frieze, Mikhail Lavrov bounds: How large can the volume ratio V (L) be? And Carnegie Mellon University what other valid comparisons can be made? While these [email protected], [email protected] questions are posed in the setting of convex geometric analysis, a fundamental role in their resolution is played by Helly’s Theorem. MS22 Connectivity and Giant Components in Stochastic Dan Klain Kronecker Graphs UMass Lowell daniel [email protected] A stochastic Kronecker graph is a random graph model wherein each vertex is represented by a string in an alpha- Christina Chen bet Γ, and the probability that two vertices are adjacent is Newton North High School determined by comparing their corresponding strings. We - use matrix concentration inequalities to develop conditions under which a stochastic Kronecker graph is asymptotically Tanya Khovanova connected, and under which it has a giant component. MIT Mary Radcliffe [email protected] UC San Diego mradcliff[email protected] MS23 Perturbation of Transportation Polytopes MS22 We introduce a perturbation method that can be used to The Power and Weakness of Two Choices: Unbal- reduce the problem of finding the multivariate generating anced Allocations function (MGF) of a non-simple cone to computing the The ”power of two choices” (generating several random MGF of simple cones. We then give a universal perturba- options and choosing among them) has been used success- tion that works for any transportation polytope. We apply fully in load balancing algorithms, queuing theory, and this perturbation to the family of central transportation Achlioptas processes. Here an allocation algorithm for gen- polytopes of order kn × n, and obtain formulas for the erating an unbalanced distribution using this technique is MGFs of the feasible cones of vertices of the polytope and presented and analyzed. This type of algorithm has con- the MGF of the polytope. The formulas we obtain are enu- nections to physics, biology, and economics, and arose merated by combinatorial objects. A special case of the originally from a real-world cost-minimization question. formulas recovers the results on Birkhoff polytopes given Its analysis uses ideas from differential equations, random by the author and De Loera and Yoshida. walks, and some new concepts. Fu Liu Amanda Redlich UC Davis Rutgers [email protected] [email protected] MS23 MS23 Monotone Hurwitz Numbers: Polynomiality and Symmetric chain decomposition for necklace posets Explicit Forms Monotone Hurwitz numbers are a desymmetrized version 74 DM12 Abstracts

of the classical Hurwitz numbers from enumerative alge- Stefan Gruenewald braic geometry. They appeared recently in the context of PICB random matrix theory. In this talk I will focus on the al- Shanghai gebraic and combinatorial structure of monotone Hurwitz [email protected] numbers, which is to a surprising extent similar to the classical case. In particular, I will discuss polynomiality and Toda equations for monotone Hurwitz numebers, as well MS24 as low-genus formulas. This is joint work with I. Goulden Switchings, Extensions, and Reductions in Central and M. Guay-Paquet (Waterloo). Digraphs Jonathan I. Novak A directed graph is called central if, for each ordered pair MIT u, v of (not necessarily distinct) vertices, there is a unique [email protected] vertex z such that the digraph contains the arcs uz, zv. Equivalently, the adjacency matrix A satisfies the matrix equation A2 = J,whereJ is the matrix with a 1 in each MS23 entry. It has been conjectured that every central directed Orientations, Semiorders, Arrangements, and graph can be obtained from a standard example by a se- Parking Functions quence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We It is known that the Pak-Stanley labeling of the Shi hy- disprove these conjectures and present a general extension perplane arrangement provides a bijection between the re- result which, in particular, shows that each counterexam- gions of the arrangement and parking functions. For any ple extends to an infinite family of central digraphs. graph G, we define the G-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all G- Andre Kundgen parking functions. California State University, San Marcos [email protected] David Perkinson Reed College Gregor Leander [email protected] Techincal University of Denmark [email protected] MS24 Distances Between Evolutionary Trees Carsten Thomassen Technical University of Denmark Phylogenetic (leaf-labelled) trees are used in computational [email protected] biology to show the evolutionary relationships within a collection of species. A common way to measure the dissim- ilarity between phylogenetic trees with identical leaf sets MS24 is the minimum number of certain tree rearrangement op- Greedy Trees and the Extremal Distances erations that is necessary to transform one tree into the other. In other words, we use the graph distance of the We show a “universal property’ of the greedy tree with a graph where the vertices are all phylogenetic trees with given degree sequence, namely that the number of pairs given leaf set and two trees are adjacent if they are a single of vertices whose distance is at most k is maximized by rearrangement operation apart. I will summarize recent re- the greedy tree for all k. This rather strong assertion im- sults on the tree bisection and reconnection (tbr) and the mediately implies, and is equivalent to, the minimality of subtree prune and regraft (spr) distances. the greedy trees with respect to graph invariants of the form Wf (T )= {u,v}⊆V (T ) f(d(u, v)) for any nonnega- Stefan Gruenewald tive, nondecreasing function f. With different choices of f, PICB one directly solves the minimization problems of distance- Shanghai based graph invariants including the classical Wiener in- [email protected] dex, the Hyper-Wiener index and the generalized Wiener index.

MS24 Nina Schmuck On t-Path Closed Graphs Technische Universit ”at Graz A t-path closed is a graph with girth t+1 such that for each [email protected] pair of distinct vertices there is a path of length t connect- ing them. In this talk we will give several constructions Stephan Wagner of such graphs. Also we will give relations with cages and Stellenbosch University Moore graphs. [email protected] Jacobus Koolen Korea Advanced Institute of Science and Technology Hua Wang Dept Mathematics,POSTECH, Georgia Southern University, GA [email protected] [email protected]

Hwang Rae Lee MS24 Dept Math Walks and homomorphisms of digraphs POSTECH [email protected] In this talk we try to discuss some connections between DM12 Abstracts 75

symbolic dynamics and digraph homomorphisms and then University of Alberta apply them to some problems on walks in digraphs posed [email protected], [email protected] by Delorme, Tvrdik, Harbane and Heydemann et al.

Yaokun Wu MS25 Shanghai Chiao Tong Univ., Unit Interval Graphs of Mixed Intervals Shanghai, China [email protected] The class of unit interval graphs has a lovely characterization as those interval graphs with no induced claw K1,3. Yaokun Wu The characterization remains the same whether the inter- Shanghai Jiao Tong University vals used in the intersection representation are all open Shanghai, China intervals or all closed intervals. In recent work, Rauten- [email protected] bach and Szwarcfiter characterize the broader class that arises when both open and closed intervals of unit length are permitted. In this talk we consider the same problem MS25 when unit length mixed intervals of the form (x, x +1]and Title Not Available at Time of Publication [x, x + 1) are also allowed. We give a structural characterization of this class of graphs. Abstract not available at time of publication. Ann N. Trenk Andreas Brandstädt Wellesley College FB Informatik, University of Rostock, Germany [email protected] [email protected] Alan Shuchat Mathematics Department MS25 Wellesley College Convexity and Graph Classes [email protected] We survey graph classes that arise from convexity in graphs with emphasis on convex geometries. Randy Shull Department of Computer Science Ortrud R. Oellermann Wellesley College The University of Winnipeg [email protected] [email protected] Lee West Wellesley College MS25 [email protected] Atomic Structure, Hyperbolicity, and Recognition of AT-free Graphs with no Induced 4-cycles MS26 An atom of a graph is a maximal induced subgraph with The Cost of 2-Distinguishing Cartesian Powers no clique cutset. We characterize atoms of C4-free AT-free graphs. This class of graphs is of interest as it coincides A graph is said to be 2-distinguishable if there is a ver- with 1-hyperbolic AT-free graphs while all other AT-free tex labeling with two labels so that only the trivial au- graphs are 2-hyperbolic. Based on the structure of atoms, tomorphism preserves the labels. Define the cost of 2- we describe an efficient algorithm for the recognition of the distinguishing G, denoted ρ(G), to be the minimum size of class. a label class in such a labeling. A determining set is a subset of vertices with trivial pointwise stabilizer; its minimum Derek Corneil size is denoted Det(G). The main result of this talk is that University of Toronto under mild hypotheses, ρ(Gk) ∈{Det(Gk), Det(Gk)+1}. [email protected] Debra L. Boutin Juraj Stacho Hamilton College University of Warwick [email protected] [email protected] MS26 MS25 Groups That Are Transitive on All Partitions of a Polygon Numbers of Circle Graphs Finite Set

Circle graphs and k-polygon graphs are intersection graphs We study the question of which subgroups of Sn are of chords of a circle and chords of a convex k-sided poly- transitive on the set of all ordered partitions of the set gon where each chord has its endpoints on distinct sides, [n]={1,...,n} and all unordered partitions of [n]. This respectively. Determining ψ(G), the minimum number of work can be considered a generalization of well known work sides in a polygon representation of a circle graph G,is in which subgroups of Sn that are transitive on the set of NP-hard, whereas determining whether G is a k-polygon all subsets of [n]ofsizek were determined. As an appli- graph for ﬁxed k can be done in polynomial time. We give cation of our results, we determine which Johnson graphs bounds on ψ(G)whenG is an arbitrary circle graph and are Cayley graphs. identify graph classes for which ψ(G) can be computed in polynomial time. Ted Dobson Mississippi State University Lorna Stewart, Richard Valenzano [email protected] 76 DM12 Abstracts

Aleksander Malnic MS27 University of Ljubljana Fast Algorithms for Phylogenetic Reconstruction [email protected] We present two recent very fast algorithms to infer phylogenetic trees in o(n2) time when the input consists of MS26 data from n species. Our first algorithm uses a search Decomposing Hypergraphs on Finite Fields tree much like a randomly balanced binary search tree to place each new species into its correct place. Our second We examine edge decompositions of complete uniform hy- algorithm uses nearest-neighbour search to speedily find pergraphs whose parts are permuted transitively by some roughly where each new taxon goes, and then uses more permutation of the vertex set. We present an algebraic robust methods to exactly place the new species correctly. method for constructing such a hypergraph decomposition on a finite field which is related to the Paley graph con- Dan Brown struction and is derived from a partition of the cosets of Cheriton School of Computer Science the multiplicative group of units of the field. We discuss University of Waterloo the symmetry and other properties of the hypergraphs we [email protected] obtain.

