36 DM14 Abstracts

IP0 University of Oxford Hot Topic Session: The Existence of Designs [email protected] We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More IP4 generally, we show that the natural divisibility conditions Submodular Functions and Their Applications are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. Submodular functions, a discrete analogue of convex functions, have played a fundamental role in combinatorial optimization since the 1970s. In the last decade, there has Peter Keevash been renewed interest in submodular functions due to their University of Oxford role in algorithmic game theory, as well as numerous appli- [email protected] cations in machine learning. These developments have led to new questions as well as new algorithmic techniques. I will discuss the concept of submodularity, its unifying role IP1 in combinatorial optimization, and a few illustrative appli- The Graph Regularity Method cations. I will describe some recent algorithmic advances, in particular the concept of multilinear relaxation, and the Szemerdi’s regularity lemma is one of the most powerful role of symmetry in proving hardness results for submod- tools in graph theory, with many applications in combina- ular optimization. I will conclude by discussing possible torics, number theory, discrete geometry, and theoretical extensions of the notion of submodularity, and some fu- computer science. Roughly speaking, it says that every ture challenges. large graph can be partitioned into a small number of parts such that the bipartite subgraph between almost all pairs Jan Vondrak of parts is random-like. Several variants of the regularity IBM Almaden Research Center lemma have since been established with many further ap- [email protected] plications. I will discuss recent progress in understanding the quantitative aspects of these lemmas and their applications. IP5 Approximation Algorithms Via Matrix Covers Jacob Fox MIT I will describe a general technique for obtaining approx- [email protected] imate solutions of hard quadratic optimization problems using economical covers of high dimensional sets by small cubes. The analysis of the method leads to intriguing al- IP2 gorithmic, combinatorial, geometric, extremal and proba- Rota’s Conjecture bilistic questions. Based on joint papers with Troy Lee, Adi Shraibman and Santosh Vempala. In 1970, Gian-Carlo Rota posed a conjecture giving a suc- cinct combinatorial characterization of the linear depen- Noga Alon dencies among a finite set of vectors in a vector space over Tel-Aviv University any given finite field. I will discuss the conjecture as well [email protected] as joint work with Bert Gerards and Geoff Whittle that led to a solution. IP6 Jim Geelen Interlacing Families and Kadison–Singer University of Waterloo [email protected] This talk will focus on the resolution of two conjectures that have been shown to be equivalent to open problems in a number of fields, most famously a problem due to Kadi- IP3 son and Singer concerning the foundations of mathematical Covering Systems of Congruences physics. This will include introducing a new technique for establishing the existence of certain combinatorial objects A distinct covering system of congruences is a collection that we call the ”method of interlacing polynomials.” The ai mod mi,1

regular tournament has a Hamilton decomposition. We Jozsef Balogh prove this conjecture for large tournaments. In fact, we University of Illinois prove a far more general result, which has further appli- [email protected] cations. Another example is the Hamilton decomposition conjecture, which was posed by Nash-Williams in 1970. It states that every regular graph on n vertices with mini- CP1 mum degree at least n/2 has a Hamilton decomposition Percolation in the Weighted Random Connection (joint work with Bela Csaba, Allan Lo, Deryk Osthus and Model Andrew Treglown). We propose the weighted random connection model, with Daniela Kuhn the goal to predict and better understand the spread of in- University of Birmingham fectious diseases. Each individual of a population is charac- [email protected] terized by two parameters: its position and risk behavior, where the probability of transmissions among individuals increases in the individual risk factors, and decays in their IP8 Euclidean distance. Our main result is on the almost sure The Complexity of Graph and Hypergraph Expan- existence of an infinite component in the weighed random sion Problems connection model, which corresponds to the outbreak of an infectious disease. We use results on the random connec- The expansion of a graph is a fundamental parameter with tion model and site percolation in Z2. many connections in mathematics and applications in computer science. Two basic variants of expansion are edge- Milan Bradonjic expansion (minimum ratio of the number of edges leav- Mathematics of Networks and Communications ing a subset of vertices to the total number of edge inci- Bell Labs, Alcatel-Lucent dent to the subset) and vertex expansion (minimum ratio [email protected] of number of vertices adjacent to a subset to the size of the subset). Both are NP-hard to compute exactly and CP1 the current√ best polynomial-time algorithms provide either an O( log n)OPT√ solution these problems, or, for Degeneracy in R-Mat Generated Graphs edge-expansion, an O( OPT ) solution. The latter is via Cheeger’s inequality relating edge expansion to the second The degeneracy of a graph G is the minimum possible value smallest eigenvalue of the graph Laplacian, and provides of d such that every subgraph of G has a vertex with dean efficient method to check if a graph is an edge expander gree at most d. It is known that the degeneracy of an (i.e., has edge expansion is at least some constant). In this Erd˝os-Rényi random graph is bounded with high probabil- talk, we survey recent developments on the following lines: ity. We examine degeneracy in the popular recursive ma- 1. What is the analog of expansion for partitioning a graph trix (R-MAT) random graph model, of which Erd˝os-Rényi into multiple parts (more than 2), and how does Cheeger’s is a special case, and determine whether its asymptotic inequality generalize? 2. Is there a Cheeger-type inequal- behavior parallels that of Erd˝os-Rényi graphs. ity for vertex expansion? How to efficiently verify whether Alex Chin a graph is a vertex expander? 3. Do there exist algorithms North Carolina State University with better performance? 4. How do these parameters, [email protected] inequalities and algorithms extend to hypergraphs?

Santosh Vempala Timothy Goodrich Georgia Institute of Technology Valparaiso University [email protected] [email protected]

Blair Sullivan SP1 North Carolina State University 2014 Dnes Knig Prize Lecture-The number of Ks,t- [email protected] free graphs

The problem of enumerating H-free graphs with a given CP1 number of vertices has been studied for almost forty years. Identifying Codes and Searching with Balls in As a result, a great deal is known about this problem. Graphs When H is not bipartite, we have not only good estimates of the number of H-free graphs, but also fairly precise We address the following search theory problem: what is structural characterisation of a typical such graph. On the the minimum number of queries that is needed to deter- contrary, not much is known in case when H is bipartite. mine an unknown vertex in a graph G if queries are balls A few years ago, we proved asymptotically optimal upper with respect to the graph distance and the answer to a bounds on the number of graphs not containing a fixed query is YES if the unknown vertex lies in the ball and complete bipartite graph Ks,t. This was the first infinite NO otherwise. We study adaptive or non-adaptive versions family of bipartite graphs for which such a bound was es- of this problem for the following classes of graphs: hyper- tablished. Since then, the method developed in our proof cubes, the Erd˝os-Rényi random graph model and graphs has found several interesting applications to a large class of bounded maximum degree. of related problems. Younjin Kim Wojciech Samotij KAIST Tel Aviv University [email protected] School of Mathematical Sciences [email protected] Mohit Kumbhat 38 DM14 Abstracts

Sungkyunkwan University that if n>rk,thenar(n, r, k)=ex(n, r, k−1)+2. We prove Suwon, South Korea this conjecture for k =2,k= 3, and suﬃciently large n,as [email protected] well as provide upper and lower bounds.

Zoltan Nagy Lale Ozkahya Renyi Institute of Math Hacettepe University Budapest, Hungary [email protected] [email protected] Michael Young Balazs Patkos Iowa State University Renyi Institute of Math, Budapest, Hungary [email protected] [email protected]

Alexey Pokrovskiy CP1 Mathematics department De-Preferential Attachment Random Graphs Freie Universitat Berlin, Germany [email protected] We consider a random graph sequence where a new vertex prefers attachment to a vertex with low degree. We Mate Vizer call such a sequence a “de-preferential attachment random Renyi Institute of Math, Budapest, Hungary graph”. We consider two models, “inverse de-preferential” [email protected] and “linear de-preferential”. If each new vertex comes with√ exactly one half-edge, the degree of a ﬁxed vertex is Θ( log n) for the inverse de-preferential case and Θ(log n) CP1 forthelinearcase.Weshowthatfortheinversede- Hyperbolicity of Random Intersection Graphs preferential model the tail of the limiting degree distribution is faster than exponential while that for the linear Graphs are a popular way to model a wide range of ap- 1 de-preferential model is exactly the Geometric 2 distri- plications, including rapidly-growing data drawn from so- bution. If each new vertex comes with m>1 half-edges, cial and information sciences. There is a large body of similar asymptotic results hold for ﬁxed vertex degree in empirical evidence suggesting that complex networks have both models. ”tree-like” properties at intermediate to large size-scales (e.g., hierarchical structures in biology, hyperbolic routing Subhabrata Sen in the Internet, and core-periphery behavior in social net- Stanford University works). We look the hyperbolicity of a promising model [email protected] for real world networks: the random intersection graph. Antar Bandyopadhyay Matthew Farrell Theoretical Statistics and Mathematics Unit Department of Mathematics Indian Statistical Institute, Delhi and Kolkata Cornell University [email protected] [email protected]

Timothy Goodrich CP2 Department of Mathematics and Computer Science Valparaiso University A Predictive Model for Gis [email protected] Any map in an Euclidean plane may be represented by one graph, as follows. A nation is represented by a node. If Nathan Lemons two nations are adjacent; i.e., if they share a common bor- Theoretical Division der (other than a single point), then they are joined by an Los Alamos National Laboratory edge. We assume that nations are contiguous; i.e., that a [email protected] nation may not be split up into smaller unconnected areas. If we want to predict the value for a given statistic that Blair Sullivan varies cyclically, and depends upon geographical location Department of Computer Science for a given sample, we have to determine ﬁrst its index of North Carolina State University incidence, and its index of prevalence. By performing el- blair [email protected] ementary alghebraic calculations, based on our graph, we will arrive at an extimate of the nex indexes of incidence and prevalence, respectively. However, this calculation will CP1 not be the same for every situation; and will vary accord- Anti-Ramsey Number of Matchings in Hyper- ing to the sample size, and the corresponding indexes of graphs incidence and prevalence. A k-matching in a hypergraph is a set of k edges such that no two edges intersect. The anti-Ramsey number of Jorge Diaz-Castro a k-matching in a complete r-uniform hypergraph H on University of P.R. School of Law n vertices, denoted by ar(n, r, k), is the smallest integer c University of P.R. at Rio Piedras such that in any coloring of the edges of H with exactly [email protected] c colors, there is a k-matching whose edges have distinct colors. The Tur´an number, denoted by ex(n, r, k), is the maximum number of edges in an r-uniform hypergraph on CP2 n vertices with no k-matching. For k ≥ 3, we conjecture Statistical and Hierarchical Graph Analysis for Cy- DM14 Abstracts 39

ber Security [email protected]

Motivated by the need for more resilient networks, we de- CP2 velop methods to understand cyber data through graph analysis. We model network traffic as a graph connecting Online Streaming of Intersection Graphs IP addresses when there is communication between them. Using entropy based measures on vertices we identify when We will present an online-streaming algorithm for inter- there is a significant change in behavior. Additionally, section graph processing. The graphs are generated as a we study these graphs hierarchically to quickly obtain es- stream from VLSI layout data comprising of 10e12 congru- timates for measurable quantities on graphs such as the ent shapes. The intersection graph of such a large number shortest path length between two IPs. congruent shapes has many interesting theoretical properties which can be efficiently exploited for producing efficient algorithms and techniques. Real world problems motivated Emilie Hogan from layout analysis and verifications will be presented. Pacific Nortwest National Lab [email protected] Sandeep Koranne Mentor Graphics Corporation Sutanay Choudhury, Peter Hui, Cliff Joslyn sandeep [email protected] Pacific Northwest National Laboratory [email protected], [email protected], cliff[email protected] CP2 Accumulation Behavior and Product Forecasting

CP2 Patterns are observed from buyers buying and accumulat- ing products. These patterns can be used in Cross Sell- The Balance Optimization Subset Selection (boss) ing and Inventory Management analysis. We introduce a Model for Causal Inference Markov Chain variant to model the accumulation behavior of a population and provide a forecasted confidence set Matching is used to estimate treatment effects in observa- of items to be accumulated in a future period which may tional studies. One mechanism for overcoming its restric- differ in length from the derived period. In non-sequential tiveness is to relax the exact matching requirement to one accumulation it will be established that the eigenvalues of of balance on the covariate distributions for the treatment this model are always real. and control groups. The Balance Optimization Subset Se- Thomas Morrisey lection (BOSS) model is introduced to identify a control Infosys group featuring optimal covariate balance. This presenta- [email protected] tion discusses the relationship between the matching and BOSS model, providing a comprehensive picture of their relationship. CP2 Various Notions and Generalizations of Alge- Sheldon H. Jacobson braically Coded Security System with Applications University of Illinois to Security and Privacy Dept of Computer Science [email protected] In this paper, we present new results on algebraically coded security system with their applications to security and pri- Jason Sauppe vacy of data. Moreover, we give various notions and gener- University of Illinois alizations of algebraic coding. Lastly, we provide practical [email protected] examples with applications to data security.