Shonda Gosselin MS27 University of Winnipeg Gene Family Evolution by Duplication and Loss - Department of Mathematics and Statistics Reconciliation and Species Tree Inference [email protected] Almost all genomes which have been studied contain genes that are present in two or more copies. They are usu- MS26 ally identified through sequence similarity, and grouped Infinite Motion and Distinguishing Number 2 into a single “Gene Family’. From a functional point of view, such grouping is not sufficient to infer a common Given a group A acting faithfully on a set X, define function for genes. Indeed, it is important to distinguish the motion (also known as minimal degree) of A,tobe between “orthologs’ which are copies in different species |{ a }| m(A)=mina=1 x : x = x . A general principle for related through speciation, and thus likely to have simi- finite A is that if m(A) >> |A|, then the action has dis- lar functions, and “paralogs’, which are copies that have tinguishing number 2, that is, there is a subset Y whose evolved by duplication, and more likely to have acquired set-wise stabilizer is trivial. For infinite A and countable new functions. Understanding the evolution of gene fami- X,wemaketheInfinite Motion Conjecture:ifm(A)isin- lies through speciation, duplication, and loss is thus a fun- finite, then the action has distinguishing number 2. We damental question in functional genomics, and also in evo- consider various contexts for the conjecture in the case of lutionary biology and phylogenomics. Reconciliation is the locally finite graphs. commonly used method for inferring the evolutionary scenario for a gene family. It consists in “embedding’ inferred Thomas Tucker gene trees into a a species tree. When a species tree is not Colgate University known, a natural algorithmic problem is to infer a species [email protected] tree from a set of gene trees, such that the corresponding reconciliation minimizes the number of duplications and/or Wilfriied Imrich losses. The main drawback of reconciliation is that the in- University of Leoben ferred evolutionary scenario is strongly dependent on the [email protected] considered gene trees, as few misplaced leaves may lead to a completely different history. In this talk, we will clarify Simon Smith, Mark E. Watkins several theoretical questions and present various algorith- Syracuse University mic issues related to reconciliation and species tree infer- [email protected], [email protected] ence. In particular, we will present a strategy for “correct- ing’ or preprocessing a gene tree prior to reconciliation or species tree inference, and give various complexity results MS26 and algorithmic solutions, in both cases of a known and an Finite Subgraphs of d-Distinguishable, Locally Fi- unknown species tree. nite Graphs Nadia S. El-Mabrouk It is known (M.E. Watkins and X. Zhou, 2007) that every Département d’informatique et de recherche infinite, locally finite tree T with finite distinguishing num- opérationnelle ber d(T )=d0 contains a finite subtree with distinguishing UniversitédeMontréal number d0. It is not difficult to prove more generally that [email protected] if every finite subgraph Φ of an infinite, locally finite graph Γsatisfiesd(Φ) ≤ d0,thend(Γ) ≤ d0 + 1. We investigate conditions subject to which this bound may be sharp. MS27 Tree Compatibility, Character Compatibility, and Mark E. Watkins, Simon Smith Graph Triangulation Syracuse University [email protected], [email protected] We consider two classic NP-complete problems in phylogenetic tree construction, tree compatibility and character compatibility. In each case, the “yes’ instances are precisely those for which a specific kind of triangulation exists for a certain graph. The relationship between triangulations and character compatibility has been known for over DM12 Abstracts 77

three decades (Buneman, 1974); an analogous relationship MS28 with tree compatibility was identified more recently (Vakati Quotients of the Boolean Lattice by Wreath Prod- and Fernández-Baca, 2011). We use these characteriza- ucts tions to gain insight into the fundamental differences in tractability between the two problems. Rod Canfield has asked if all quotients of a Boolean lattice Bn induced by subgroups of its automorphism group, David F. Fernandez-Baca essentially Sn, are symmetric chain orders. K. Jordan ver- Computer Science ified this in the special case of a subgroup generated by Iowa State University an n-cycle, answering a question raised earlier by Griggs, [email protected] Killian and Savage. In earlier work we found an especially straightforward proof of Jordan’s result based on the Sudheer Vakati Greene-Kleitman symmetric chain decomposition. Hersh Department of Computer Science and Schilling, and Dhand have presented other proofs of Iowa State University the same result, and the latter has the substantial general- n [email protected] ization that P /Zn is an SCO whenever P is an SCO. Here we continue in the same vein as our earlier work, describing circumstances under which subgroups G of the symmetric MS27 group which are wreath products are guaranteed to induce Aspects of Fractionation; A Fundamental Evolu- quotients that are symmetric chain orders. This is done by tionary Process carefully winnowing Greene-Kleitman SCDs. In the evolutionary history of virtually all eukaryotic Dwight Duffus species is at least one whole genome duplication (WGD). Emory University Each WGD event is followed by a a period of fractiona- Mathematics and Computer Science Department tion whereby only one of the copies of almost all gene pairs [email protected] is shed, sometimes on one of the two homeologous chromosomes, and sometimes on the other. This effectively Kyle Thayer scrambles the gene order on chromosomes and introduces Emory University an extreme level of noise into gene order comparisons be- [email protected] tween genomes descending from the WGD and those that diverged before this event, and also between genomes in sister lineages descending from the event radiating soon MS28 after the WGD. In these cases, fractionation is responsible Poset-free Families of Sets for more gene order disruption than rearrangements such as inversion and translocation. We discuss three aspects of Given a finite poset P , we consider the largest size La(n, P ) fractionation: of a family F of subsets of [n]:={1,...,n} that contains no (weak) subposet P .ForfixedP it can be very difficult • the time course of fractionation on scales of tens to to determine La(n, P ), even asymptotically. We continue n hundreds of millions of years, →∞ to believe that π(P ) := limn La(n, P )/ n/2 exists for general posets P , and, moreover, it is an integer that we • a recurrence for the run lengths of deleted and un- can predict. The existence of the limit remains open, for deleted genes on a chromosome, and instance, for the four-element diamond poset and the six- • a model for gene family size, where a genome origi- element crown. However, we are developing methods that nates from more than one successive WGD or other allow us to solve the asymptotic problem, and sometimes polyploidization event. even the exact problem for all n,forincreasinglymany posets P . These methods involve studying the Lubell function of a family F , which is the average number of times a David Sankoff, Chunfang Zheng random full chain meets F . Department of Mathematics and Statistics University of Ottawa Jerry Griggs sankoff@uottawa.ca, [email protected] University of South Carolina, Columbia [email protected]

MS27 Wei-Tian Li Reconstruction of Certain Phylogenetic Networks Department of Mathematics from the Tree-Additive Distances Between Their Univeristy of South Carolina Leaves [email protected]

Suppose N is a rooted directed graph and each leaf corre- Linyuan Lu sponds to an extant biological species. Suppose each arc University of South Carolina of N has a nonnegative weight and inheritance of a char- [email protected] acter at a hybrid vertex occurs with a certain probability from each parent. The tree-additive distance between two leaves is the expected value of their distances in the dis- MS28 played trees. Suﬃcient conditions are investigated so that, Chopping Celery and the Lattice of Integer Parti- given the tree-additive distances between all leaves, it is tions possible to reconstruct the network. The chop vector of a set S of celery sticks of positive integer Stephen J. Willson lengths is the inﬁnite vector vS =(v1,v2,v3,...), where Iowa State University each vw is the minimum number of cuts needed to chop S [email protected] 78 DM12 Abstracts

into unit pieces, using a knife that can cut up to w sticks at Dalhousie University, Canada a time. In this talk we see a connection (found jointly with Dept of Maths, Stats, & CS Thao Do) between the set of chop vectors and the lattice [email protected] of integer partitions.

Bill Sands,ThaoDo MS29 University of Calgary Graph Colouring and the Topological Penrose [email protected], [email protected] Polynomial The Penrose polynomial of a plane graph first appeared MS28 implicitly in Roger Penroses 1971 work on diagrammatic Diamond-Free Collections of Subspaces tensors, but was discovered to have a number of remarkable graph theoretical properties, particularly with respect to Let V be a finite-dimensional vector space over a finite graph colouring. We now extend the Penrose polynomial, field, and let A, B, C,andD be subspaces of V .Wesay originally defined only for plane graphs, to graphs embed- that {A, B, C, D} is a diamond if A is a subspace of both ded in arbitrary surfaces. This leads to new identities and B and C which in turn are subspaces of D. Inspired by the relations for the Penrose polynomial which cannot be re- recent work on the Boolean Lattices, we ask: What is the alized within the class of plane graphs. In particular, by size of the largest collection of diamond-free subspaces of exploiting connections with the transition polynomial and V ? The corresponding question in the Boolean Lattices has the ribbon group action, we find a deletion-contraction- not been completely settled and neither is the question in type relation for the Penrose polynomial and its relation the Linear Lattices. We report on partial progress. Joint to the ribbon graph polynomial of Bollobas and Riordan. work with Ghassan Sarkis and the undergraduate math We relate the Penrose polynomial of an orientable checker- research circle at Pomona College. board colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial Shahriar Shahriari interpretations of the Penrose polynomial. We also show Pomona College that the Penrose polynomial of a plane graph G can be ex- [email protected] pressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the MS28 Four Colour Theorem. Linear Discrepancy of Partially Ordered Sets Joanna Ellis-Monaghan Saint Michaels College The linear discrepancy of a P , denoted ld(P ), is the min- Department of Mathematics imum over all linear extensions of the maximum differ- [email protected] ence between the positions of two incomparable elements. Tanenbaum, Trenk, and Fishburn conjectured that always ld(P ) ≤(3r − 1)/2,wherer is the maximum number of Iain Moffatt elements incomparable to any single element. We disprove University of South Alabama this conjecture by showing that appropriately randomized imoff[email protected] bipartite posets have linear discrepancy asymptotic to the − trivial upper bound, 2r 1. On the other hand, we show MS29 that the conjecture does hold for posets having width 2. Matroids, Greedoids and the Tutte Polynomial Jeong Ok Choi GIST College We present some conjectures and counterexamples related Gwangju Institute of Science and Technology to the greedoid generalization of the Tutte polynomial of [email protected] a matroid. In particular, we give simple examples to show the Tutte polynomial does not distinguish greedoids. We also study the affine relations satisfied by the coefficients Kevin Milans of the Tutte polynomial in this more general setting. University of South Carolina [email protected] Gary Gordon Lafayette College Douglas B. West Department of Mathematics University of Illinois, Urbana [email protected] Department of Mathematics [email protected] MS29 Some Recent Results on Chromatic and Tutte MS29 Polynomials for Families of Graphs Roots of Combinatorial Polynomials We discuss some recent results on chromatic and Tutte Polynomials arise in a variety of ways in combinatorics, polynomials (equiv. to Potts model partition functions from applications like graph colourings and network relia- in statistical physics) and some generalizations for various bility, to theoretical investigations of sequences related to families of graphs. These include weighted-set chromatic independent sets of graphs and open sets of finite topolo- and Tutte polynomials, lower bounds on Potts ground- gies. We discuss some recent results about the roots of state entropy, and chromatic polynomials for one- and such polynomials, both in terms of location and nature, as multi-parameter families of planar triangulation graphs. well as some connections to fractals. References include S.-C.Chang and R. Shrock, J. Phys. A (JPA) 42, 385004 (2009); J. Stat. Phys. (JSP) 138, 496 Jason Brown (2010); R. Shrock and Y. Xu, JSP 139, 27 (2010); JSP DM12 Abstracts 79

141, 909 (2010); Phys. Rev. E 81, 031134 (2010); JPA {1,...,k} such that f(v) = f(w) for all vw ∈ E.The 45, 055212 (2012); arXiv:1201.4200; R. Shrock, JPA 44, Grundy number of a graph is the largest integer k such 145002 (2011). that there is a k-coloring f of G such that if 1 ≤ i

A k-coloring of a graph G =(V,E) is a function f : V → Jacob Fox 80 DM12 Abstracts

··· k MIT h =(a1, ,a( )) ,andNt(H) the matrix whose columns [email protected] t are the images of h under the symmetric group Sk.We determine a diagonal form (Smith normal form) of Nt(H) Benny Sudakov for a very general class of H. Motivated by Y. Caro’s Department of Mathematics results, we show that if H is simple, the zero-sum Ramsey UCLA number (mod 2) of H is almost surely k as k →∞. [email protected] Wing Hong Tony Wong, Richard Wilson California Institute of Technology MS31 [email protected], [email protected] On Restricted Ramsey Numbers

A classical Ramsey theorem states that in any 2-coloring MS32 of the edges of a sufficiently large complete graph, one will Non-extendible Latin Cuboids always find a monochromatic complete subgraph. In 1970, Folkman extended this result showing that for any graph There is a celebrated result due to Marshall Hall that every G there exists a graph H with the same clique number latin rectangle is completable to a latin square. However, as G such that any 2-coloring of the edges of H yields a the equivalent statement in higher dimensions is not true. monochromatic copy of G. In this talk, we present some old A 3-dimensional array has lines of cells in three directions, and recent developments concerning Folkman-type results called rows, columns and stacks.Ann×n×k latin cuboid is for vertex colorings. a 3-dimensional array containing n different symbols posi- tioned so that every symbol occurs exactly once in each row Andrzej Dudek and column and at most once in each stack. An n × n × n Department of Mathematics latincuboidisalatin cube of order n.Ann × n × k latin Western Michigan University cuboid is extendible if it is contained in some n×n×(k +1) [email protected] latin cuboid and it is completable if it is contained in some latin cube of order n. In this talk I will present some results on non-extentible and non-completable n×n×k latin MS31 cuboids where k is approximately half of n. Ramsey Problems on Non-Complete Graphs Darryn Bryant A typical question in Ramsey Theory is, for given graphs University of Queensland G and H, to determine the Ramsey number R(G, H): the [email protected] smallest integer N such that, in every colouring of the edges of the complete graph KN on N vertices with red and blue, Nicholas Cavenagh we either find a red copy of G or a blue copy of H.One University of Waikato naturally wonders whether the complete graph KN in the [email protected] above definition can be replaced by some other graph F on N vertices. It would be naive to think we can take any Barbara Maenhaut graph on N vertices, therefore, we put some restrictions University of Queensland on such a graph F . In this we shall discuss the following [email protected] problem: For given graphs G and H, is there a constant 0 < c<1 such that if a graph F on N = R(G, H)verticeshas minimum degree at least (1 − c)N, then in every colouring Kyle Pula of the edges of F with red and blue, we either find a red University of Denver copy of G or a blue copy of H. It is not even clear why [email protected] such c should even exist. We determine the maximum c when G and H are cycles or paths, and show that c exists Ian Wanless for a rather large class of pairs (G, H). Monash University [email protected] Jozef Skokan London School of Economics [email protected] MS32 Block Colourings of Designs Revisited MS31 Block colourings of designs, both classical and its variants, Title Not Available at Time of Publication and also the so-called specialized block colourings, offer an extensive variety of interesting problems. We will discuss Abstract not available at time of publication. some old and new concepts, results and open problems in colourings of block designs, mostly for Steiner triple sys- TBA TBA tems and Steiner systems S(2, 4,v). IBM Corporation TBA Alexander Rosa Department of Mathematics McMaster University MS31 [email protected] Diagonal Forms for Incidence Matrices and Zero- Sum (mod 2) Ramsey Theory

Let H be a t-uniform hypergraph on k vertices, with ai ≥ 0 ≤ ≤ k denotes the multiplicity of the i-th edge, 1 i ( t ). Let DM12 Abstracts 81

MS32 morphisms (SNPs) from a large number of measured can- Skolem and Rosa Rectangles and Related Designs didates that are associated with a speciﬁc observable trait. Identiﬁcation of associated genetic variants is particularly A Skolem sequence of order n is a sequence S = hard due to very large genotype sizes in comparison to (s1,s2,...,s2n)of2n integers such that (1) for all k = case-control group sizes. We formally frame this problem 1,...,n there are exactly two terms si, sj such that as Bayesian variable selection in a logistic regression model si = sj = k,and(2)ifsi = sj = k and i

Vaclav Linek University of Winnipeg MS33 [email protected] Encoding Memories in Neuronal Networks Various mechanisms for encoding memories in networks of MS32 neurons have been studied in the past few decades, most Cyclic Block Designs with Block Size 3 from notably since Hopﬁeld’s famous papers in the early 1980s. Skolem-Type Sequences Given a prescribed list of memory patterns, however, it is still an unsolved problem how to arrange connections be-

A Skolem-type sequence is a sequence (s1,...,st)ofposi- tween neurons in a recurrent network such that these - and tive integers i ∈ D such that for each i ∈ D there is exactly only these - memories are encoded via stable ﬁxed points one j ∈{1,...,t− i} such that sj = sj+i = i. Positions in of the dynamics. I will discuss some recent progress on this the sequence not occupied by integers i ∈ D contain null problem, which exploits connections to discrete geometry. elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ≡ 1, 3(mod 6),v =9. Carina Curto Using the same technique in 1981, Colbourn and Colbourn Department of Mathematics extended the solution to all admissible λ>1. It is known University of Nebraska-Lincoln that Skolem-type sequences may be used to construct cyclic [email protected] Steiner triple systems as well as cyclic triple systems with λ = 2. The main result of this talk is an extension of former MS33 results onto cyclic triple systems with λ>2. In addition we introduce a new kind of Skolem-type sequence. This is Assembling Helices in RNA Junctions by Using 3D joint work with my supervisor, Dr. Nabil Shalaby. Graphs Daniela Silvesan, Nabil Shalaby The structure of RNA junctions is important to under- Memorial University of Newfoundland stand its function. A novel 3D tree graph approach is in- [email protected], [email protected] troduced to predict the geometrical helical arrangements of junctions in space. Tree graphs are implemented to represent junctions, and biological assumptions combined MS33 with graph combinatorics are used to study the geometri- Combinatorics of Splice Graphcs cal/topological conﬁgurations of native-like junctions. For a given target junction, statistical analysis is then used to Splice graphs compactly represent the possible isoforms of predict the tree graph that best resemble its RNA native a gene which undergoes multple splicing. I will talk about structure. the combinatorics of splice graphs and how this relates to the problem of inferring them from RNA-Seq data. Christian Laing Departments of Biology, Mathematics and Computer Dustin Cartwright Science Yale University Wilkes University [email protected] [email protected]

MS33 MS33 Bayesian Centroid Estimation for Genome-Wide Asymptotic Distribution of Substructures in a Association Studies Stochastic Context-free Grammar Model of RNA Folding Genome-Wide Association Studies (GWAS) attempt to identify a (possibly small) subset of single nucleotide poly- Some recent methods for predicting RNA secondary structures are based on stochastic context-free grammars 82 DM12 Abstracts

(SCFGs). We analyze one of the most notable SCFGs MS34 which is used in the prediction program Pfold. In par- Title Not Available at Time of Publication ticular, we show that the distribution of base pairs, helices and various types of loops in RNA secondary structures ABstract not available at time of publication. generated by this SCFG is asymptotically Gaussian, for a generic choice of the grammar probabilities. Our proofs Prasad Raghavendra are based on singularity analysis of probability generating Georgia Tech functions. [email protected] Svetlana Poznanovikj Schoolg of Mathematics MS34 Georgia Institute of Technology The Entropy Rounding Method in Approximation [email protected] Algorithms

Christine E. Heitsch Let A be a matrix and x be a fractional vector, say an School of Mathematics LP solution to some discrete optimization problem. We give a new randomized rounding procedure for obtaining Georgia Tech ≈ [email protected] an integral vector y such that Ay Ay, provided that A has bounded Δ-approximate entropy. This property means that for uniformly chosen random signs χ(j) ∈{±1} on MS34 any subset of the columns, the outcome Aχ can be approximately described using at most m bits in expectation Improving Christofides’ Algorithm for the Metric 5 s-t Path Traveling Salesman Problem (with m being the number of selected columns). To achieve this result, we modify well-known techniques from the field The Metric s-t Path Traveling Salesman Problem (TSP) of discrepancy theory, especially we rely on Beck’s entropy asks for a minimum-cost Hamiltonian path with specified method. We demonstrate our procedure by providing an 2 start and end points in a finite metric space. Christofides’ OPT + O(log OPT) approximation for Bin Packing With 3/2-approximation algorithm for the Metric TSP was Rejection. shown by Hoogeveen in 1993 to be a 5/3-approximation for the Metric s-t Path TSP. We present a polynomial-time φ- Thomas Rothvoss approximation algorithm for the s-t Path TSP, where φ is Massachusetts Institute of Technology the golden ratio, the first improvement upon Christofides’ Department of Mathematics algorithm in general metrics. [email protected]

Hyung-Chan An, Robert Kleinberg, David Shmoys Cornell University MS34 [email protected], [email protected], Semidefinite Programming Hierarchies and the [email protected] Unique Games Conjecture We survey recent results about semidefinite programming MS34 (SDP) hierarchies in the context of the Unique Games Con- Electrical Flows, Laplacian Systems, and Faster jecture. This conjecture has emerged as a unifying ap- Approximation of Maximum Flow in Undirected proach towards settling many central open problems in the Graphs theory of approximation algorithms. It posits the hardness of a certain constraint satisfaction problem. We show both In this talk, I’ll describe a new technique for approximat- upper and lower bounds on the complexity of this problem ing the maximum flow in capacitated, undirected graphs. within restricted computational models defined by SDP hi- I’ll then use this technique to develop the asymptotically erarchies. fastest-known algorithm for solving this problem. Our approach is based on treating the graph as a network of re- David Steurer sistors and solving a sequence of electrical flow problems Microsoft Research New England, USA with varying resistances on the edges. Each of these may [email protected] be reduced to the solution of a system of linear equations in a Laplacian matrix, which can be solved in nearly-linear MS35 time. Some Problems and Results on the On-line Chain Paul Christiano, Jonathan Kelner Partitioning of Posets MIT [email protected], [email protected] On-line chain partitioning of posets has been a widely researched topic since Kierstead’s seminal paper on the subject in 1981. In this talk we describe a generalization of Aleksander Madry Kierstead’s auxiliary order, and we prove many powerful Microsoft Research New England properties, in hopes of providing a linear algorithm for the [email protected] chain partitioning problem.