Edward Sewell Benard O. Nyaare Southern Illinois University Edwardsville JARAMOGI OGINGA ODINGA UNIVERSITY OF [email protected] SCIENCE AND TECHNOLOGY STUDENT [email protected] CP2 CP3 Qubit Unextendible Product Bases Aliens Vs Zombies: An Introduction To Bipartite Dot Product Graphs Qubit unextendible product bases are a useful tool in the theory of quantum entanglement. We show that the exis- A bipartite dot product representation of a bipartite graph tence of qubit unextendible product bases can be rephrased G with V (G)=X ∪ Y is a function f : V (G) → IRk such as a question of whether or not there exists an edge col- that for vertices x ∈ X and y ∈ Y , xy ∈ E if and only oring of the complete graph with certain properties. We if f(x)T f(y) ≥ 1. We illustrate how these representations use this characterization to find the minimum size of qubit may be used to illuminate structure in social and biological unextendible product bases and to find all of them on four networks, especially where there are specified sets within or fewer qubits. which relationships are ignored. Aliens and Zombies, for example. We will also characterize the (bipartite) graphs NathanielD.Johnston with a representation f : V (G) → IR1; i.e., the graphs Institute for Quantum Computing with bipartite dot-product-dimension 1, and present results University of Waterloo for bipartite graphs with bipartite dot-product-dimension 40 DM14 Abstracts

k>1. [email protected], [email protected], guol- [email protected] Sean Bailey Utah State University Department Of Mathematics CP3 [email protected] Maps on Surfaces and Matroids

David E. Brown Maps on surfaces are graphs embedded onto closed ori- Utah Sate University entable surfaces such that the complement of the graph [email protected] is a disjoint union of topological disks. To such graphs a bond matroid can be associated and also a slightly more general structure: a symmetric or Lagrangian matroid. I CP3 will discuss some properties of Lagrangian matroids (such as duality and representability) and if time permits, intro- The Abstract Algebra of Big Data and Associative duce a more general structure: the Coxeter matroid. Arrays Goran Malic Spreadsheets, databases, hash tables and dictionaries; School of Mathematics these are the fundamental building blocks of big data stor- University of Manchester age, retrieval and processing. The MIT Dynamic Dis- [email protected] tributed Dimensional Data Model (D4M) is an interface to databases and spreadsheets that allow developers to write analytics in the language of linear algebra, significantly re- CP3 ducing the time and effort required. The core of D4M are On the Type(s) of Minimum Size Subspace Parti- associative arrays that allow big data to be represented as tions large sparse matrices. This sparse matrix D4M schema has been widely adopted in a number of domains. We explore Let V = V (kt + r, q) be a vector space of dimension kt + r the mathematics behind the D4M system by formulating over the finite field with q elements. Let σq(kt+r, t)denote an axiomatic definition for associative arrays in terms of the minimum size of a subspace partition P of V in which semi-modules over semi-rings, then exploring the algebraic t is the largest dimension of a subspace. We denote by ndi properties of the latter. We are primarily concerned with the number of subspaces of dimension di that occur in P n n the linear systems theory and spectral theory of a three d1 dm P and we say [d1 ,...,dm ]isthetype of . We show that particular families of semi-rings. We present a structure a partition of minimum size has a unique partition type if theorem for the solutions of linear systems of an important t + r is an even integer and we give partial results for the family of semi-rings. more intricate case when t + r is an odd integer.

Jeremy Kepner Esmeralda L. Nastase MIT Lincoln Laboratory Xavier University [email protected] [email protected]

Julian Chaidez MIT CP3 [email protected] Necessarily Flat Polytopes

Call a d-dimensional polytope P necessarily flat if every CP3 embedding of its (d − 1)-skeleton is affinely d-dimensional. The Global Invariant of Graphic Hyperplane Ar- For example, a connected sum of polytopes cannot be nec- rangements essarily flat, whereas every cone over a polytope is necessarily flat. Such polytopes are instrumental in Richter- The fundamental group of the complement of a hyperplane Gebert’s proof of universality for 4-dimensional convex arrangement in a complex vector space is an interesting polytopes. We demonstrate strong evidence of a conjecture and complicated invariant. The third rank of successive concerning necessarily flat polytopes in the case d =3. quotients in the lower central series of the fundamental Joshua B. Tymkew group was called the global invariant of the arrangement Western Michigan University by Falk. Falk gave a general formula to compute the global [email protected] invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the CP3 number of cliques with three or four vertices in the graph On the Number of Bases in a Matroid with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements, but their ar- Given a triple of integers (n, r, b), 0

Qiumin Guo, Wentao Hu, Ling Guo Wing Hong Tony Wong Beijing university of chemical technology Kutztown University of Pennsylvania DM14 Abstracts 41

[email protected] [email protected]

Sin Tsun Edward Fan California Institute of Technology CP4 [email protected] Strong Double Graphs: Spectral Properties, En- ergy and Laplacian Energy

CP4 AgraphG is a finite, undirected, simple graph with m ∞ n vertices and m edges having vertex set V (G)= On the Matrix Sequence {Γ(A )}m for a Boolean =1 {v1,v2, ···,vn}. The adjacency matrix A =(aij )ofG Matrix A Whose Digraph Is Linearly Connected is a (0, 1)-square matrix of order n whose (i, j)-entry is equal to one if vi is adjacent to vj and equal to zero, oth- In this talk, we study the convergence of the matrix se- m ∞ erwise. The spectrum of the adjacency matrix is called the quence {Γ(A )}m for a matrix A ∈Bn the digraph =1 A-spectrum of G.Ifλ1,λ2, ···,λn is the adjacency spec- of which is linearly connected. We completely character- n | | m ∞ trum of G, the energy of G is defined as E(G)= i=1 λi . ize A for which {Γ(A )}m converges and find its limit. =1 Let D(G)=diag(d1,d2, ···,dn) be the diagonal matrix We go further to characterize A for which the limit of associated to G,wheredi is the degree of vertex vi.The { m }∞ Γ(A ) m=1 is a J block diagonal matrix. All of these re- matrices L(G)=D(G)-A(G)andQ(G)=D(G)+A(G)are sults are derived by studying the m-step competition graph respectively called Laplacian and signless Laplacian ma- of the digraph of A. trices and their spectras are respectively called Laplacian spectrum (L-spectrum) and signless Laplacian spectrum Jihoon Choi (Q-spectrum) of G.If0=μn ≤ μn−1 ≤ ··· ≤ μ1 and Seoul National University, South Korea ≤ + ≤ + ≤ ··· ≤ + 0 μn μn−1 μ1 are respectively the L- Department of Mathematics Education spectrum and Q-spectrum of G, the Laplacian energy of a [email protected] n | − 2m | graph G is defined as LE(G)= μi n .Foragraph i=1 Suh-Ryung Kim G with vertex set V (G)={v1,v2, ···,vn}, the strong dou- Seoul National University ble graph SD(G) is a graph obtained by taking two copies [email protected] of G and joining each vertex vi in one copy with the closed neighbourhood N[vi]=N(vi) ∪{vi} of corresponding vertex in another copy. We study some basic properties like CP4 the bipartiteness, Hamiltoniancity, strong regularity, chro- Forcing Sets in Self-Folding Origami matic number and spectra of the graph SD(G) together with its energy and Laplacian energy. As an application, We describe some of the combinatorial geometry problems we obtain some new families of equienergetic and Lapla- based on microscale self-folding polymer gels, especially cian equienergetic graphs, an infinite family of graphs G the concept of a forcing set: a subset of the creases whose for which LE(G)

of a base and a set of digits has a long history, new ques- [email protected], tions arise even in the binary case - digits 0 and 1. A binary [email protected], saman- positional number system (binary√ radix system) with base tha.graff[email protected], [email protected] equal to the golden ratio (1 + 5)/2isfairlywellknown. The main result of this paper is a combinatorial characterization of all binary radix systems for the real numbers. CP5 Hamiltonian Cycles in {1, 4}-Leaper Graphs Andrew Vince Department of Mathematics A knight can travel to every square of a standard 8 × 8 University of Florida chessboard exactly once and return to the starting square, avince@ufl.edu a tour which corresponds to a Hamiltonian cycle in a graph with 64 vertices and with edges defined by the knight’s le- gal moves. Many variations of this well-known Knight’s CP5 Tour problem have been studied, including boards of var- Cycle Extendability of Chordal Graphs and Tour- ious dimensions, boards on tori and other surfaces, and naments variations on the traditional knight’s move. In this talk, we present preliminary results about which n × m rectan- S is a subset of {1, 2,...,k}.AcycleC in a graph or di- gular boards admit a knight’s tour with the variation that graph is S-extendable if there is a cycle C∗ which contains the knight moves 1 square either vertically or horizontally the vertices of C and is of length i greater than C for some and 4 squares in the perpendicular direction, that is, a i ∈ S. A graph or digraph is S-cycle-extendable if every {1, 4}-leap. cycle is S-extendable. We prove that Hamiltonian chordal graphs (graphs with a Hamiltonian cycle and no induced Grady Bullington cycle of length greater than three) and strong tournaments University of Wisconsin Oshkosh (orientations of complete graphs in which there is a path bullington@uwosh from x to y and from y to x for every pair of distinct vertices x and y)are{1, 2}-cycle-extendable. Also, in the case Linda Eroh of tournaments, we characterize those regular and nearly Department of Mathematics regular tournaments which are not {1}-extendable. We University of Wisconsin at Oshkosh present and suggest directions for future research on S- [email protected] cycle-extendability. Steven J. Winters David E. Brown University of Wisconsin Oshkosh Utah Sate University [email protected] [email protected]

LeRoy Beasley, Deborah Arangno CP5 Utah State University Near Perfect Matchings in k-uniform Hypergraphs [email protected], [email protected] Let H be a k-uniform hypergraph on n vertices where n is a sufficiently large integer not divisible by k.Weprovethat CP5 if the minimum (k − 1)-degree of H is at least n/k ,then Loose Cycles in 3-Uniform Hypergraphs H contains a matching with n/k edges. This confirms aconjectureofRödl, Ruciński and Szemerédi, who proved Let H be a 3-uniform hypergraph. In this talk, we that the minimum (k − 1)-degree n/k + O(log n)suffices. will discuss sufficient conditions for minimum co-degree More generally, we show that H contains a matching of of H which guarantee the existence of a spanning sub- size d if its minimum codegree d is less than n/k,whichis hypergraph of H consisting of independent copies of a loose also best possible. This extends a result of Rödl, Ruciński cycle. In particular, using the stability approach, we give and Szemerédi. a tight condition for the existence a Hamilton loose cycle. Jie Han Andrzej M. Czygrinow Georgia State University Arizona State University [email protected] [email protected]

CP5 CP5 Nonempty Intersection of Longest Paths in Series- On 2-Factors with a Bounded Number of Odd Cy- Parallel Graphs cles In 1966 Gallai asked whether all longest paths in a con- In this talk we consider conditions that guarantee that a nected graph have nonempty intersection. Even though 2-factor in a graph has a bounded number of odd cycles. this is not true in general, the answer to Gallais question is In particular, we consider various conditions on claw-free positive for several well-known classes of graphs, e.g. out- graphs, extending a result from Ryj´aˇcek, Saito and Schelp. erplanar graphs, split graphs, and 2-trees. We present a We also consider conditions that ensure the existence of a proof that all series-parallel graphs (i.e. connected K4- pair of disjoint 1-factors in a claw-free graph, as the union minor free graphs) have a vertex that is common to all of of such a pair is a 2-factor with no odd cycles. its longest paths.

Jennifer Diemunsch, Michael Ferrara, Samantha Graﬀeo, Carl Georg Heise Timothy Morris Technische Universit¨at Hamburg-Harburg University of Colorado Denver Germany DM14 Abstracts 43

[email protected] Rank-width at Most k

Julia Ehrenmueller Linear rank-width is a graph width parameter, which is a Hamburg University of Technology variation of rank-width by restricting its tree to a caterpil- [email protected] lar. It is known for each k, there is a ﬁnite obstruction set Ok of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomor- Cristina Fernandes phic to a graph in Ok. However, no attempts have been University of So Paulo made to bound the number of Ok for k ≥ 2. We show [email protected] k that for each k,thereareatleast2Ω(3 ) pairwise locally non-equivalent graphs in Ok. To prove this theorem, we CP5 generalize a theorem of Bouchet, which states that if two Cycle Decompositions with a Certain Degree of trees are locally equivalent, then they are isomorphic, into Symmetry block graphs without simplicial vertices.

A cycle decomposition of a graph Γ is a set C of cycles O-Joung Kwon whose edges partition the edge-set of Γ; partitioning C into Department of Mathematical Sciences, KAIST factors of Γ results in a 2-factorization. Cycle decomposi- [email protected] tions having a certain automorphism group with (possibly) a prescribed action on the vertex-set or on the edge-set Sang-Il Oum, Jisu Jeong have attracted much attention. In this talk I will select KAIST some of the most recent results on this topic. [email protected], [email protected]

Tommaso Traetta Dipartimento di Matematica e Informatica, CP6 Universita degli Studi di Perugia N-Flips in 4-Connected Eulerian Triangulations on [email protected], [email protected] the Sphere

It is known that any two Eulerian triangulations on the CP6 sphere with the same tripartition can be transformed into New Characterizations of Proper Interval Bigraphs each other by N-flips. In this paper, we shall prove that any two 4-connected Eulerian triangulations on the sphere In this paper, we define astral triple of edges and charac- with the same tripartition can be transformed into each terize proper interval bigraphs via absence of astral triple other by N-flips, preserving the 4-connectivity, if they are of edges. We also characterize proper interval bigraphs in not isomorphic to the exception. terms of dominating pair of vertices as defined by Corneil et al. Sen and Sanyal characterized biadjacency matrix of a Naoki Matsumoto proper interval bigraph in terms of monotone consecutive Research Fellow of Japan Society arrangement(MCA). We have shown an interrelation be- for the Promotion of Science, Yokohama National tween MCA and circularly compatible 1’s of binary matrix University as defined by Tucker. [email protected]

Ashok K. Das Dept. of Pure Mathematics, University of Calcutta. India CP6 [email protected] Characterizing Graph Classes Using Twin Vertices in Regular Induced Subgraphs

CP6 A graph is chordal and only if every two vertices are Analytical Properties of Horizontal Visibility twins in every connected regular induced subgraph (delet- Graphs Generated by the Feigenbaum Unversal ing “connected’ characterizes the split graphs). Conjec- Route to Chaos in Discrete Models ture: A graph is weakly chordal if and only if twin vertices exist in every nontrivial regular induced subgraph (equiva- This paper establishes some analytical properties of Hori- lently with “connected’ inserted). I describe the new class zontal Visibility Graphs along the lines of Feigenbaum pe- of graphs in which every vertex has a twin in every non- riod doubling route from regular system to chaos, time trivial regular induced subgraph (equivalently with “con- series analysis, Lyapunov exponents and fractal dimen- nected’ inserted). sions in discrete models like Rickers Population model : f(x)=xe(r(1−x/k)),wherer is the control parameter and k Terry McKee is the carrying capacity. This also leads us to build bridges Wright State University between time series analysis, nonlinear dynamics and graph Department of Mathematics & Statistics theory in chaotic discrete systems. [email protected]

Tarini K. Dutta PROFESSOR OF MATHEMATICS, GAUHATI CP6 UNIVERSITY, GUWAHATI, Weak Universality for 2-Dimensional Parallel ASSAM STATE: 781014, INDIA Drawings [email protected] Let G be a d-regular graph equipped with a d-edge- coloring. A parallel drawing of G is a 2-dimensional draw- CP6 ing of G such that edges sharing a common color are mutu- Excluded Vertex-minors for Graphs of Linear ally parallel (disregarding crossings). It is an open problem 44 DM14 Abstracts

to ﬁnd a graph-theoretic characterization of graphs which [email protected] have such a drawing in which the edges are represented by distinct lines. The purpose here is to demonstrate that in the case when d = 4, there is an “algebraic’ obstruction for CP7 the classiﬁcation of such graphs. Chromatic-Choosability of the Power of Graphs

The kth power Gk of a graph G is the graph defined on David Richter k V (G) such that two vertices u and v are adjacent in G if Western Michigan University the distance between u and v in G is at most k. A graph [email protected] H is called chromatic-choosable if χl(H)=χ(H). A natural question raised by Xuding Zhu is whether there exists a constant integer k such that Gk is chromatic-choosable CP6 for every graph G. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) asked whether Almost Isometric Drawings of Graphs G2 is chromatic-choosable for every graph G. In 2013, Kim and Park answered the Kostochka and Woodall’s question Every connected undirected (not necessarily finite) graph in the negative by finding a graph G such that G2 is a G induces a metric space (V,d)onitsvertexsetV.For complete multipartite graph. In this paper, we answer an injective mapping f from V into the Euclidean plane Zhu’s question by showing that for every integer k ≥ 2, let the distortion of f be the smallest number q such that there exists a graph G such that Gk is not chromatic- 1 ≤ d(x,y) ≤ q d f x ,f y q for all vertices x = y of G,whered2 choosable. Moreover, for any fixed k we show that the 2( ( ) ( )) k k denotes the Euclidean distance. In the talk we study upper value χl(G ) − χ(G ) can be arbitrarily large. bounds on q in terms of the structure of G. Seog-Jin Kim Konkuk University Jens Schreyer, Thomas Boehme [email protected] Technical University Ilmenau Germany Young Soo Kwon [email protected], thomas.boehme@tu- Yeungnam University, Korea ilmenau.de [email protected]

Boram Park CP7 National Institute for Mathematical Sciences, Korea Colorings with Fractional Defect [email protected]

Consider a coloring of a graph such that each vertex is as- CP7 signed a fraction of each color, with the total amount of Equitable Coloring of Graphs with Intermediate colors at each vertex summing to 1. We define the frac- Maximum Degree tional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the AgraphG is equitably k-colorable if the vertex set of G graph to be the maximum of the defects over all vertices. can be properly colored with k colors such that the sizes of Note that this coincides with the usual definition of de- any two color classes differ by at most one. We show that fect if every vertex is monochromatic. We provide results a connected G is equitably Δ(G)-colorable when (|G| + on the minimum fractional defect of 2-colorings of some 1)/3 ≤ Δ(G) < |G|/2, where |G| and Δ(G)denotethe graphs. order and the maximum degree of G, respectively.

Wayne Goddard Ko-Wei Lih School of Computing Institute of Mathematics, Academia Sinica, Taipei Clemson University [email protected] [email protected] Bor-Liang Chen Honghai Xu Department of Business Administration Clemson University National Taichung University of Science and Technology [email protected] [email protected]

Kuo-Ching Huang Department of Financial and Computational Mathematics CP7 Providence University Coloring Planar Digraphs [email protected]

A conjecture of Neumann-Lara (1985) states that the ver- CP7 tex set of any planar oriented graph G can be 2-colored A Distance-Two Coloring with Application to such that each color class induces an acyclic subdigraph. Wireless Sensor and Actor Networks We show that the conjecture holds when one further as- sumes that G is of digirth ﬁve, and discuss some related Navarra et al. in 2010 proposed a new virtual infrastruc- results. This is joint work with Bojan Mohar. ture for the coarse-grained localization of wireless sensor and actor networks, and proposed distance-two coloring for Ararat Harutyunyan the adjacency graph of their virtual infrastructure, named Mathematical Institute, University of Oxford G. Navarra et al. proposed optimal coloring for G4, G5, DM14 Abstracts 45

and G3·2x , x ≥ 0. Our main contribution of this work is to [email protected] propose optimal colorings for all the remaining G’s.

CP8 Wu-Hsiung Lin, Well Chiu, Yen-Cheng Chao, Chiuyuan Chen On the Centralizing Number of Graphs Departement of Applied Mathematics, National Chiao Tung University Hedetniemi observed that every graph G is the center of some graph. More formally, for every graph G there is a [email protected], [email protected], ∼ [email protected], [email protected] graph H such that C(H) = G,whereC(H) is the subgraph of H induced by the vertices in the center of H. Buckley, Miller and Slater introduced the centralizing number A(G) of a graph G, which is the minimum number of CP7 vertices that must be added to G to form a supergraph ∼ H where C(H) = G. Hedetniemi’s original construction Steinberg’s Conjecture, the Bordeaux Coloring ≤ Conjecture and Near-Coloring showed that A(G) 4 for any graph G, and it is straight- forward to see A(G) = 1. We investigate the distribution of graphs with respect to A(G)asthenumbern of ver- An important result in the theory of graph coloring is tices increases, showing that the number of graphs with Grotzsch’s theorem, which states that every triangle-free fixed centralizing number is growing with n.Ourresults planar graph is 3-colorable. A famous related question is derive from a closer examination of the centralizing num- due to Steinberg and states that any planar graph with- ber, including the effect on this parameter from cloning out 4- or 5-cycles is 3-colorable. In this talk, we will dis- and twinning vertices. cuss some of the recent progress made towards proving Steinberg’s conjecture and discuss joint work with Ken- Sarah L. Behrens ichi Kawarabayashi that planar graphs with no 5-cycles, University of Nebraska - Lincoln 6-cycles or intersecting triangles are 3- colorable. In ad- [email protected] dition, we discuss recently completed senior thesis work based on near-coloring with Kyle Yang. Stephen Hartke University of Nebraska-Lincoln Carl Yerger [email protected] Department of Mathematics Davidson College [email protected] CP8 Deciding the Bell Number for Hereditary Graph Kyle Yang Classes Stanford University [email protected] The speed of a hereditary graph class is the number of (la- belled) graphs on {1, 2,...,n} in the class, as a function of n. It is known that not every function can be obtained as the speed of some such class; e.g., if the speed grows CP8 faster than any polynomial, than it is at least exponential. A Result on Clique Convergence for Join of Graphs Another such jump was identified by Balogh–Bollobás– Weinreich (2005): if the speed is at least n(1−o(1))n,thenit is bounded from below by the nth Bell number (the num- Let G be a (p, q)-graph and KG be the set of all maximal cliques of G, then the K-graph (clique graph) operator of ber of partitions of an n-element set). We study the computational problem to decide for a given hereditary class G denoted by K(G) is the graph with vertex set KG and whether its speed is below or above* the Bell number. We two elements Qi,Qj ∈KG form an edge if and only if 0 provide an algorithm that solves this problem for classes Qi ∩ Qj = ∅. We define iterated clique graphs by K (G)= G,andKn(G)=K(Kn−1(G)) for n>0. We study the described by a finite list of forbidden induced subgraphs. It is based on a characterisation of minimal classes with speed dynamical behavior of some simple graphs G = G1 + G2 under the iterated application of the clique graph operator above* the Bell number. * By “above’ I mean “greater than K. We determine the number of cliques in K(G)whenG = or equal to’. G1+G2. Also, we prove a necessary and sufficient condition Jan Foniok for clique graph K(G)whenG = G1 + G2 is complete, i.e., Queen’s University let G = G1 +G2 such that K(G) is a complete graph if and [email protected] only if either K(G1)orK(G2) is a complete graph. Further we gave the characterization for convergence of the join of graphs using clique graph operator. Aistis Atminas, Andrew Collins, Vadim Lozin University of Warwick [email protected], [email protected], V. V. P. R. V. B. Suresh Dara [email protected] Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, CP8 INDIA Modified Regular Line Digraphs for Optimal Con- [email protected] nectivity and Small Diameters