Daniel Spielman Csaba Biro Yale University University of Louisville [email protected] [email protected]

Shang-Hua Teng Linyuan Lu USC University of South Carolina [email protected] [email protected] DM12 Abstracts 83

MS35 invented by Pemmaraju, Raman, Varadarajan et al. News About Semiantichains and Unichain Cover- ings Peng Li Shanghai Jiao Tong University Saks and West ask if for any product of posets the size [email protected] of the minimum unichain covering equals of the size of a maximum semiantichain, which would be a nice generaliza- Yaokun Wu tion of Greene–Kleitman Theorem. We found several new Shanghai Chiao Tong Univ., classes of posets for which the conjecture is true. However, Shanghai, China we found an example showing that in general this min-max [email protected] relation is false. This ﬁnally disproofs 30 year old conjecture. Yaokun Wu Shanghai Jiao Tong University Bartlomiej E. Bosek Shanghai, China Jagellion University, Poland [email protected] [email protected]

Stefan Felsner MS35 Technical University of Berlin Semiorders and Ascent Sequences Institute for Mathematics [email protected] We analyze the recent bijection of Bousquet-M´elou, Claes- son, Dukes and Kitaev between interval orders and ascent Kolja Knauer sequences restricted the the case of semiorders, resulting in TU-Berlin a new structural result on the canonical representation of [email protected] semiorders. Stephen J. Young, Jeﬀ Remmel Grzegorz Matecki University of California, San Diego Jagiellonian University [email protected], [email protected] [email protected]

MS36 MS35 Algorithms and Existential Polytime? An Improved Bound for First-Fit on Posets With- out Two Long Incomparable Chains An EP theorem asserts the existence of something for which any instance is ‘easy’ to recognize. Often an EP It is known that the First-Fit algorithm for partitioning a theorem can be proved by a polytime algorithm for finding poset P into chains uses relatively few chains when P does an instance of what is asserted to exist. We discuss favorite not have two incomparable chains each of size k.Inpartic- open problems about the following two EP theorems: (1) If ular, if P has width w then Bosek, Krawczyk, and Szczypka there is a Hamiltonian cycle H containing edge e in graph (SIAM J. Discrete Math., 23(4):1992–1999, 2010) proved 2 G such that G minus the edges of H is connected, then an upper bound of ckw on the number of chains used by there is another one H’, different from H. (2) If a surface First-Fit for some constant c, while Joret and Milans (Or- 2 triangulation S contains a set T of triangles which parti- der, 28(3):455–464, 2011) gave one of ck w.Weprovean tion the vertices of S, then S contains another such set T’, upper bound of the form ckw, which is best possible up to different from T. the value of c. Jack Edmonds Vida Dujmovic Kitchener Carleton University [email protected] [email protected]

Gwenael Joret MS36 Universite Libre de Bruxelles Some Graph Theory Problems I Would Like to See [email protected] Solved

David R. Wood I will describe a number of open graph theory problems University of Melbourne which arise primarily out of questions in Ramsey theory. [email protected] Ron Graham UCSD MS35 [email protected] First-Fit Coloring of Interval Graphs Let FF(k) be the maximum number of colors to be used MS36 when we adopt the First-Fit algorithm to color an interval Euler’s Rigidity Conjecture graph with maximum clique size k. We will discuss the estimate of FF(k) in this talk. Especially, we show that In 1766, Euler conjectured that every polyhedron was rigid. 4k − 5 ≤ FF(k) ≤ 8k − 11 when k>1, improving a Connelly ﬁnally disproved the conjecture in 1977. The in- little bit the current known best record. The upper bound vestigations of the conjecture and rigidity in general in- is obtained basically by the column construction method cluded many other interesting conjectures; perhaps the 84 DM12 Abstracts

most interesting has yet to be formulated. MS37 Seepage in Directed Acyclic Graphs Jack Graver Mathematics Department Seepage is a vertex-pursuit game played on directed acyclic Syracuse University graphs (or dags), introduced by Nowakowski et al. We [email protected] consider applications of Seepage to the modelling of hierarchical social networks. A stochastic model is introduced which generates dags with a prescribed degree sequence. MS36 We play a variant of Seepage on the model, and contrast Ringel and Kotzig after Fifty Years results on the green number in the case of a constant versus a power law degree sequence. Next year will mark the 50th anniversary of the first truly large international graph theory conference which Anthony Bonato took place in Smolenice, Czechoslovakia. Many outstand- Ryerson University ing problems were posed there, including the celebrated [email protected] Ringel’s problem on decomposing the complete graph into isomorphic trees, and Kotzig’s problem on perfect one- factorizations of the complete graph. We will discuss some MS37 developments related to these problems which remain still Ambush Cops and Robbers wide open today. A variation of the game with two robbers is introduced. Alexander Rosa The cops win by moving onto the same vertex as one of the Department of Mathematics robbers after a finite number of moves. The robbers win McMaster University by avoiding capture indefinitely or by both moving onto [email protected] the same vertex as the cop. (Otherwise, the robbers are on distinct vertices.) We present a recognition theorem (and cop strategy) for graphs on which one cop can guarantee a MS37 win. (Joint work with M. Creighton) Cops and Robbers on Geometric Graphs Nancy E. Clarke In the game of cops and robbers, one robber is pursued Acadia University by a set of cops on a graph G. In each round, these [email protected] agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We MS37 study the cop number of geometric graphs. For points Cops and Scared Robber 2 + x1,x2,...,xn ∈ R ,andr ∈ R , the geometric graphs G(x1,...,xn; r) is the graph on these n points, with xi,xj In a traditional game of Cops and Robber, the cops and adjacent when xi − xj ≤r.Weprovethatc(G) ≤ 9 robber move alternately on a graph, where each player can for any connected geometric graph G ∈ R2.Weimprove elect to move to an adjacent vertex or remain at his current on this bound for random geometric graphs that are suf- position. In this case, the Scared Robber has the restriction ficiently dense. Let G(n, r) denote the probability space that he can not move closer to any cop. The motivation of geometric graphs with n vertices chosen uniformly and for adding this restriction is to gain insight into the rob- independently from [0, 1]2.ForG ∈ G(n, r), we show that ber’s strategy in the traditional game, and determine when with high probability (whp), if nr4 log n then c(G) ≤ 2 moving toward a cop would actually be to the robber’s ad- and if nr4 log n then c(G) = 1. Finally, we provide vantage. a lower bound near the connectivity regime of G(n, r): if nr2 log2 n then c(G) > 1 whp. Shannon L. Fitzpatrick University of Prince Edward Island Andrew J. Beveridge sfi[email protected] Department of Mathematics, Statistics and Computer Science Stephen Finbow Macalester College Math, Stats, and CS [email protected] St. Francis Xaxier University sfi[email protected] Andrzej Dudek Department of Mathematics Western Michigan University MS37 [email protected] Chasing Cops on Random Graphs

Alan Frieze In the game of cops and robber, the cops try to capture a Department of Mathematical Sciences robber moving on the vertices of the graph. The minimum Carnegie Mellon University number of cops required to win on a given graph G is called [email protected] the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected Tobias Mueller | | Centrum voor Wiskunde en Informatica graph G is at most C V (G) . We show that Meyniel’s [email protected] conjecture holds asymptotically almost surely for a random d-regular graph, d ≥ 3, as well as in the standard random graph model G(n, p). (Joint work with Nick Wormald.) We also investigate the game played on a (percolated) random DM12 Abstracts 85

geometric graph. (Joint work with Noga Alon.) t(t − 1) contains a bidirected clique of order t as an immersion. Joint work with Matt DeVos, Bojan Mohar and Pawel Pralat Diego Scheide. Ryerson University [email protected] Jessica McDonald Simon Fraser University Nick Wormald jessica [email protected] Department of Combinatorics and Optimization University of Waterloo, Waterloo [email protected] MS38 Hamiltonicity of Graphs on Surfaces Noga Alon In 1956, Tutte proved that every 4-connected plane graph Schools of Mathematics and Computer Science has a hamiltonian cycle. Beginning with the result by Tel Aviv University Tutte, many results on the hamiltonicity of graphs on sur- [email protected] faces were shown. However, the following conjecture by Grünbaum and Nash-Williams has been unsolved for more MS38 than 40 years; every 4-connected graph on the torus has a hamiltonian cycle. In this talk, we will mention recent An Analogue of the Harer-Zagier Formula for Gen- results around the conjecture by Grünbaum and Nash- eral Surfaces Williams. We consider the different ways of gluing the edges of a 2n- Ken-ichi Kawarabayashi, Kenta Ozeki gon in pairs so as to create a surface without boundary. Up National Institute of Informatics, Japan to homeomorphism, the surface obtained is characterized k [email protected], [email protected] by its genus and by its orientability. The Harer-Zagier formula characterizes the generating function of the genus of the orientable surfaces obtained from the gluings of the MS39 2n-gon. In this work we give an analogue of the Harer- Distinguishing Numbers and Regular Orbits Zagier for general (locally orientable) surfaces. The proof is bijective. For a group G acting on a set X,thedistinguishing number is the least number of parts in a partition of X which is Olivier Bernardi preserved only by the identity element of G. Various results MIT about distinguishing numbers can be expressed in terms of [email protected] orbits of G on the power set of X. We survey some known results from the group theory literature, and consider some MS38 problems of a combinatorial nature. Rooted K2,4 Minors in Planar Graphs Robert Bailey University of Regina Given 3 terminal vertices in a graph G, Robertson and [email protected] Seymour characterized whether we could find a rooted K2,3 minor in G using the terminals for the larger side of the bipartition. We provide a characterization for when G has MS39 4 terminals and is planar. On the Orders of Symmetric Graphs

Lino Demasi In this talk I will describe some recent developments con- Simon Fraser University cerning finite symmetric (arc-transitive) graphs. The first [email protected] is the computer-assisted determination of all cubic (3- valent) symmetric graphs on up to 10000 vertices, which MS38 produced some new large graphs with given degree and diameter. The second is a new approach for classifying all Coloring Graphs on Surfaces connected symmetric cubic graphs of given orders, show- We will review new progress on coloring and list-coloring ing for example that for any fixed k and any integer s>1, embedded graphs. This talk is based on works with Ken- there are only finitely many s-arc-transitive cubic graphs ichi Kawarabayashi, Bernard Lidicky, Bojan Mohar, Luke of order kp where p is prime. In turn, these results have led Postle and Robin Thomas. to some further new discoveries about the orders of symmetric graphs of higher valency (in some recent joint work Zdenek Dvorak with Caiheng Li and Primoz Potocnik). Department of Applied Mathematics Charles University, Prague Marston Conder [email protected]ff.cuni.cz University of Auckland, New Zealand [email protected]