S. M. Hegde Regular digraphs of optimal connectivity and O(log n) di- Department of Mathematical and Computational Sciences ameter are designed from a seed by recursively generating National Institute of Technology Karnataka, Surathkal, modiﬁed line graphs based on n and d. To our knowledge, INDIA this result is the best known in literature for this class of 46 DM14 Abstracts

graphs. Shortest paths are obtained in O(log n) steps with Kuwait University and without the presence of faulty nodes. Applications [email protected] are to fault tolerant networks, low diameter degradation, rerouting for transient faults, and modifications for perma- Harun Aydilek, Asiye Aydilek nent faults. Gulf University for Science and Technology [email protected], [email protected] Prashant D. Joshi Intel Corporation [email protected] CP9 Said Hamdioui Finding the Minimum Value of a Vector Function Delft University of Technology Using Lipschitz Condition [email protected] Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find CP8 the global minimum of a function over a compact domain. Some Problems on Paths and Cycles These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this In 1976, Thomassen conjectured that every longest cycle paper, we consider Lipschitz functions of order α and give in a 3-connected graph has a chord. In 1979, Kotzig con- a new minimization condition for the exclusion algorithm. jectured that there exists no graph in which each pair of Furthermore, convergence and complexity results are pre- vertices is connected by a unique path of length k (k ≥ 3). sented for such algorithm. In this talk, some results on these problems and related conjectures are summarized. Ibraheem Alolyan Chunhui Lai King Saud University Zhangzhou Normal University [email protected] [email protected]; [email protected] CP9 CP8 On the Stable Matchings That Can Be Reached Signed Edge Domination Numbers of Complete When the Agents Go Marching in One by One Tripartite Graphs The Random Order Mechanism (ROM) is a sequential ver- The closed neighborhood NG[e]ofanedgee in a graph G is the set consisting of e and of all edges having an end-vertex sion of Gale and Shapley’s deferred-acceptance (DA) algorithm where agents are arriving one at a time, and each in common with e.Letf be a function on E(G), the edge {− } ≥ newly arrived agent has an opportunity to propose. Sur- set of G,intotheset 1, 1 .If x∈N[e] f(x) 1foreach ∈ prisingly, it is possible for ROM to output a stable match- edge e E(G), then f is called a signed edge domination ing that is different from the man-optimal and woman- function (SEDF) of G. The signed edge domination num- optimal stable matchings. In this talk, we consider the ber γ (G)ofG is defined as γs(G)=min{ f(e) | sk e∈E(G) following question: Given a stable matching μ of instance f is an SEDF of G}. In this presentation, we find the I, can ROM output μ? signed edge domination number for the complete tripartite ≤ ≤ ≤ graph Km,n,p,wherem n p and p m + n. Christine T. Cheng Abdollah Khodkar University of Wisconsin-Milwaukee Department of Mathematics, University of West Georgia [email protected] [email protected]

Arezoo Nazi Ghameshlou CP9 Department of Irrigation & Reclamation Engineering Characterization of Convex Sets Which Are Faculty Semidefinite Representable at Infinity University of Tehran [email protected] We characterize a convex set which is semidefinite representable (SDR) at ∞. We prove : a closed convex set is SDRiffitisSDRateachpointofK ∪{∞}. We define new CP9 class of sets which are SDR away from a point. We prove Scheduling Problem with Uncertain Processing :letφ ∈ K. K is SDR iff K and K0 both are SDR away Times from φ. Scheduling problem with uncertain processing times is ad- dressed with the objective of minimizing mean completion Anusuya Ghosh time. Some polynomial time heuristics, utilizing the lower PhD student, Industrial Engineering & Operations and upper bounds of processing times, are proposed. The Research, proposed and existing heuristics are compared by extensive Indian Institute of Technology Bombay. computational experiments. The computational results in- [email protected] dicate that the proposed heuristics perform significantly better than the existing heuristics in terms of both error Vishnu Narayanan and computation time. Assistant professor, Industrial Engineering and Operations Ali Allahverdi Research, Indian Institute of Technology Bombay DM14 Abstracts 47

[email protected] http://arxiv.org/abs/1308.2365 for the full manuscript.

Easton Li Xu CP9 University of Hong Kong Discrete Tropical Optimization Problems [email protected]

Multidimensional optimization problems are presented, Guangyue Han which are formulated and solved in the framework of trop- Department of Mathematics ical (idempotent) mathematics. The problems consist of University of Hong Kong minimizing nonlinear objective functions defined on finite- [email protected] dimensional semimodules over integer and rational semi- fields with idempotent addition, subject to linear inequality constraints. We offer explicit solutions to the problems CP10 in a compact vector form suitable for both further analy- On Minimal Partial Words with Subword Complex- sis and practical implementation. Application examples of ity 2n problems in operations research are provided to illustrate the results. Recently, it was shown that compressed binary De Bruijn sequences can be built using minimal partial words of sub- Nikolai Krivulin word complexity 2n.Moreprecisely,a2n-complex partial St Petersburg State Univ word of a positive integer order N has 2n distinct subwords Faculty of Math and Mechanics for all n =1, 2,...,N. Such a partial word is minimal if [email protected]; Nikolai [email protected] there are no shorter 2n-complex partial words also of order N. In this paper, we show how to construct minimal 2n- complex partial words of a given order N with a given num- CP9 ber of holes h. We provide some insights into the lengths Deriving Compact Extended Formulations via LP- of such partial words, as well as how many there are, as based Separation Techniques functions of h and N.Inthecasewhenh =0,wealso provide an algorithm to generate them. Furthermore, we Some combinatorial optimization problems can modeled discuss several conjectures and open problems towards the via Linear Programs with an exponential number of rows classification of the minimal 2n-complex partial words. and/or columns, which can be solved by generating rows Francine Blanchet-Sadri and columns via separation techniques. A crucial question University of North Carolina concerns the possibility of having alternative formulations [email protected] which require additional variables but only a polynomial number of inequalities. These formulations are called compact extended. We show how to derive compact extended Mark Heimann formulations if the separation problem can be solved by University of Washington in St. Louis a linear program. We illustrate the technique by provid- [email protected] ing examples of several known combinatorial optimization problems. Andrew Lohr, Katie McKeon Rutgers Giuseppe Lancia [email protected], [email protected] University of Udine [email protected] CP10 Paolo Serafini Keeping Avoider’s Graph Almost Acyclic University of Udine, Italy paolo.serafi[email protected] In this talk we consider biased (1 : b)Avoider-Enforcer games in the monotone and strict version. In particular, we show that Avoider can keep his graph acyclic for every CP9 but the last round of the game if b ≥ 200n log n. We hereby improve previous upper bounds on the threshold biases for The Minimum Number of Hubs Needed in Net- the non-planarity game, the non-k-colorability game, and works the Kt-minor game. Moreover, we give a slight improve- ment for the lower bound in the non-planarity game. A hub refers to a non-terminal vertex of degree ≥ 3. We study the minimum number of hubs needed in a network Julia Ehrenmueller to guarantee certain flow demand constraints imposed be- Hamburg University of Technology tween multiple pairs of sources and sinks. We prove that [email protected] under the constraints, regardless of the size of the network, such minimum number is always upper bounded and we derive tight upper bounds for some special parameters. In Dennis Clemens particular, for two pairs of sources and sinks, we present FU Berlin a novel path-searching algorithm, the analysis of which is [email protected] instrumental for the derivations of the tight upper bounds. Our results are of both theoretical and practical interest: Yury Person in theory, they can be viewed as generalizations of the clas- Universitaet Frankfurt sical Menger’s theorem to a class of undirected graphs with [email protected] multiple sources and sinks; in practice, our results, roughly speaking, suggest that for some given flow demand con- Tuan Tran straints, not too many hubs are needed in a network. See FU Berlin 48 DM14 Abstracts

[email protected] tions in partial words over any alphabet size called squares, which consist of two adjacent occurrences of compatible substrings. We establish tight upper and lower bounds on CP10 the number of occurrences of primitively-rooted squares in Minimal Partial Languages and Automata a given partial word of a fixed length and number of holes. These bounds have applications to the analysis of efficient Partial words are sequences of characters from an alpha- algorithms for detecting all repetitions in partial words. bet in which some positions may be marked with a “hole’ symbol, .Wecancreatea-substitution mapping this Francine Blanchet-Sadri symbol to a subset of the alphabet, so that applying such University of North Carolina a substitution to a partial word results in a set of full words [email protected] (ones without holes). This setup allows us to compress regular languages into smaller partial languages. Determinis- Jordan Nikkel tic finite automata for such partial languages, referred to Vanderbilt University as -DFAs, employ a limited non-determinism that can al- [email protected] low them to have lower state complexity than the minimal DFAs for the corresponding full languages. Our paper fo- James Quigley cuses on algorithms for the construction of minimal partial University of Illinois at Urbana-Champaign languages, associated with some -substitution, as well as [email protected] approximation algorithms for the construction of minimal -DFAs. Xufan Zhang Francine Blanchet-Sadri Princeton University University of North Carolina [email protected] [email protected] CP10 Kira Goldner Pattern Occurrence Statistics and Applications to Oberlin College the Ramsey Theory of Unavoidable Patterns [email protected] As suggested by Currie, we apply the probabilistic method Aidan Shackleton to problems regarding pattern avoidance. Using techniques Swarthmore College from analytic combinatorics, we calculate asymptotic pat- [email protected] tern occurrence statistics and use them in conjunction with the probabilistic method to establish new results about the Ramsey theory of unavoidable patterns in the full word CP10 case (both nonabelian sense and abelian sense) and in the The Relationships Between the Bernaulli Numbers partial word case. We suggest additional possible uses of B2n &An and the Euler Numbers E2n &Bn these data in applications such as cryptography and musi- cology. Traditionally, the values of Riemann zeta function at the even positive integers have been formulated in terms of Francine Blanchet-Sadri Bernoulli numbers B2n and the sums of the alternating University of North Carolina series of odd powers of the reciprocal of odd positive inte- [email protected] gers have been calculated in terms of Euler numbers E2n. However, the present author reproduced the sum of the Jim Tao same series using different procedures in terms of two ra- Princeton University tional numbers an and bn. In this paper, the relationships [email protected] between B2n &an and E2n &bn have been established. Consequently, some of the theorems and corollaries in [5] have been restated in terms of B2n and E2n. CP10 A Variation of the Instant Insanity Puzzle Mohammed R. Karim Department of Mathematics The Instant Insanity puzzle consists of four cubes whose Alabama A&M University, Normal, Alabama faces are colored with one of four colors: red, blue, green [email protected] and white. The object of the game is to stack the four cubes on top of one another so that each side of the stack has one face of each color. A variation of this puzzle will CP10 be presented in this talk. Tight Bounds on Number of Primitively-Rooted Squares in Partial Words Steven J. Winters University of Wisconsin Oshkosh Partial words have been subject of much recent investi- [email protected] gation, both combinatorial and algorithmic. They are sequences that may have undefined positions, called holes, that match, or are compatible with, any letter of the al- MS1 phabet over which they are defined. These sequences al- Cycle Packing low for data that is incomplete or corrupted. The known algorithms for detecting all repetitions in strings without In the 1960s, Erd˝os and Gallai conjectured that the edge holes cannot be easily extended to strings with holes, one set of every graph on n vertices can be partitioned into of the most important culprits being that compatibility O(n) cycles and edges. They observed that one can easily is not transitive. This paper deals with a type of repeti- get an O(n log n) upper bound by repeatedly removing the DM14 Abstracts 49