MS38 MS39 Clique Immersion in Digraphs A Catalog of Self-Dual Plane Graphs with Max De- Immersion is a containment relation between graphs (or gree 4 digraphs) which is deﬁned similarly to the more familiar notion of minors, but is incomparable to it. This talk In this paper, we produce a catalog of self-dual plane will begin with a gentle introduction, and eventually show graphs with maximum degree 4 (self-dual spherical grids). that every Eulerian digraph with minimum degree at least Based on their automorphism groups, self-dual spherical 86 DM12 Abstracts

grids fall into a finite number of parameterized, infinite [email protected] families. The individual self-dual spherical grids in a family have the same basic shape, differing only in size. MS40 Elizabeth Hartung Sensitive Graph-theoretic Approaches for Extract- Massachusetts College of Liberal Arts ing Function from Biological Networks [email protected] Sequence-based computational approaches have revolu- Jack Graver tionized our understanding of biology. Since proteins ag- Mathematics Department gregate to perform function instead of acting in isolation, Syracuse University the connectivity of a protein interaction network (PIN) will [email protected] deepen biological insights over and above sequences of individual proteins. The advancement relies on developing sensitive graph-theoretic methods for extracting biological MS39 knowledge from PINs. Analogous to sequence alignment, Hamiltonian Cycles in Cayley Graphs alignment of PINs will impact biological understanding. Network alignment maps nodes of different networks with I will provide an overview of general results on hamilto- the goal of ”fitting” one network into the other well to nian cycles in Cayley graphs, focusing on some recent work identify topologically similar network regions and transfer that deals with graphs whose order has few prime factors. function between them. Since network alignment is com- Some of this is based on work produced jointly in various putationally infeasible, owing to the NP-completeness of combinations with Steve Curran, Klavdija Kutnar, Dragan the underlying subgraph isomorphism problem, heuristic Maruˇsiˇc, Dave Witte Morris, and Primoz Sparl.ˇ algorithms are sought. We designed a sensitive measure of the topological position of a node in the network, graphlet Joy Morris degree vector (GDV), to measure the number of 2-5-node Mathematics and Computer Science graphlets (induced subgraphs) that the node participates University of Lethbridge in. Hence, GDV of a node is a detailed topological de- [email protected] scriptor of its extended network neighborhood. Our GDV- similarity measure compares GDVs of two nodes to quantify similarity of their extended networks neighborhoods. MS40 Using these measures, we developed two network alignment Title Not Available at Time of Publication algorithms: a greedy ”seed and extend” approach that quickly finds approximate alignments and an approach that Abstract not avaialable at time of publication. finds optimal alignments by using the ”expensive” Hungar- Joel S. Bader ian algorithm. Their alignments expose surprisingly large Johns Hopkins University regions of network and functional similarity even in dis- Department of Biomedical Engineering tant species, suggesting broad similarities in cellular wiring [email protected] across all life on Earth. Also, we used these measures to show that aging, cancer, pathogen-interacting, drug-target and genes involved in signaling pathways are ”central” in MS40 the network, occupying dense network regions and ”domi- Network Archaeology: Uncovering Ancient Net- nating” other genes in the network. works from Present-Day Interactions Tijana Milenkovic Abstract not available at time of publication. Department of Computer Science and Engineering University of Notre Dame Carl Kingsford [email protected] Department of Computer Science University of Maryland, College Park [email protected] MS40 Biological Networks in Three Dimensions

MS40 Two studies are highlighted where three-dimensional mod- Understanding Phenome-genome Association by eling of protein networks provides insights into the bio- Co-clustering and Graph Matching physics and evolution of biological systems. First, a structural analysis of the yeast protein-protein interaction net- Graph algorithms are playing an increasingly important work reveals that protein evolution is constrained by the role in studying phenotype-gene relations in the context biophysics of protein folding, the mutational robustness of of systems biology and phenome (the whole collection of proteins, and the biophysics and function of protein-protein phenotypes) analysis. I will introduce two methods that interactions. Second, a structural analysis of human-virus utilize network information in phenome network and gene- protein-protein interaction networks reveals distinct prin- relation network to study phenome-genome associations, a ciples governing antagonism versus cooperation in host- random-walk based graph-matching algorithm (Bi-RW) for pathogen interactions. unveiling associations between phenome and genes, and a regularized NMF algorithm (R-NMTF) to co-cluster phe- Yu (Brandon) Xia notypes and genes, and detect associations between phe- Bioinformatics Program & Dept. of Chemistry notype clusters and gene clusters. Boston University [email protected] Rui Kuang Dept Computer Science University of Minnesota DM12 Abstracts 87

MS41 remedy is to consider the complement of the branch cuts, Advances on Quantifier Elimination and Applica- decomposing this space by Cylindrical Algebraic Decompo- tions sition and checking validity on each connected component. This process will be discussed and how it is possible to im- In this talk, we present a new approach for computing prove this algorithm’s efficiency with pre-conditioning and cylindrical algebraic decomposition of real space via trian- choice of variable order. gular decomposition of polynomial systems. Based on it, a general quantifier elimination method is proposed. The David J. Wilson, Russell Bradford, James Davenport usage of our method is illustrated by several application University of Bath examples. [email protected], [email protected], [email protected] Changbo Chen University of Western Ontario [email protected] MS41 A Solver for Linear Algebraic Systems Resulting Marc Moreno Maza from Selecting Discrete Structures Computer Science Department University of Western Ontario A computer algebra program LSSS (Linear Selective Sys- [email protected] tems Solver) has been developed to solve linear systems efficiently that are sparse, overdetermined and have many variables that take the value zero in the solution. Ap- MS41 plications include the determination of first integrals and Automatic Quantifier Elimination Proves a Key Lie-symmeties of systems of non-commutative ODEs re- Result in Submodular Function Minimization quiring the solution of linear systems with over one billion equations, the computation of Poisson homologies and co- We show how symbolic computation can automatically homologies of Poisson algebras and the study of compatible prove an important and well-known result in combinatorial Poisson structures. optimization. Specifically, we address the question “Which objective functions can be minimized by reduction to s-t Thomas Wolf min cut?” This question has been answered by a number of Department of Mathematics highly-cited papers in discrete mathematics and computer Brock University vision. By posing this same question to a computer algebra [email protected] system, we automatically identify submodularity as being the key condition for reduction to s-t min cut. MS42 Andrew Delong A De Bruijn - Erdos Theorem in Connected The University of Western Ontario, Canada Graphs? [email protected] De Bruijn and Erd˝os proved that every noncollinear set of n points in the plane determines at least n distinct lines. MS41 Chen and Chvátal suggested that this might generalize to On Newton Polytopes, Tropisms, and Puiseux Se- all finite metric spaces with appropriately defined lines. ries to Solve Polynomial systems As for metric spaces defined by connected graphs, the analogue of the De Bruijn - Erd˝os theorem is known to hold Sparse polynomial systems in several variables are charac- for all bipartite graphs, all chordal graphs, and all graphs terized by a set of exponents of monomials that appear with of diameter two. nonzero coefficient. The convex hull of this set of exponents is the Newton polytope. To a system of sparse polynomial Vasek Chvatal equations corresponds a tuple of Newton polytopes. Vec- Concordia University tors perpendicular to a tuple of edges of all polytopes are [email protected] tropisms when they define the leading powers of a Puiseux series development for a solution of the polynomial sys- MS42 tem. In this talk we outline a polyhedral approach to solve polynomial systems by means of Puiseux series. On stan- Matthews Sumner Conjecture on Claw-Free dard benchmark problems as the cyclic n-roots systems we Graphs obtained exact representations for surfaces of solutions. The Matthews Sumner Conjecture is that every 4- Jan Verschelde,DankoAdrovic connected claw-free graph is Hamiltonian. There are a Department of Mathematics, Statistics and Computer large number of conjectures, some seemingly weaker and Science some seemingly stronger, that are equivalent to this con- University of Illinois at Chicago jecture that will be discussed. For example, the Carsten [email protected], [email protected] Thomassen Conjecture that every 4-connected line graph is Hamiltonian is an equivalent conjecture.

MS41 Ralph Faudree University of Memphis Algebraic Representations of Branches of Func- Department of Math Sciences tions [email protected] Apparently-simple function deﬁnitions may hide subtleties from branch cuts in the complex plane. Hence formulae that initially seem correct may fail for various values. One 88 DM12 Abstracts

MS42 tions. How to Recognize a Good Conjecture Jan Foniok The speaker will present some challenges in dealing with Queen’s University open problems and conjectures. He may discuss some spe- [email protected] ciﬁc graph theory conjectures and demonstrate some ratio- nale behind the Open Problem Garden. MS43 Bojan Mohar A Randomized Version of Ramsey’s Theorem Simon Fraser University [email protected] For every integer r ≥ 2wecallak-uniform hypergraph H on n vertices r-Ramsey-forcing if every r-edge-coloring of Kn contains a monochromatic copy of Kk whose ver- MS42 tices form a hyperedge in H. We determine the threshold Some of my favorite Conjectures for a random k-uniform hypergraph on n vertices and edge probability q to be r-Ramsey-forcing. Further we discuss We will present and brieﬂy discuss some open questions in common features of the above problem with the more clas- two areas: (1) Cyclic Coloration of plane graphs and (2) sical Ramsey’s theorem for random (hyper-)graph G(n, p). Well-covered graphs. Michael D. Plummer Luca Gugelmann Department of Mathematics Institute of Theoretical Computer Science ETH Zurich Vanderbilt University 8092 Zurich, Switzerland [email protected] [email protected]

Yury Person MS42 Freie Universitat, Berlin The Implicit Representation of Graphs Conjecture [email protected] A class of graphs P is called a hereditary property if it is closed under taking induced subgraphs and isomor- Angelika Steger phism. Common examples include being acyclic, being Institute of Theoretical Computer Science planar, and being bipartite. Given a hereditary property ETH, Zürich P, we seek a method of labeling the vertices of graphs in [email protected] P with short labels [that is, O(log n)bitsforeachver- tex] such that we can determine adjacency of two ver- Henning Thomas tices just by considering their labels. Formally, a prop- Institute of Theoretical Computer Science ETH Zurich erty P admits an implicit representation provided there 8092 Zurich, Switzerland is a function A : Z+ × Z+ →{0, 1} and a positive inte- [email protected] ger k such that for every G ∈Pwith there is a mapping : V (G) → [nk](wheren = |V (G)|) such that u ∼ v if and only if A((u),(v)) = 1. It is easy to create such MS43 a labeling for acyclic graphs: Pick a root vertex in each General Deletion Lemmas via the Harris Inequality component, number the vertices from 1 to n,andlabel vertex v as (v, w)wherew is the unique neighbor of v on For a given graph H let XH denote the random variable apathfromv to the root (or use the label (v, v)ifv is that counts the number of copies of H in a random graph the root). However, bipartite graphs admit no such repre- Gn,p. In order to control the upper tail of XH ,Rödl and sentation simply because there are too many. That is, for Ruciński showed that with ‘Janson-like’ probability, delet- a property P,thespeed of P is a function that gives for ing a small fraction of all edges suffices to reduce the num- each n the number of labeled graphs in P with V =[n]. ber of copies of H to at most (1+ ε)E[XH ]. This approach If a property has an implicit representation, then neces- is known as the ‘deletion method’ and appears in many sarily its speed is bounded by an expression of the form proofs of Ramsey properties of random structures. In this ncn for a positive constant c.TheImplicit Representation talk I will present a variant of the deletion method that is Conjecture of Kannan, Naor, and Rudich posits that this based on the Harris inequality and extends to more general necessary condition is also sufficient. graph properties.