edges of the longest cycle. We make the first progress on [email protected] this problem, showing that O(n log log n)cyclesandedges suffice. We also prove the Erd˝os-Gallai conjecture for random graphs and for graphs with linear minimum degree. MS1 Proof of Two Conjectures of Thomassen on Tour- naments David Conlon University of Oxford We prove the following two conjectures of Thomassen on [email protected] highly connected tournaments: (i) For every k,thereis an f(k) so that every strongly f(k)-connected tournament Jacob Fox contains k edge-disjoint Hamilton cycles (joint work with MIT Kühn, Lapinskas and Patel). (ii) For every k,thereisan [email protected] f(k) so that every strongly f(k)-connected tournament has a vertex partition A, B for which both A and B induce Benjamin Sudakov a strongly k-connected tournament (joint with Kühn and Dept. of Mathematics Townsend). Our proofs introduce the concept of ‘robust UCLA dominating structures’, which will hopefully have further [email protected] applications. I will also discuss related open problems on cycle factors and linkedness in tournaments.

MS1 Daniela Kuhn, John Lapinskas, Deryk Osthus University of Birmingham A Refinement of the Ore-type Version of the [email protected], [email protected], Corrádi-Hajnal Theorem [email protected] This result is joint with H. Kierstead, T. Molla and E. Yea- ger. In 1963, Corrádi and Hajnal proved that for all k ≥ 1 Viresh Patel and n ≥ 3k,everyn-vertex graph G with minimum degree Queen Mary, University of London δ(G) ≥ 2k contains k disjoint cycles. The degree bound is [email protected] sharp. Enomoto and Wang independently proved the Ore- type refinement of the result: the same conclusion holds un- Timothy Townsend der the weaker assumpion that for all distinct nonadjacent University of Birmingham vertices x, y, d(x)+d(y) ≥ 4k−1. We describe for all k ≥ 4 [email protected] and n ≥ 3k the n-vertex graphs G with d(x)+d(y) ≥ 4k−3 for all distinct nonadjacent x, y that do not contain k disjoint cycles. In particular, the result describes all sharpness MS1 examples for the Corrádi-Hajnal Theorem. Packing Trees of Bounded Degree Alexandr Kostochka Motivated by a conjecture of Gyarfás and Lehel, recently University of Illinois, Urbana-Champaign Böttcher, Hladký, Piguet and Taraz showed that every col- [email protected] lection of trees can be packed into a complete graph with at (1 + o(1))n vertices, provided each tree has at most n vertices, bounded maximum degree, and all trees have at MS1 most n2/2 edges all together. We give an alternative proof Robustness of Graphs - Case Study: Dirac’s Theo- and discuss some extensions of this result. rem Mathias Schacht A typical result in graph theory can be read as following: University of Hamburg under certain conditions, a given graph G has some prop- [email protected] erty P. For example, a classical theorem of Dirac asserts that every n-vertex graph G of minimum degree at least Messuti Silvia n/2 is Hamiltonian, where a graph is called Hamiltonian Universit if it contains a cycle that passes through every vertex of [email protected] the graph. Recently, there has been a trend in extremal graph theory where one revisits such classical results, and P Vojtech Rödl attempts to see how strongly G possesses the property . Emory University In other words, the goal is to measure the robustness of [email protected] G with respect to P. In this talk, we discuss several measures that can be used to study robustness of graphs with respect to various properties. To illustrate these measures, MS2 we present several extensions of Dirac’s theorem. Electroid Varieties and a Compactification of the Choongbum Lee Space of Electrical Networks MIT We construct a compactification of the space of circu- cb [email protected] lar planar electrical networks studied by Curtis-Ingerman- Morrow and De Verdière-Gitler-Vertigan, using cactus net- Michael Krivelevich works. We embed this compactification as a linear slice of Tel Aviv University the totally nonnegative Grassmannian, and relate Kenyon [email protected] and Wilson’s grove measurements to Postnikov’s boundary measurements. Intersections of the slice with the positroid Benny Sudakov stratification leads to a class of electroid varieties, indexed ETH Zurich by matchings. The partial order on matchings arising from 50 DM14 Abstracts

electrical networks is shown to be dual to a subposet of from the perspective of a toggle group action on the alter- aﬃne Bruhat order. The analogues of matroids in this set- nating sign matrix poset, thereby simplifying some aspects ting are certain distinguished collections of non-crossing of their proof. partitions. Jessica Striker Thomas Lam North Dakota State University University of Michigan [email protected] [email protected]

MS3 MS2 Generating Random Graphs in Biased Maker- Asymptotics of Symmetric Polynomials with Ap- Breaker Games plications to Statistical Mechanics Models In this talk we present a general approach connecting bi- We develop methods for asymptotic analysis of normal- ased Maker-Breaker games and problems about local re- ized symmetric polynomials (Schur functions and other Lie silience in random graphs. Using this method, we investi- group characters). We show how these asymptotics, to- gate the threshold bias b for which Maker can win certain gether with some combinatorial interpretations, give the (1 : b) games. We utilize this approach to prove new re- distributions of certain observables in the scaling limit of sults about biased games and also to derive some known grid models – lozenge tilings, 6-vertex model with domain results in novel ways. We also present some related and wall boundary conditions (i.e. ASMs), the O(1) dense challenging open problems. loop model. Based on joint work with Vadim Gorin (MIT, IITP). Ferber Asaf ETH Zurich Greta Panova Switzerland UCLA [email protected] [email protected] Michael Krivelevich Dept. of Mathematics, Tel Aviv University MS2 [email protected] Title Not Available at Time of Publication

Abstract not available at time of publication. Humberto Naves ETH Zurich Pavlo Pylyavskyy [email protected] University of Minnesota [email protected] MS3 Picker-Chooser Fixed Graph Games MS2 Arctic Curves of the Octahedron Equation Given a fixed graph H and a positive integer n,aPicker- Chooser H-game is a biased game played on the edge set We study the octahedron relation, which solution corre- of Kn in which Picker is trying to force many copies of H sponds to the partition function for dimer coverings of the and Chooser is trying to prevent him from doing so. In this Aztec Diamond graph. We find exact solutions for a par- paper we conjecture that the value of the game is roughly ticular class of periodic initial conditions. We show that the same as the expected number of copies of H in the the density function, that measures average dimer occupa- random graph G(n, p) and prove our conjecture for special tion of a face of the Aztec graph, obeys a linear system cases of H such as complete graphs and trees. with periodic coefficients. We explore the thermodynamic limit of the dimer models and derive exact ”arctic” curves Malgorzata Bednarska-Bzdega separating various phases. Adam Mickiewicz University [email protected] Rodrigo A. Soto Garrido UIUC Dan Hefetz [email protected] University of Birmingham [email protected] Philippe Di Francesco Department of Mathematics Tomasz Luczak University of Illinois at Urbana-Champaign Adam Mickiewicz University [email protected] [email protected]

MS2 MS3 Rowmotion, Homomesy, and the Razumov- Does the Random Graph Intuition Help when Ori- Stroganov Ex-conjecture entations are Involved?

The Razumov-Stroganov ex-conjecture, an important link We study Maker-Breaker games involving orientations. between statistical physics and combinatorics, relates the Two players, say Red and Blue, claim and direct edges ground state eigenvector of the O(1) dense loop model to of the complete graph alternately. It is of interest to de- a reﬁned counting of fully-packed loops on a square grid, termine the largest k such that Red has a strategy so that which are in bijection with alternating sign matrices. We (1) the Red digraph contains a copy of a predeﬁned tour- interpret Cantini and Sportiello’s proof of this conjecture nament on k vertices, (2) the Red digraph contains a copy DM14 Abstracts 51

of every tournament on k vertices. We also discuss connec- [email protected] tions to the random graph intuition.

Dennis Clemens MS3 FU Berlin Walker vs Breaker [email protected] We introduce the concept of Walker-Breaker games on Heidi Gebauer graphs, a variant of Maker-Breaker games where Maker’s ETH Zurich edge selections are required to ”walk” around the underly- [email protected] ing graph in various ways. In particular, we will consider games where Walker’s goal is to span as large a subgraph Anita Liebenau as possible. University of Warwick [email protected] Lisa Espig Carnegie Mellon University [email protected]

MS3 Alan Frieze Carnegie Mellon University Maker-Breaker Games on Random Geometric Department of Mathematical Sciences Graphs [email protected]

In a Maker-Breaker game on a graph G, Breaker and Maker Michael Krivelevich alternately claim edges of G. Maker wins if, after all edges Dept. of Mathematics, Tel Aviv University have been claimed, the graph induced by his edges has [email protected] some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our Wesley Pegden four games we show that if we add edges between n points Courant Institute chosen uniformly at random in the unit square by order [email protected] of increasing edge-length then, with probability tending to one as n →∞, the graph becomes Maker-win the very mo- ment it satisfies a simple necessary condition. In particular, MS4 with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins σ-Polynomials and Their Roots the Hamilton cycle game as soon as the minimum degree Given a graph G of order n,theσ-polynomial of G is the is at least four; Maker wins the perfect matching game as i soon as the minimum degree is at least two and every edge generating function σ(G, x)= aix where ai is the num- has at least three neighbouring vertices; and Maker wins ber of partitions of the vertex set of G into i nonempty the H-game as soon as there is a subgraph from a finite list independent sets. Such polynomials arise in a natural way of “minimal graphs”. These results also allow us to give from chromatic polynomials, and have strong connections precise expressions for the limiting probability that G(n, r) to several other well known polynomials in combinatorics. isMaker-winineachcase,whereG(n, r) is the graph on n Brenti proved that σ-polynomials of graphs with chromatic points chosen uniformly at random on the unit square with number at least n − 2 had all real roots, and conjectured an edge between two points if and only if their distance is the same held for chromatic number n − 3. We affirm this at most r. conjecture.

Jason Brown, Aysel Erey Andrew J. Beveridge Dalhousie University Department of Mathematics, Statistics and Computer [email protected], [email protected] Science Macalester College [email protected] MS4

Andrzej Dudek Independent Set, Induced Matching, and Pricing: Department of Mathematics Connections and Tight (Subexponential Time) Ap- Western Michigan University proximation Hardnesses [email protected] We present almost settled inapproximability results for three fundamental problems. The ﬁrst one is the Alan Frieze subexponential-time inapproximability of the maximum in- Carnegie Mellon University dependent set problem, a question studied in the area of Department of Mathematical Sciences parameterized complexity. The second one is the hardness [email protected] of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last one is the Tobias Muller tight hardness of approximating the k-hypergraph pricing Utrecht University problem, a fundamental problem in the area of algorithmic Netherlands game theory. [email protected] Bundit Laekhanukit Milos Stojakovic School of Computer Science University of Novi Sad McGill University 52 DM14 Abstracts

[email protected] [email protected]

MS4 MS5 A Geometric Foundation to Information Networks Subgraphs in Random Non-uniform Hypergraphs

We analyze the geometric properties underlying informa- For a set of positive integers R := {k1,k2,...,kr} and r R tion networks modeled as large-scale finite graphs G = probabilities p =(p1,p2,...,pr) ∈ [0, 1] ,letG (n, p) (V,E), where the vertex set V defines the information lo- be the random hypergraph G on n vertices so that for cation and the edge set E labeled with their logical dis- 1 ≤ i ≤ r each ki-subset of vertices appears as an edge tance. We determine the required conditions for the result- of G with probability pi independently. The threshold of ing quasi-geodesic metric space to define k-local geodesics. the occurence of a fixed graph H in the Erd˝os-Rényi model These properties are then exploited as the foundation of G(n, p)(hereR = {2}) is well-studied in literature. We a distributed information localization algorithm which en- generalize these results to non-uniform hypergraphs. This ables to retrieve information from their distance. is joint work with Linyuan Lu.

Dimitri Papadimitriou Edward Boehnlein,LinyuanLu Bell Labs University of South Carolina [email protected] [email protected], [email protected]

MS4 MS5 Unique Vector Coloring Networks with Similar Assortativity - JDMs and PAMs Strict vector coloring is one formulation of an optimization program for Lovász theta function: assign vectors to ver- The partition adjacency matrix (PAM) encodes the num- tices so that adjacent vertices obtain vectors with large an- ber of edges connecting vertices of two partition classes in a gle in-between. We study when this assignment is unique. given partition of the vertices. It is a generalization of the We find analog of a classical result on uniqueness of col- joint degree matrix, which encodes the information neces- oring of graph products and generalize it. We use recent sary to compute the assortativity of the network. In this results of Laurent and Varvitsiotis on universal rigidity, talk we will explore what is the size of edge-swaps needed and properties of eigenspaces of Kneser graphs. to walk the space of graphs with given degree sequence and PAM, and related questions. Robert Samal Charles University Eva Czabarka Computer Science Institute Department of Mathematics [email protected]ff.cuni.cz University of South Carolina [email protected] David Roberson Nanyang Technological University [email protected] MS5 Sampling Graphs Using Markov Chains

MS4 The problem of efficiently sampling from a set of graphs (bipartite graphs, directed graphs) with a given degree se- 4-coloring Graphs with No Induced 5-cycle and 6- quence has many applications. One approach to this prob- vertex Path lem uses Markov chains to perform the sampling. The dif- We show that the 4-coloring problem can be solved in poly- ficulty lies in establishing rigorously that the Markov chain converges rapidly to its stationary distribution. I will re- nomial time on graphs with no induced 5-cycle C5 and no induced 6-vertex path P . The more general 4-coloring on view the known results in this area and discuss ongoing 6 extensions. P6-free graphs is one of the remaining cases left to complete the complexity characterization of the c-coloring problem Catherine Greenhill on Pk-free graphs for fixed c and k. While this is our ulti- University of New South Wales, School of Mathematics mate goal, the case with no induced 5-cycles already reveals and interesting structure and requires non-trivial arguments. Statistics, Sydney NSW AUSTRALIA 2052 Maria Chudnovsky [email protected] Columbia [email protected] MS5 Hypergraph Problems Arising from Networks in Peter Maceli Algebraic Statistics Columbia University [email protected] Social networks and other large sparse data sets pose significant challenges for statistical inference, as many stan- Juraj Stacho dard statistical methods for testing model/data fit are not DIMAP and University of Warwick applicable in such settings. Algebraic statistics offers a the- [email protected] oretically justified approach to goodness-of-fit testing that relies on the theory of Markov bases and is intimately con- Mingxian Zhong nected with the geometry of the model as described by its Columbia University fibers. Most current practices require the computation of DM14 Abstracts 53

the entire basis, which is infeasible in many practical set- naturally defined in settings like digraphs, edge-coloured tings. We present a dynamic approach to explore the fiber graphs, signed graphs, and general relational systems. One of a model, which bypasses this issue, and is based on the may also place restrictions on the function f, for example combinatorics of hypergraphs arising from the toric alge- we may require f to be locally injective or restrict its val- bra structure of log-linear models. This algebraic statis- ues to particular allowed images. In this introductory talk tics problem is intimately tied with graph and hypergraph we examine homomorphisms in settings mentioned above: sampling problems, which will be explained in this setting. edge-coloured, signed, and directed graphs. We survey Joint work with Elizabeth Gross and Despina Stasi. some results with particular interest given to complexity and good characterizations. Sonja Petrovic Illinois Institute of Technology, Department Richard Brewster of Applied Mathematics, Chicago IL 60616 Department of Mathematics and Statistics [email protected] Thompson Rivers University [email protected] MS5 Constrained Graph Construction Problems for MS6 Network Modeling Purposes Complexity of Signed Graph Homomorphisms Here I will discuss some of the fundamental questions related to graph ensemble based modeling of empirical net- We present recent advances in the study of the computa- works. In particular I will focus on degree based and tional complexity of the signed graph homomorphism prob- joint-degree based graph existence, construction, sampling lem, in an attempt of a full classification based on the tar- and enumeration problems and to a lesser extent, on get signed graph. In this talk, a signed graph is a graph soft-constraints based modeling using exponential random whose edges are either positive or negative, and switch- graph ensembles. ing (i.e. locally changing the signs of the edges incident to a given vertex) is allowed. A homomorphism of signed Zoltan Toroczkai graphs is a mapping between the vertex sets that preserves University of Notre-Dame the edge signs up to switching. [email protected] Richard Brewster MS6 Department of Mathematics and Statistics Thompson Rivers University Geometric Homomorphisms and the Geochromatic [email protected] Number

A geometric graph G is a simple graph G together with a Florent Foucaud fixed straightline drawing in the plane. A geometric homo- University Dauphine morphism ϕ : V (G) → V (H) is a homomorphism of the fl[email protected] underlying abstract graphs that also preserves edge crossings. That is, ϕ is a geometric homomorphism if whenever Pavol Hell v, u are adjacent vertices in G then ϕ(v),ϕ(u)areadjacent School of Computer Science in H and whenever e, f are crossing edges in G then ϕ(e) Simon Fraser University and ϕ(f) are crossing in H. Analogous to the chromatic [email protected] number of an abstract graph, the geochromatic number of a geometric graph is defined using geometric homomor- Reza Naserasr phisms. That is, the geochromatic number of a geometric Paris-Sud graph G is the smallest integer n so that there is a geomet- [email protected] ric homomorphism from G to some geometric realization of Kn. This talk will consider conditions under which we can find or bound the geometric chromatic number of a MS6 geometric graph. Colourings of Edge-Coloured Graphs and the Ver- Debra L. Boutin, Sally Cockburn tex Switching Operation Hamilton College [email protected], [email protected] The operation of switching at a vertex x of an oriented graph reverses the orientation of every arc incident with Alice M. Dean x. Similarly, switching at a vertex x of a 2-edge coloured Skidmore College graph changes the colour of each edge incident with x.We [email protected] will properties of vertex colourings and homomorphisms of 2-edge coloured graphs which are analogous to properties that hold in the case of oriented graphs, and which gener- MS6 alize to m-edge coloured graphs for m>2. Graph Homomorphisms: Edge Colours, Signs, and Crossings, An Introduction Gary MacGillivrary University of Victoria Given graphs G and H,ahomomorphism of G to H is a [email protected] function f : V (G) → V (H) such that uv ∈ E(G) implies f(u)f(v) ∈ E(H). We write G → H. Graph homomor- J. Maria Warren phisms generalize colourings in the sense that G → Kn Mathematics and Statistics is simply an n-colouring of G. Homomorphisms can be University of Victoria 54 DM14 Abstracts

[email protected] [email protected]

MS6 MS7 Signed Graph Homomorphisms Estimation of the Size of Families Not Containing the Given Posets A signed graph, denoted (G, Σ), is a graph where each edge has a sign, Σ is the set of negative edges. The no- We are interested in how large a family of subsets of the tion ﬁrst was introduced by F. Harary to study social net- set [n]:={1,...,n} that avoids a given partially ordered works where relationships can be of two sorts: friendship set (poset) P =(P, ≤). Let La(n, P ) be the largest size of or enmity. The notion of minor for signed graphs then families of subsets of [n] that do not contain the poset P .A allowed graph theorists to formulate a better language to conjecture posed by Griggs and Lu (2009), independently already exisiting results on coloring graphs with forbid- by Bukh (2009), is that La(n, P ) is asymptotically equal to n den structures. For example, Catlin’s proof of odd-K4 free e n ,wheree is an integer determined by P when n is 2 graphs being 3-colorable can be read as: if (G, E(G)) has large enough. The best general upper bound for La(n, P ) n no (K4,E(K4))-minor, then G is 3-colorable. The aim of 1 | | − is ( P + h(P )) 1 n , due to Bursi and Nagy. In this talk is to introduce a further generalization of this sort 2 2 of coloring results using the language of homomorphisms. the talk we will see how to improve the upper bound to be Reza Naserasr 1 1 2 n |P | + (m +3m − 2)(h(P ) − 1) − 1 , Paris-Sud n m +1 2 2 [email protected] n where m can be any positive integer less than 2 . MS7 Hong-Bin Chen, Wei-Tian Li Packing Posets in the Boolean Lattice Institute of Mathematics Academia Sinica We consider maximizing the number of pairwise unre- [email protected], lated copies of a poset P in the family of all subsets of [email protected] [n]:={1,...,n}. When P is a chain on k elements, the 1 n →∞ answer is asymptotic to 2k−1 n/2 ,asn ,byare- sult of Griggs, Stahl, and Trotter. While it remains open MS7 to determine asymptotically the largest size La(n, P )ofa Extremal Problems on Posets and Hypergraphs family of subsets of [n] that contains no subposet P ,we solve this new problem: The maximum is ∼ 1 n , We are interested in how large a family of subsets of the n- c(P ) n/2 { } where the integer c(P ) relates to embeddings of P into the set [n]:= 1,...,n there is that avoids certain patterns. Boolean lattice. This problem was independently posed The foundational result of this sort, Sperner’s Theorem and solved by Katona and Nagy. from 1928, solves this problem for families that contain no two-element chains. Here we will present some recent Andrew Dove results on extremal families avoiding a given (weak) sub- University of South Carolina poset, such as crowns, diamonds, etc. When viewing the [email protected] family of subsets as a hypergraph, we generalized several properties of Tur´an density, such as supersaturation, blow- Jerry Griggs up, and suspension, from uniform hypergraphs to non- University of South Carolina, Columbia uniform hypergraphs. The Lubell function plays a key role [email protected] in both areas. Linyuan Lu, Travis Johnston MS7 University of South Carolina [email protected], [email protected] Largest Union-intersecting Families