Ed Scheinerman Reto Spöhel Johns Hopkins University Max-Planck-Institut für Informatik [email protected] Saarbrücken [email protected]

MS43 Angelika Steger Ramsey Classes Defined by Forbidden Homomor- Institute of Theoretical Computer Science phisms ETH, Zürich [email protected] I will talk about Ramsey properties of classes of digraphs (or more generally, relational structures) defined by forbid- Lutz Warnke ding homomorphisms from a fixed set of digraphs. Cases University of Oxford with infinitely many forbidden structures are of particu- [email protected] lar interest. These classes have several interesting applica- DM12 Abstracts 89

MS43 plementation in dominant and undominated strategies are On Ramsey Multiplicities of Graphs Containing naturally extended to this setting, but their power is vastly Triangles different. Namely, (1) We prove that no dominant-strategy mechanism can guarantee more social welfare than by as- AgraphG is k-common (k>1) if the minimum number of signing the good at random; but (2) We prove a much bet- monochromatic copies of G over all k edge colorings of Kn ter performance can be obtained via undominated-strategy is asymptopic to the expected number of monochromatic mechanisms, and actually provide tight upper and lower copies of G in a random k edge coloring. Jagger, S˘˘tov´ı˘cek bounds for the fraction of the maximum social welfare and Thomason defined the class of k-common graphs, and guaranteable by such mechanisms, whether deterministic showed among other results that every graph containing or probabilistic. K4 as a subgraph is not 2-common. We prove that every Silvio Micali graph containing K3 as a subgraph is not 3-common. Massachusetts Institute of Technology Michael Young [email protected] Iowa State University [email protected] Alessandro Chiesa, Zeyuan Zhu MIT James Cummings [email protected], [email protected] Carnegie Mellon University [email protected] MS44 Online Procurement MS44 Multidimensional Algorithmic Mechanism Design We present results for online procurement markets. We introduce mechanisms with desirable approximation guar- In his seminal paper, Myerson [1981] provides a revenue- antees in various models of online procurement. We also optimal auction for a seller who is looking to sell a single present new models of strategic bidding in such markets, item to multiple bidders. Extending this auction to simul- and discuss new mechanism design frameworks for these taneously selling multiple heterogeneous items has been bidding models. one of the central problems in Mathematical Economics. We provide such an extension that is also computationally Yaron Singer efficient. University of California at Berkeley [email protected] Costis Daskalakis Massachusetts Institute of Technology [email protected] MS44 Bayesian Multi-Parameter Scheduling

MS44 We study the makespan minimization problem with unre- Combinatorial Walrasian Equilibria lated selfish machines. Specifically, our goal is to schedule agivensetofjobsonm machines; the machines are strate- One way to resolve a combinatorial auction is via Wal- gic agents who hold the private information of the run-time rasian equilibrium: setting a price for each item so that, of each job on them. The goal is to design a mechanism when bidders take their most demanded sets, each object that minimizes makespan — the time taken for the last job is chosen by exactly one bidder. Such equilibria have de- to complete. In the strategic setting, strong impossibility sirable properties, but they exist only in special cases. In results are known that show that no truthful (anonymous) this work we introduce the alternative notion of combina- mechanism can achieve better than a factor m approxima- torial Walrasian equilibrium, in which one first partitions tion. We show that under mild Bayesian assumptions it is the objects into indivisible bundles and then sets a price for possible to circumvent such negative results and obtain a each bundle. We design mechanisms that generate combi- constant approximation. natorial Walrasian equilibria for various classes of bidder valuations, with the goal of maximizing social welfare. Our Balasubramanian Sivan study also touches upon revenue and truthfulness in such University of Wisconsin-Madison mechanisms. [email protected]

Brendan Lucier Shuchi Chawla Microsoft Research New England Department of Computer Sciences [email protected] University of Wisconsin-Madison [email protected] Michal Feldman Hebrew University of Jerusalem Jason Hartline [email protected] Northwestern U [email protected] MS44 David Malec Knightian Auctions University of Wisconsin-Madison [email protected] In an auction, a player may not exactly know his own val- uation, or its distribution. We thus study single-good auctions whose players know their own valuations only within a multiplicative factor (e.g., 10%). The notions of im- 90 DM12 Abstracts

MS45 department of Applied Mathematics Graph Homomorphism Counts Mod M Charles University [email protected]ff.cuni.cz I will talk about the numbers hom(F, G) mod m as the graphs F and G vary. MS45 Swastik Kopparty Semilattice and NU Polymorphisms on Reflexive Massachusetts Institute of Technology Graphs [email protected] The structures H for which arc-consistency solves H-COL are exactly those that admit TSI polymorphisms. The two MS45 main sources of TSI polymorphisms are near unanimity Frozen Vertices in Colourings of a Random Graph (NU) and semilattice (SL) polymorphisms. Much has been done to understand the nature of NU polymorphisms, but Over the past decade, much of the work on random k-SAT, SL polymorphisms are not so well understood. In this talk random graph colouring, and other random constraint sat- we describe the structure of symmetric reflexive graphs ad- isfaction problems has focussed on some foundational un- mitting various types of SL polymorphisms. Further we re- proven hypotheses that have arisen from statistical physics. late the existence of SL polymorphisms on these graphs to Some of the most important such hypotheses concern the the existence of NU polymorphisms of various types, and “clustering’ of the solutions. It is believed that if the prob- to related classes of graphs. This is preliminary work in in lem density is sufficiently high then the solutions can be understanding SL polymorphisms, and includes joint work partitioned into clusters that are, in some sense, both well- with Pavol Hell. connected and well-separated. Furthermore, the clusters contain a linear number of “frozen variables”, whose val- Mark Siggers ues are fixed within a cluster. The density where such Kyungpook National University clusters arise corresponds to an algorithmic barrier, above Daegu, Republic of Korea which no algorithms have been proven to solve these prob- [email protected] lems. In this talk, we prove that frozen vertices do indeed arise for k-colourings of a random graph, when k is a sufficiently large constant, and we determine the exact density MS46 threshold at which they appear. Near Factorisations of the Complete Graph Michael Molloy A w-near k-factor of a graph G on n vertices is a spanning University of Toronto subgraph of G with w vertices of degree 0 and n − w ver- [email protected] tices of degree k. In this talk, we introduce the concept of a w-near k-factorization of a graph G,whichisadecom- position of G into w-near k-factors. Thus, for example, MS45 a k-factorization is equivalent to a 0-near k-factorization, Colorings and Homomorphisms of Sparse Graphs and a near 1-factorization is equivalent to a 1-near 1- factorization. We focus on w-near 2-factorizations of the chromatic number and low tree-width / low tree-depth col- complete graph Kn and Kn −I; in the case that the partial orings. Dichotomy of sparsity by means of chromatic num- 2-factors are required to be pairwise isomorphic, this may bers. Applications to distance coloring, circular chromatic be viewed as a generalization of the Oberwolfach problem. number of large girth graphs with sub-exponential expan- We discuss some constructions of w-near 2-factorizations in sion, etc. which all cycles in the near-factors have the same length. Jaroslav Nesetril Peter Danziger, Andrea Burgess department of Applied Mathematics Ryerson University Charles University [email protected], [email protected] [email protected]ff.cuni.cz

Patrice Ossona de Mendez MS46 CNRS Paris Bounds for Covering Arrays with Row Limit Centre d’Analyse et de Mathematiques Sociales [email protected] Covering arrays with row limit (CARLs) are a generalization of covering arrays. They have a new parameter row weight, w, representing the number of non-empty cells MS45 in a row. Considering w as a function of the number of Fast Algorithms for Sparse Graphs columns, k, there are two extremal cases: when w is a constant, CARLs are equivalent to group divisible cover- Algorithms for sparse graphs: subgraph isomorphism prob- ing designs and we know their optimal size; when w = k, lem, connection with bounded local tree-width. Low Tree CARLs are covering arrays, and their optimal size is still Depth Decomposition and ﬁrst-order properties. Applica- unknown in most cases. We present two probabilistic mod- tion to graph invariants. els to determine the upper bounds on the size of a CARL when w = f(k) and we look into critical points where the Patrice Ossona de Mendez change in the behaviour occurs. CNRS Paris Centre d’Analyse et de Mathematiques Sociales Nevena Francetic [email protected] University of Toronto [email protected] Jaroslav Nesetril DM12 Abstracts 91

Peter Danziger attained in this bound whenever q is an even power of Ryerson University two. For a Steiner√ triple system on v points, we prove that [email protected] s ≤ (2v +5− 24v +25)/2 and we determine necessary and sufficient conditions for equality to be attained in this Eric Mendelsohn bound. University of Toronto Department of Mathematics Douglas R. Stinson [email protected] University of Waterloo David R. Cheriton School of Computer Science [email protected] MS46 A new construction of strength-3 covering arrays MS47 using primitive polynomials over finite fields Optimality of the Neighbor Joining Algorithm and In this talk, we present a new construction of covering ar- Faces of the Balanced Minimum Evolution Poly- rays based on Linear Feedback Shift Register (LFSR) se- tope quences constructed using primitive polynomials over finite fields. For any prime power q, this construction gives a cov- Balanced minimum evolution (BME) is a statistically con- ering array of strength 3 with q2 + q + 1 factors/columns sistent distance-based method to reconstruct a phyloge- over q levels/symbols that has size 2q3−1(numberofrows). netic tree from an alignment of molecular data. In 2000, The construction can be extended to non-prime powers v Pauplin showed that the BME method is equivalent to op- by dropping levels from a larger prime power. This re- timizing a linear functional over the BME polytope, the sults in significant reductions on known upper bounds for convex hull of the BME vectors obtained from Pauplins covering array sizes in most cases covered by this construc- formula applied to all binary trees. The BME method is tion. In particular, for the values of v ≤ 25 kept in Col- related to the popular Neighbor Joining (NJ) algorithm, bourn’s covering array tables, this construction improves now known to be a greedy optimization of the BME prin- upper bounds for all v =2 , 3, 6. ciple. In this talk I will elucidate some of the structure of the BME polytope and strengthen the connection be- Lucia Moura tween the BME method and NJ Algorithm. I will show University of Ottawa that any subtree-prune-regraft move from a binary tree to Information Technology another binary tree corresponds to an edge of the BME [email protected] polytope. Moreover, I will describe an entire family of faces parametrized by disjoint clades. Finally, given a phy- Sebastian Raaphorst logenetic tree T, I will show that the BME cone and every University of Ottawa NJ cone of T have intersection of positive measure. [email protected] David Haws University of Kentucky Brett C. Stevens [email protected] Carleton University School of Mathematics and Statistics Terrell Hodge [email protected] Western Michigan University [email protected] MS46 New Areas in Covering Arrays Ruriko Yoshida University of Kentucky In recent years, with Gary Bazdell, I have had increased [email protected] communication and collaboration with industrial users of combinatorial designs, predominantly covering arrays. Coming out of their needs and interests, Gary and I have MS47 been motivated to think about new generalisations of cov- Finding the Max Cut of the Genetic Interactions ering arrays, specifically ones that introduce order to the Graph in Yeast Finds Compensatory Pathways. rows and columns. I will discuss these definitions, the mathematics and their connections to applications. By using a randomized local search method to find the max cut in the graph of genetic interactions in yeast we generate Brett C. Stevens potential compensatory networks. Unlike other methods, Carleton University this method works solely on genetic interaction data from School of Mathematics and Statistics epistatic miniarray profiles (E-MAP) and synthetic genetic [email protected] array analysis (SGA), this enables us to use physical interactions as an additional validation source. We finish with some open questions regarding the behavior of our tech- MS46 nique on different graphs and weights. Nonincident Points and Blocks in Designs Benjamin Hescott We study the problem of finding the largest possible set Tufts University of s points and s blocks in a balanced incomplete block [email protected] design, such that that none of the s points lie on any of the s blocks. We investigate this problem for two types Mark Leiserson of BIBDs: projective planes and Steiner triple systems. Brown University For a projective√ plane of order q,weprovethats ≤ 1+ [email protected] (q +1)( q − 1) and we also show that equality can be 92 DM12 Abstracts