János Körner asked the following question. Let [n]= MS7 { } F⊂ [n] 1, 2,...,n and let 2 be a family of its subsets. It Recent Progress on Diamond-free Families is called union-intersecting if (F1 ∪ F2) ∩ (F3 ∪ F4)isnon- empty whenever F1,F2,F3,F4 ∈F and F1 = F2,F3 = F4. In the Boolean lattice, a diamond is a subposet of four What is the maximum size of a union-intersecting family? distinct subsets A, B, C, D such that A ⊂ B,C and D ⊃ This question is answered in the present paper. The opti- B,C. One of the most well-studied problems in extremal mal construction when n is odd consists of all subsets of poset theory is determining the size of the largest diamond- n−1 size at least while in the case of even n it consists of free family in the n-dimensional Boolean lattice. We will 2 n n − all sets of size at least 2 and sets of size 2 1 containing discuss some recent progress on this problem. a fixed element, say 1. We also proved some extensions, variants and analogues of this statement. The following Lucas J. Kramer, Ryan R. Martin, Michael Young one is an example. Suppose that F is a union-intersecting Iowa State University family of k-element subsets of [n]. We found that the opti- [email protected], [email protected], my- mal construction for this problem consists of all k-element [email protected] subsets of size k containing the element 1, and one more additional set, if for n>n(k). The results were jointly achieved with Dániel T. Nagy. MS8 Ordered Ramsey Numbers Gyula Katona Renyi Institute, Hungarian Academy of Sciences Given a labeled graph H with vertex set {1, 2,...,n},the DM14 Abstracts 55

ordered Ramsey number r<(H) is the minimum N such [email protected] that every two-coloring of the edges of the complete graph on {1, 2,...,N} contains a copy of H with vertices appear- Jacob Fox ing in the same order as in H. The ordered Ramsey number MIT of a labeled graph H is at least the Ramsey number r(H) [email protected] and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the Benny Sudakov ordered Ramsey number is superpolynomial in the number ETH Zurich of vertices. We prove several additional results regarding [email protected] ordered Ramsey numbers, including an ordered analogue of the Burr-Erd˝os conjecture. We also show a close connection between ordered Ramsey numbers of graphs and MS8 the ordinary Ramsey number for 3-uniform hypergraphs. Sharp Thresholds for Van Der Waerden’s Theorem David Conlon It was shown by Rödl and Ruciński that for m = University of Oxford (k−2)/(k−1) [email protected] Crn almost every m-element subset M of the first n integers has the property that every r-coloring of M yields a monochromatic arithmetic progression of length k, Jacob Fox, Choongbum Lee (k−2)/(k−1) MIT while sets of size o(n ) fail to have this property. [email protected], cb [email protected] We discuss sharp thresholds for this an related properties. This is joint work with E. Friedgut, H. Hàn, and Y. Person. Benny Sudakov ETH Zurich Mathias Schacht [email protected] University of Hamburg [email protected] MS8 Ehud Friedgut On the Size Ramsey Numbers and their Variations Weizmann Institute [email protected] For a given graph G,itssize-Ramsey number is the minimum number of edges in a graph H such that any 2- coloring of the edges of H yields a monochromatic copy Hiep Han of G. This concept was introduced by Erd˝os, Faudree, Emory University Rousseau, and Schelp and considered by several other re- Universidade de Sao Paulo searchers. Somewhat surprisingly this idea has not yet [email protected] been extended to hypergraphs, even though classical Ram- sey numbers for hypergraphs have been studied extensively. Yury Person In this talk, we discuss this new direction of research. Universitaet Frankfurt [email protected] Andrzej Dudek Department of Mathematics Western Michigan University MS8 [email protected] On the Ramsey Number of the Clique and the Hy- percube

MS8 Abstract not available at time of publication. Solution to the Erd˝os-Gyárfás Conjecture on Gen- eralized Ramsey Numbers Jozef Skokan London School of Economics ≤ ≤ p Fix positive integers p and q with 2 q 2 .Anedge- [email protected] coloring of the complete graph Kn is said to be a (p, q)- coloring if every Kp receives at least q different colors. The function f(n, p, q) is the minimum number of colors that MS9 are needed for Kn to have a (p, q)-coloring. This function Acyclic List Coloring on Planar Graphs was introduced by Erd˝os and Shelah about 40 years ago, and Erd˝os and Gyárfás in 1997 initiated the systematic Let G =(V,E) be a graph. A proper vertex coloring of G study of this function. In particular, they were interested is acyclic if G contains no bicolored cycle. Namely, every how for each fixed p, the asymptotic behavior of f(n, p, q) cycle of G must be colored with at least three colors. G (as n goes to infinity) changes for different values of q. is acyclically L-list colorable if for a given list assignment They proved several interesting results, but the question L = {L(v):v ∈ V }, there exists a proper acyclic coloring π of determining the maximum q for which f(n, p, q) is sub- of G such that π(v) ∈ L(v) for all v ∈ V .IfG is acyclically polynomial in n remained open. In this talk, we prove that L-list colorable for any list assignment with |L(v)|≥k for the answer is p − 1. all v ∈ V ,thenG is acyclically k-choosable. In 1976, Stein- berg conjectured that every planar graph without 4- and Choongbum Lee 5-cycles is 3-colorable. This conjecture cannot be improved MIT to 3-choosable basing on the examples given by Voigt and cb [email protected] independently, by Montassier. In this talk, We will show that planar graphs without 4- and 5-cycles are acyclically David Conlon 4-choosable. This result is also a new approach to the con- University of Oxford jecture proposed by Montassier, Raspaud and Wang, which 56 DM14 Abstracts

says that every planar graph without 4-cycles is acyclically Jan Volec 4-choosable. IUUK Charles University [email protected] Min Chen Zhejiang Normal University [email protected] MS9 Algorithms to Color Sparse Graphs Andre Raspaud Universite Bordeaux I Abstract not available at time of publication. [email protected] Matthew Yancey University of Illinois at Urbana-Champaign MS9 [email protected] Colorings and Independent Sets of Triangle-Free Planar Graphs MS10 We use coloring arguments to give a sufficient condition Erd˝os–Szekeres-type Statements in Dimension 1 for the size of the largest independent set in an n-vertex triangle-free planar graph to be at least an n/3+k for a Consider a geometric property of k-tuples of points express- fixed constant k, and use it to show the fixed parameter ible by a system of polynomial equations and inequalities. tractability of the corresponding decision problem. For some properties, every large set contains a subset each k-tuple of which satisfies the property. For example, every Zdenek Dvorak very large set contains a large subset in convex position Computer Science Institute (Erd˝os–Szekeres theorem). We treat the general problem Charles University, Prague in dimension 1, and show partial results in higher dimen- [email protected]ff.cuni.cz sions.

Matthias Mnich Boris Bukh MaxPlanckInstitutfür Informatik Princeton University Saarbrücken [email protected] [email protected] Jiˇr´ı Matouˇsek Institute for Theoretical Computer Science - ITI MS9 Charles University On Thue Colourings of Graphs [email protected]ff.cuni.cz

Here we deal with different versions of nonrepetitive total colourings of graphs and define the minimum number of MS10 colours required in every strong (weak) total Thue colour- Simplifying Inclusion - Exclusion Formulas ing. We prove some general upper bounds for the normal as well as the list version of the problem and show some The classical inclusion-exclusion formula expresses the better bounds for special classes of graphs. measure of a union of sets from the measures of their intersections. For n sets this formula has 2n − 1terms,and Erika Skrabulakova a significant amount of research, originating in applied ar- Technical University of Kosice eas, has been devoted to constructing simpler formulas for [email protected] particular families. We show that every family of n sets with Venn diagram of size m admits an inclusion-exclusion 2 Jens Schreyer formula with mO(log n) terms, with ±1 coefficients, com- 2 Technical University Ilmenau putable in mO(log n) expected time. Joint work with J. Germany Matousek,Z.Safernova,M.TancerandP.Patak [email protected] Xavier Goaoc Laboratoire de l’Institut Gaspard Monge MS9 Université Paris-Est Marne-la-Vallée Subcubic Triangle-free Graphs have Fractional [email protected] Chromatic Number at most 14/5

We prove that every subcubic triangle-free graph has frac- MS10 tional chromatic number at most 14/5, thus confirm- Splitting Points and Hyperplanes ing a conjecture of Heckman and Thomas [A new proof of the independence ratio of triangle-free cubic graphs. Dis- In this talk, we will discuss some of the discrete structures crete Math. 233 (2001), 233–237]. that arise while studying finite families of points and hyperplanes in IRd. Given a set of points and hyperplanes Zdenek Dvorak in IRd, we aim to find a large subset of them such that no Computer Science Institute selected hyperplane splits two selected points. This will Charles University, Prague be related to classic convex partition results of Yao and [email protected]ff.cuni.cz Yao which were originally motivated by geometric range queries. Jean-Sébastien Sereni CNRS (LORIA) Pablo Soberón [email protected] University of Michigan DM14 Abstracts 57

[email protected] spaces.

Edgardo Roldán-Pensado Christopher Hoffman UNAM University of Washington [email protected] hoff[email protected]

Matt Kahle MS10 Ohio State University The Cylindrical Crossing Number of the Complete [email protected] Bipartite Graph Elliot Paquette A cylindrical drawing of the complete bipartite graph on Weizmann Institute (n,m) vertices consists of n vertices placed on the upper [email protected] rim and m vertices on the lower rim of a straight circular cylinder where every two vertices from diﬀerent rims are joined by an edge on the surface. We present a for- MS11 mula for the minimum number of edge-crossings among all Interlacing Families and Bipartite Ramanujan such drawings for each (n,m) and an algorithm for creating Graphs drawings that achieve this crossing number. Abstract not available at time of publication. Athena C. Sparks, Silvia Fernandez California State University, Northridge Adam Marcus [email protected], [email protected] Yale University [email protected]

MS11 Gaps in Eigenfunctions of Graphs MS11 An Alon-Boppana Result for the Normalized We will discuss some problems and results on various prop- Laplacian erties of eigenfunctions of graphs. For, we will describe methods to bound the ‘stretches’ of edges which depend We prove a Alon-Boppana style bound for the spectral gap on the eigenfunctions. of the normalized Laplacian for general graphs via a bound on the weighted spectral radius of the universal cover graph Fan Chung-Graham Department of Mathematics Stephen J. Young University of California at San Diego University of California, San Diego [email protected] [email protected]

MS11 MS12 Construction of Matrices with a Given Graph and Generalized Chromatic Polynomials Prescribed Interlaced Spectral Data We show that many counting functions of graph proper- A result of Duarte [Linear Algebra Appl. 113 (1989) ties give rise to graph polynomials similar to the chromatic 173182] asserts that for real λ1,...,λn and μ1,...,μn−1 polynomial of graphs. We survey what is known so far and with λ1 ≤ μ1 ≤ λ2 ≤···≤μn−1 ≤ λn and each tree T on formulate problems and conjectures. n vertices there exists an n × n, real symmetric matrix A whose graph is T such that A has eigenvalues λ1,...,λn Tomer Kotek and the principal submatrix obtained from A by deleting Institut f¨ur Informationssysteme its last row and column has eigenvalues μ1,...,μn−1.This Technical University, Vienna, Austria result is extended to connected graphs through the use of [email protected] the implicit function theorem. Janos Makowsky Keivan Hassani Monfared Technion, – Israel Institute of Technology University of Wyoming Haifa, Israel [email protected] [email protected]

Bryan L. Shader Department of Mathematics MS12 University of Wyoming Algebraic Properties of the Chromatic Polynomial [email protected] There is increasing interest in algebraic properties of chromatic polynomials including which algebraic numbers are MS11 roots, chromatic factorisation (where a chromatic polyno- The Spectrum of Erdos-Renyi Random Graphs mial is the product of chromatic polynomials) and rela- Near the Connectivity Threshold tionships between chromatic polynomials with the same splitting field. Morgan and Farr introduced certificates for We give sharp estimates for the spectral gap of an Erdos- chromatic factorisation. Certificates are a powerful tool in Renyi random graph near the connectivity threshold. We proving algebraic properties of graph polynomials without show applications of these results to random topological the cost of computing the polynomial. We give an intro- 58 DM14 Abstracts

duction to using certiﬁcates in these proofs. into the model.