Andrew Gallant, Diana Tatar MS48 Tufts University New Results on D-Optimal Matrices [email protected], [email protected] D-optimal matrices are 2v 2v (-1,+1)-matrices that have Lenore Cowen maximal determinant among all 2v 2v (-1,+1)-matrices, Dept. of Computer Science where v is an odd positive integer. The value of the max- Tufts University imal determinant is given by Ehlich’s bound. We present [email protected] new theoretical and computational results on D-optimal matrices of circulant type. Such D-optimal matrices are constructed via two circulant submatrices of orders v each. MS47 In particular, we construct new D-optimal matrices of or- Statistics and Applications of Tree Space ders 206, 242, 262, 482. Joint work with D. Z. Djokovic.

The space of metric n-trees is a polyhedral complex, in which each polyhedron corresponds to a different tree Ilias S. Kotsireas topology. Under the construction of Billera, Holmes, and Wilfrid Laurier University Vogtmann, this space is non-positively curved, so there is Department of Computing a unique geodesic (shortest path) between any two trees. [email protected] We will look at what it means to do statistics on this space, as well as some biological applications. MS48 Megan Owen Matroid Base Polytope Decomposition and Appli- Fields Institute cations mowen@fields.utoronto.ca Let P (M) be the matroid base polytope of a matroid M.A matroid base polytope decomposition of P (M)isadecom- MS47 t position of the form P (M)= P (M )whereeachP (M ) Comparing Biological and Mathematical Ap- i i i=1 proaches to Modeling Tissue Development in C. is also a matroid base polytope for some matroid Mi,and elegans for each 1 ≤ i = j ≤ t, the intersection P (Mi) ∩ P (Mj ) is a face of both P (M )andP (M ). In this talk, we shall Gene networks are often inferred from experimental data i j using single-method approaches; however, an individual present some new results on the above decomposition. We method may not sufficiently recover the network. We de- give sufficient conditions on M in order that P (M)hasa veloped a modeling pipeline that combines statistical and hyperplane split (that is, a decomposition with t =2)and algebraic methods. We built a mathematical model of tis- show that P (M) has not a hyperplane split if M is binary. sue development from wildtype C. elegans data. We com- We finally discuss an application in relation with Tutte and pared it to an existing biological model on a gold-standard Ehrhart polynomials. network. While both models identified many interactions, the mathematical model was enriched for positive predic- Jorge L. Ramirez Alfonsin tions compared to biological model. Montpellier 2, France [email protected] Brandilyn Stigler Southern Methodist University [email protected] MS48 Generating Program Invariants via Interpolation

MS47 This talk focuses on automatically generating polynomial Profiling RNA Secondary Structures equations that are inductive loop invariants of computer programs. A new method which is based on polynomial Efficient methods for sampling from the Boltzmann distri- interpolation is proposed. Though the proposed method is bution have led to an emergence of ensemble-based ap- not complete, it is efficient and can be applied to a broader proaches to RNA secondary structure prediction. We range of problems compared to existing methods targeting present a novel combinatorial method, RNA structure pro- similar problems. The efficiency of our approach is testified filing, for identifying patterns in structural elements across by experiments on a large collection of programs. a Boltzmann sample. Our approach is based on classifying structures according to features chosen from well-defined Rong Xiao structural units called helix classes. We show that combi- University of Western Ontario natorial profiling is straightforward, stable and surprisingly [email protected] comprehensive. Marc Moreno Maza Emily Rogers, M. Shel Swenson Computer Science Department School of Mathematics University of Western Ontario Georgia Institute of Technology [email protected] [email protected], [email protected]

Christine E. Heitsch MS48 School of Mathematics Parallel Computation of the Minimal Elements of Georgia Tech a Poset and Applications [email protected] Computing the minimal elements of a partially ordered ﬁ- nite set (poset) is a fundamental problem in combinatorics DM12 Abstracts 93

with numerous applications. In a previous work, we have MS49 developed for this task a divide-and-conquer algorithm Title Not Available at Time of Publication which is also cache-efficient and which can be efficiently parallelized free of determinacy races. We adapted this Abstract not available at time of publication. technique to questions arising in computer algebra (polynomial expression optimization) and combinatorics (transver- Fan Chung Graham sal hypergraph generation). In this talk, we present new University of California, San Diego applications in both computer algebra (coprime factoriza- [email protected] tion) and combinatorics (orthogonal hypergraph generation). We have implemented the corresponding algorithms in Cilk++ targeting multi-cores. For our test problems of MS51 sufficiently large input size our code demonstrates a linear On Metric Dimension of Functigraphs speedup on 32 cores. AsetS of vertices in a graph G is called a resolving set for Yuzhen Xie G if for every pair of vertices x, y ∈ V (G), there is a vertex University of Western Ontario s ∈ S such that d(x, s) = d(y,s). The minimum number [email protected] of vertices in a resolving set for G is the metric dimension dim(G)ofG. Given two copies G1 and G2 of a graph G and → Marc Moreno Maza a function f : V (G1) V (G2), the functigraph C(G, f)is the graph with vertices V (G1) ∪ V (G2)andedgesE(G1) ∪ Computer Science Department ∪{ | } University of Western Ontario E(G2) xy y = f(x) . In this talk, we give bounds on [email protected] the metric dimension of functigraphs, particularly in the cases when G is a cycle or path and when the function f is a constant function or permutation. MS49 Linda Eroh Conjectures on Thickness, Connectivity, and Other Department of Mathematics Things University of Wisconsin at Oshkosh Various conjectures that I’ve found intriguing will be dis- [email protected] cussed. These will include a conjecture on thickness and chromatic number, a mixed-connectivity conjecture, and Cong Kang, Eunjeong Yi the total graph conjecture. Texas A&M University at Galveston [email protected], [email protected] Lowell Beineke Indiana University - Purdue University Fort Wayne [email protected] MS51 Closed K-Stop Distance in Graphs. MS49 The Traveling Salesman Problem (TSP) is still one of the Snarks and Their Relation to Conjectures about most researched topics in computational mathematics, and Graphs we introduce a variant of it, namely the study of the closed k-walks in graphs. We search for a shortest closed route In this talk I will shortly discuss the role of snarks in prov- visiting k cities in a non complete graph without weights. ing or refuting conjectures and sketch an algorithm that This motivates the following definition. Given a set of k was used to generate large lists of snarks. The results of distinct vertices S = {v1,v2,...,vk} in a simple graph G, these computations and some of the published conjectures the k-circuit-distance of set S is defined to be that could be refuted by the snarks generated will be presented. This research is joint work with Jan Goedgebeur, dk(S)= min d(θ(x1),θ(x2))+d(θ(x2),θ(x3))+...+d(θ(xk),θ(x1)) , Jonas Hägglund and Klas Markström θ∈P(S) Gunnar Brinkmann where P(S) is the set of all permutations from S onto S. University of Ghent, That is the same as saying that d (S) is the length of the Belgium k shortest circuit through the vertices {x ,...,x }.Recall [email protected] 1 k that the Steiner distance sd(S) is the number of edges in a minimum connected subgraph containing all of the vertices S MS49 of . We note some relationships between Steiner distance and circuit distance. The 2-circuit distance is twice the On Conjectures of Grafitti.pc ordinary distance between two vertices. We conjecture that ≤ ≤ k This talk will begin with a brief description of how the radk(G) diamk(G) k−1 radk(G) for any connected conjecture-making computer program, Graffiti.pc, gener- graph G for k ≥ 2. For k = 2, this formula reduces to the ates graph theoretical conjectures. Then, I will talk about well-known classical result rad(G) ≤ diam(G) ≤ 2rad(G). some of my favorite Graffiti.pc conjectures. We prove the conjecture in the cases when k =3andk =4 for any graph G and for k ≥ 3whenG is a tree. Ermelinda DeLaVina University of Houston Ralucca M. Gera [email protected] Naval Postgraduate School [email protected]

Grady Bullington, Linda Eroh, Steve Winters University of Wisconsin Oshkosh bullington@uwosh, eroh@uwosh, winters@uwosh 94 DM12 Abstracts

MS51 MS52 Distance in Rainbow Connected Graphs Graph Partitions A path (not necessarily a shortest one) between two ver- We consider partitions of the vertices of a graph into a tices in a connected graph is rainbow-colored if each edge ﬁxed number of labelled cells such that (i) the subgraph of the path is labeled with a distinct color. A connected induced by each cell belongs to a family that depends on graph is rainbow-connected if the edges of the graph are the cell, and (ii) edges joining vertices in diﬀerent cells are assigned colors so that a rainbow-colored path exists be- allowed only if the cells correspond to adjacent vertices in tween every pair of vertices. In this paper, I discuss the a given pattern graph H. Both polynomiality and NP- recent work on methods for determining rainbow-colored completeness results are presented. paths in a given rainbow-connected graph. Gary Macgillivray Garry L. Johns Mathematics and Statistics Saginaw Valley State University University of Victoria [email protected] [email protected]

Peter Dukes MS51 University of Victoria, Canada Using Graphs for Facility Location Problems [email protected] The eccentricity of a vertex v in a connected graph G is the distance from v to a vertex farthest from v in G. The Steve Lowdon distance of a vertex v in a connected graph G is the sum Mathematics and Statistics of the distances from v to the vertices of G. In this talk, University of Victoria we will consider subgraphs induced by these concepts for [email protected] facility location problems. Steven J. Winters MS52 University of Wisconsin Oshkosh Graph Partitions With Emphasis on 2K2-Partition [email protected] Pavol Hell will present some recent progress and some open questions on the problem of partitioning a graph, MS52 or digraph, according to a prescribed pattern of adjacen- Near-Unanimity Graphs and Absolute Retracts cies. Both classifications by complexity and characterizations by forbidden induced subgraphs will be discussed. We describe a generating set for the variety of reflexive Then Daniel Paulusma will describe a recent solution (joint graphs that admit a k-ary near-unanimity operation; we with Barnaby Martin) of the problem of complexity of further delineate a very simple subset that generates the the disconnected cut, or equivalently the problem of 2K2- variety of absolute retracts with respect to j-holes; in par- partition. Specifically, for a connected graph G =(V,E), ticular we show that every reflexive graph with a 4-NU a subset U of V is called a disconnected cut if both U operation is an absolute retract for 3-holes. Our results and V disconnect G. The problem to decide whether a generalise and encompass several results on NU-graphs and graph has a disconnected cut is shown to be NP-complete. absolute retracts from the past 30 years. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing Tomas Feder if a graph allows a vertex-surjective homomorphism to the Stanford University reflexive 4-cycle and testing if a graph has a spanning sub- [email protected] graph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are Pavol Hell NP-complete as well. School of Computer Science Simon Fraser University Pavol Hell [email protected] School of Computer Science Simon Fraser University [email protected] Benoit Larose Concordia University [email protected] Barnaby Martin, Daniel Paulusma Durham University [email protected], Cynthia Loten [email protected] Fraser Valley [email protected] MS52 Mark Siggers Bipartite Graphs and Approximation of Minimum Kyungpook National University Cost Homomorphisms Daegu, Republic of Korea [email protected] For a fixed target graph H, the minimum cost homomorphism problem, MinHOM(H), asks, for a given graph G ∈ ∈ Claude Tardif with integer costs ci(u), u V (G), i V (H), and an in- Department of Mathematics and Computer Science teger k, whether or not there exists a homomorphism of G Royal Military College to H of cost not exceeding k. When the target graph H [email protected] is a bipartite graph, a dichotomy classification is known: MinHOM(H) is solvable in polynomial time if and only DM12 Abstracts 95

if H is a proper interval bigraph (i.e., a bipartite per- ence in ad hoc and sensor wireless networks, along with mutation graph). We suggest an integer linear program corresponding algorithmic and complexity results for the (ILP) formulation for MinHOM(H), and by rounding the interference minimization problem. ILP solution, we obtain a 2-approximation algorithm for MinHOM(H), when H is a doubly convex bipartite graph. Stephane Durocher If time permits, we explain how to modify the rounding University of Manitoba procedure to obtain a |V (H)| - approximation algorithm [email protected] for MinHOM(H) when the complement of the bipartite graph H is a circular arc graph. MS53 Arash Rafiey On the Treewidth of Random Geometric Graphs Simon Fraser University arash.rafi[email protected] We give asymptotically exact values for√ the treewidth tw(G) of a random geometric graph in [0, n]2.Morepre- cisely, we show that there exists some c1 > 0, such that for log n MS52 any constant 0 c1, such that for any r = r(n) ≥ c2, tw(G)=Θ(r n). Our proofs show that for the corre- Functors in graph theory are essentially constructions that sponding values of r the same asymptotic bounds also hold make new graphs out of input graphs. Two functors L, R for the pathwidth and treedepth of a random geometric are respectively right and left adjoint to the other if there is graph. a correspondence between the homomorphisms of L(G)to H and the homomorphisms of G to R(H). There are var- Dieter Mitsche ious places where adjoint functors arise naturally without Ryerson University necessarily being noticed: - Viewing lexicographic prod- Toronto ucts by complete graphs as right adjoints naturally defines [email protected] Kneser graphs as (cores of) left adjoints of complete graphs. - Reformulating the question of Harvey and Murty on the Guillem Perarnau existence of k-chromatic graphs with strongly independent Universitat Politecnica de Catalunya colour classes in terms of adjoint functors naturally leads [email protected] to the approach taken by Gyárfás, Jensen and Stiebitz to answer it affirmatively. In the case of directed graphs and constraint satisfaction problems, the adjoint functors arise MS53 in the arc graph construction, and they preserve tree dual- On the 2-Edge Connectivity of Geometric Planar ity and many Maltsev-type operations related to complex- Graphs With Bounded Edge Length ity classifications. All of this suggests that the power and limitations of adjoint functors in graph theory should be Abstract not available at time of publication. better understood. Oscar Morales Ponce Claude Tardif Carleton University Department of Mathematics and Computer Science Ottawa, Canada Royal Military College [email protected] [email protected]

Jan Foniok MS53 Queen’s University Some Results about Disk Graphs and Pseudocircle [email protected] Arrangements Abstract not available at time of publication. MS53 Tobias Mueller New Results and Open Problems on Random Ge- Centrum voor Wiskunde en Informatica ometric Graphs [email protected] Abstract not available at time of publication. Josep Diaz MS54 Departament de Llenguatges i Sistemes Inform`atics Generating Trees for Partitions and Permutations Universitat Polit`ecnica de Catalunya, Spain with No k-nestings [email protected] We describe a generating tree approach to the enumeration and exhaustive generation of set partitions and per- MS53 mutations with no k-nestings. Unlike previous work in the Modelling Interference in Wireless Networks using literature using connections of these objects with Young Geometric Graphs tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We de- Given a set of positions for wireless nodes, the interference tail a generation algorithm that uses only labels to describe minimization problem is to assign a transmission radius these set partitions and permutations with no k-nestings, (equivalently, a power level) to each node such that the re- and provide explicit functional equations for the generat- sulting communication graph is connected, while minimiz- ing functions, with k as a parameter, getting valuable series ing the maximum interference. In this talk I will discuss results. recent geometric graph models for representing interfer- Sophie Burrill 96 DM12 Abstracts

Simon Fraser University Algebra [email protected] Finite fields display erratic behavior when the field size is Sergi Elizalde small and for large finite fields memory usage and processor Dartmouth College power become bottlenecks. I will outline the computer al- [email protected] gebra packages containing finite fields implementations and will list drawbacks and advantages to each. I will illustrate this with examples from my research including an exhaus- Marni Mishna tive search algorithm for normal bases as well as data sets Simon Fraser University obtained for a section of the forthcoming ”Handbook of [email protected] Finite Fields”. Lily Yen David Thomson Capilano University Carleton University Simon Fraser University [email protected] [email protected]

MS55 MS54 A Theoretical Foundation for Neural Sensor Net- Bijections Between Truncated Affine Arrange- works ments and Valued Graphs Multiple recent multi-neuron recordings have uncovered We present some contructions on the set of nbcs (No Bro- structure in the statistics of mammalian retinal output ken Circuit sets) of some deformations of the braid arrange- that is well-captured by a probabilistic model from sta- ment. This leads us to some new bijective proofs for Shi, tistical physics called the Ising model. Guided by these Linial and similar hyperplane arrangements. findings and motivated by classical ideas in theoretical neu- roscience, we have developed a model of neural sensor net- David Forge works. Our approach incorporates a recently developed Université de Paris-Sud Ising model parameter fitting technique called Minimum [email protected] Probability Flow (MPF). Although a theoretical explanation for network processing of sensory stimuli, our model veronique ventos can also be used to uncover structure in many highly noisy Universite Paris 11 datasets. We discuss several such applications to neuro- [email protected] science, behavioral genetics, and machine learning. In addition, we discuss how our network can learn an exponen- sylvie corteel tial number of combinatorial structures (cliques in graphs) Universite Paris 7 from only a small number of training samples. [email protected] Christopher Hillar University of California, Berkeley MS54 [email protected] What Computer Algebra Systems Can Offer to Tackle Realizability Problems of Matroids Kilian Koepsell UC Berkeley In his PhD dissertation, Hiroki Nakayama establishes a [email protected] connection between oriented matroids and mathematical programming, by applying polynomial optimization techniques to the realizability problem of oriented matroids. In MS55 this talk, I will discuss what are the tools that computer Neural Code Topology Imposes Constraints on the algebra systems offer today to implement realizability al- Network Architecture gorithms, such as the one of Nakayama. There are two complementary perspectives on what con- Marc Moreno Maza trols neural activity in sensory systems: receptive fields, Computer Science Department and neural network dynamics. Receptive fields largely de- University of Western Ontario termine the representational function of a neuronal net- [email protected] work, while network dynamics stem from the network structure. We propose a mathematical framework where the structure of neuronal networks can be related to com- MS54 binatorial properties of receptive fields, and demonstrate Stochastic Ptri Nets and Symbolic Calculus how this can be used to relate network structure to homotopy invariants of represented stimuli. Abstract not available at time of publication. Vladimir Itskov Michel Petitot Department of Mathematics Université des Sciences et Technologies de Lille I University of Nebraska, Lincoln michel.petitot@lifl.fr [email protected]

MS54 MS55 The Interplay Between Finite Fields and Computer Stable Exponential Storage in Hopﬁeld Networks A Hopﬁeld auto-associative neural network on n nodes can DM12 Abstracts 97

store a random collection of at most 2n binary patterns as memories. However, it has been observed experimentally that random networks are typically capable of exponential storage in the form of spurious memories, which emerge mysteriously from the random network coupling. We give for the first time a parameterized family of Hopfield networks that stably store exponential numbers of binary patterns. Moreover, we give experimental evidence suggesting that these memories are attractors for the clean-up dynamics of the Hopfield network with large basins of attraction. We also present some applications. Kilian Koepsell UC Berkeley [email protected]

Ngoc Tran, Christopher Hillar University of California, Berkeley [email protected], [email protected]

MS55 Noise-Induced Dynamics in Electricaly Coupled Neuronal Networks Electrically coupled networks of excitable neurons forced by small noise are analyzed in this work, using techniques from dynamical systems and algebraic graph theories. The results are presented from two complementary perspectives of variational analysis of spontaneous network dynamics and the slow-fast analysis of synchronization. The former yields geometric interpretation of various dynamical regimes, while the latter highlights the contribution of the network architecture to the stability of the synchronous state. Georgi S. Medvedev Drexel University Department of Mathematics [email protected]

MS55 Simplex Packings of Marginal Polytopes and Mix- tures of Exponential Families

A combinatorial approach to the representational power of mixtures of discrete exponential families is presented based on combinatorics of convex polytopes and coding theory. The number of mixtures of a hierarchical model which contain another hierarchical model can be bounded by the cardinality of simplex face packings of their marginal polytope lattices. A bound for such packings is presented for the k-way interaction exponential families. Guido F. Montufar Pennsylvania State University [email protected]