Kerri Morgan,GrahamFarr Lenore J. Cowen Faculty of Information Technology Dept. of Computer Science Monash University, Australia Tufts University [email protected], [email protected] [email protected]

Mengfei Cao, Hao Zhang, Jisoo Park MS12 Tufts University Counting Cuts, Matchings, Bipartite Subgraphs, [email protected], [email protected], and Dominating Sets: The Bipartition Polynomial [email protected] of a Graph Noah Daniels We introduce a new graph polynomial, the bipartition MIT polynomial, as a common generalization of the domina- [email protected] tion polynomial and the Ising model, which is also the generating function for the number of matchings, bipartite subgraphs, and (for regular graphs) independent sets. Mark Crovella The smallest known pair of non-isomorphic graphs with the Boston University same bipartition polynomial consists of two graphs of order [email protected] 22. The bipartition polynomial can be used to prove properties of other graph polynomials and graph invariants. Ben Hescott Department of Computer Science Peter Tittmann Tufts University Department of Mathematics [email protected] University of Applied Sciences, Mittweida, Germany [email protected] MS13 Large Highly Connected Clusters in Protein- MS12 Protein Interaction Networks Caterpillars and the U-polynomial The edge (vertex) connectivity of a graph is the minimum The U-polynomial (Noble and Welsh) and the chromatic number of edges (vertices) that must be removed to discon- symmetric function (Stanley) are known to be equivalent nect the graph. Connectivity is an important property of for trees. A long-standing open question is whether there a graph but has seldom been used in the study of protein- exist non-isomorphic trees with the same chromatic sym- protein interaction (PPI) and other biological networks. metric function. Caterpillars are trees such that the graph Connectivity differs from edge density in that it is based induced by the non-leaf vertices is a path. We establish a on the number of paths between all vertices in the graph, relation between caterpillars and integer compositions and while edge density is based solely on the number of edges. thereby prove that proper caterpillars (caterpillars with- Connectivity may be a better indicator of large clusters out vertices of degree two) are distinguished by the U- than edge density: as the number of vertices in a clus- polynomial. ter increases, the edge density tends to decrease rapidly, but the connectivity remains constant. We developed algo- Jose Zamora rithms to search for subgraphs with high connectivities and Depto. de Matematicas applied these algorithms to Saccharomyces cerevisiae and Universidad Andres Bello, Santiago, Chile human PPI networks. We discovered that PPI networks [email protected] have large subgraphs (20-130 vertices) with high connectivity that were previously unrecognized. The function of these subgraphs remains unknown, but they are far more MS13 highly connected than subgraphs in random subgraphs and New Metrics for Protein Interaction Networks have significant numbers of proteins with shared biological functions, suggesting that these are biologically significant. In protein-protein interaction (PPI) networks, functional similarity is often inferred based on the function of di- rectly interacting proteins, or more generally, some notion Debra Goldberg of interaction network proximity among proteins in a local Dept. of Computer Science neighborhood. Prior methods typically measure proximity University of Colorado– Boulder as the shortest-path distance in the network, but this has [email protected] only a limited ability to capture fine-grained neighborhood distinctions, because most proteins are close to each other, and there are many ties in proximity. We introduce diffu- MS13 sion state distance (DSD), a new metric based on a graph Efficiently Enumerating All Connected Induced diffusion property, designed to capture finer-grained dis- Subgraphs of a Large Molecular Network tinctions in proximity for transfer of functional annotation in PPI networks. We present a tool that, when input a PPI In systems biology, the solution space for a broad range network, will output the DSD distances between every pair of problems is composed of sets of functionally associated of proteins. We show that replacing the shortest-path met- biomolecules, which can be identified from connected in- ric by DSD improves the performance of classical function duced subgraphs of molecular interaction networks. Ap- prediction methods across the board. If time permits, we plications typically quantify the relevance (e.g., modular- will also mention our most recent work that inccorporates ity, conservation, disease association) of connected subnet- information about confidence scores and explicit pathways works using an objective function and use a search algo- DM14 Abstracts 59

rithm to identify sets of subnetworks that maximize this [email protected] objective function. Here, we propose and comprehensively evaluate an algorithm that signiﬁcantly reduces the com- putations necessary to enumerate all maximal subgraphs MS14 that satisfy a hereditary property. Delaunay Complexes, Noneuclidean Geometry, and Complexity Mehmet Koyuturk Case Western Reserve University I will discuss Delaunay tessellations in the plane, connec- Depertment of Electrical Engineering and Computer tions to hyperbolic geometry and other kinds of geometry, Science and some surprising complexity results, some stemming [email protected] from old and some from very new work. Igor Rivin Sean Maxwell, Mark Chance Department of Mathematics Case Western Reserve University Temple University Center for Proteomics and Bioinformatics igor [email protected] [email protected], [email protected]

MS14 MS13 Improvements on the Elekes-Rónyai Method Network-Based Rnaseq Quantification and Sur- vival Analysis for Cancer Genomics Elekes and Rónyai considered the following problem. Let A, B be two sets of n real numbers, and let f be a real bivariate polynomial. They showed that if |f(A×B)|≤cn, High-throughput mRNA sequencing (RNA-Seq) is a for some constant c>0andforn ≥ n0,wheren0 depends promising technology for gene transcript quantification. on c and degf only, then f must be of one of the special We introduce a Network-based method for RNA-Seq-based forms f(u, v)=h(g(u)+k(v)), or f(u, v)=h(g(u)k(v)), for Transcript Quantification (Net-RSTQ) to integrate protein some univariate polynomials h, k, h over R. Let us denote domain-domain interaction information with short read the number of triples (a, b, c) ∈ A × B × C such that c = alignment for transcript abundance estimation. Net-RSTQ / f(a, b)byM. We show that either M = O(n11 6)where models the expression of interacting transcripts as Dirich- the constant of proportionality depends on degf,orelsef let priors on the likelihood of the observed read alignments has one of the above special forms. This improves bounds against the transcripts in one gene. Laboratory validation on various Erd˝os type problems in discrete geometry. with qPCR confirmed the performance of Net-RSTQ for transcript quantification. The transcript abundances esti- Jozsef Solymosi mated by Net-RSTQ are also more informative for cancer UBC sample classification. [email protected]

Wei Zhang, Baolin Wu Micha Sharir, Orit Raz University of Minnesota Tel Aviv University [email protected], [email protected] [email protected], [email protected]

Hui Zheng Guangzhou Institutes of Biomedicine and Health of CAS MS14 zheng [email protected] Embeddability in the 3-sphere is Decidable

Rui Kuang During the talk we sketch a proof that the following algo- Dept Computer Science rithmic problem is decidable: given a 2-dimensional simpli- University of Minnesota cial complex, can it be embedded (topologically, or equiva- [email protected] lently, piecewise linearly) in R3? This is a natural higher- dimensional analogue of the graph-planarity question. By a known reduction, it suffices to decide the embeddability 3 MS13 of a given triangulated 3-manifold X into the 3-sphere S . The main step relies on a search for a short meridian, via Computational Approaches for Analyzing Large- (marked) normal surface theory. scale Genetic Interaction Networks Jiˇr´ı Matouˇsek Many phenotypes are the result of interactions between Charles University, Prague multiple genetic variants. Our lab has been involved in an [email protected]ff.cuni.cz on-going effort to measure quantitative phenotypes for mil- lions of double mutants in yeast in an attempt to under- Eric Sedgwick stand the fundamental principles of genetic interactions. DePaul University I will describe what we can learn from these large-scale [email protected] experiments, highlight insights that contribute to our understanding and treatment of human disease, and discuss Martin Tancer where innovations in machine learning and data mining are Charles University particularly relevant in addressing these questions. [email protected]ff.cuni.cz

Chad Myers Uli Wagner University of Minnesota IST Austria 60 DM14 Abstracts

[email protected] single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph G, written cc(G). We give bounds on MS14 cc(G) in terms of the cop number of G for bipartite graphs. On the Number of Plane Graphs with Polyline Edges Anthony Bonato Itisshownthatoneveryn-element point set in the plane, Ryerson University at most exp(O(kn)) labeled planar graphs can be embed- [email protected] ded using polyline edges with k bends per edge. This is the first exponential upper bound for the number of labeled plane graphs where the edges are polylines of constant size. MS15 Several standard tools developed for the enumeration of Ambush Cops and Robbers on Graphs with Small straight-line graphs, such as triangulations and crossing Girth numbers, are unavailable in this scenario. Furthermore, the exponential upper bound does not carry over to other In this variation of the game with two robbers, the cops popular relaxations of straight-line edges: for example, the win by moving onto the same vertex as one of the robbers number of plane graphs realizable with x-monotone edges after a finite number of moves. The robbers win by avoid- on n points is already super-exponential. (Joint work with ing capture indefinitely or by both moving onto the same Andrea Francke.) vertex as the cop. (Otherwise, the robbers are on distinct vertices.) We present results on graphs with small girth. Csaba D. Toth University of Calgary Nancy E. Clarke [email protected] Acadia University [email protected] MS14 Space Curve Arrangements with Many Incidences MS15 Cops and Invisible Robbers If a collection of lines in R3 has too many pairwise intersections, then many of these lines must lie on planes or In this talk we will consider the game of cops and robbers, reguli. In this talk I will discuss a generalization of this re- as in the classic model, with the added restriction that sult to algebraic space curves; if a collection of curves has the robber is invisible to the cops. This alteration brings too many pairwise intersections, then these curves must the game more in line the pursuit-evasion model known as congregate on on lower dimensional varieties. graph searching. We will consider some classic investiga- tions along this line, including relations to monotonicity, Josh Zahl pathwidth, and complexity. MIT [email protected] Dariusz Dereniowski Gdansk University of Technology [email protected] MS15 Fighting Fires on Layered Grids Danny Dyer,RyanTifenbach In the Firefighter Problem, a fire start at a vertex of a graph Memorial University of Newfoundland at time 0, and in discrete time steps, it spreads from burn- [email protected], [email protected] ing vertices to their neighbors unless they are protected by one of the firefighters that are deployed between each Boting Yang time step. We assume that the number of firefighters per Dept. of Computer Science turn is a fixed positive integer. We also assume that once University of Regina a vertex is burning or protected, it remains in that state. [email protected] The process terminates when fire can no longer spread. In the case of locally-finite infinite graphs, the key question is to determine the minimum number of firefighters needed MS15 to stop a fire in finite time. In this talk, we will consider Edge Contraction and Cop-Win Critical Graphs graphs that are the Cartesian product of a path and an infinite 2-dimensional grid, and will give an upper bound In the game of Cop and Robber, a single cop tries to ap- on the number of firefighters needed in such a graph given prehend a robber as they move alternately along edges of that the number of firefighters that can be deployed in each a reflexive graph. Graphs in which one cop can always win layer per time unit does not exceed a fixed number. are called cop-win graphs. A graph is Cop-win Edge Crit- ical with respect to Contraction (CECC) if it is not itself Amir Barghi, Nicole Rosato cop-win, but with the contraction of any edge, the resulting Bard College graph is cop-win. In this talk, I will present characteriza- [email protected], [email protected] tions of some particular classes of CECC graphs such as bipartite, minimum degree three, and 4-regular. I will also compare these results with known results for graphs that MS15 are Cop-win Edge Critical with respect to Edge Addition. The Robber Strikes Back

We consider the new game of Cops and Attacking Robbers, Shannon L. Fitzpatrick which is identical to the usual Cops and Robbers game University of Prince Edward Island except that if the robber moves to a vertex containing a sﬁ[email protected] DM14 Abstracts 61

Ben Cameron few years dealing with chorded cycles. I will then concen- Dalhousie University trate on some new results concerned with placing elements [email protected] on chorded cycles, when a set of k independent edges can be made the chords of k vertex disjoint chorded cycles, and ﬁnally, how many chords we should be hoping to ﬁnd. MS16 Long Cycles in Graphs with Bounded Degrees Ronald Gould Dept. of Mathematics and Computer Science In 1993 Jackson and Wormald conjectured that for any Emory University positive integer d ≥ 4, there exists an positive real num- [email protected] ber α depending only on d such that if G is a 3-connected n-vertex graph with maximum degree d ≥ 4. then G has a cycle of length ≥ αnlogd−1 2. They showed that the ex- MS16 ponent in the bound is best possible if the conjecture is Vertex-Disjoint Theta Subgraphs true. In this paper, we report the recent progress toward this conjecture. A theta graph is a graph consisting of three internally disjoint paths joining the same two vertices. In 2009, Chiba Guantao Chen et al. proved that every graph with minimum degree at Georgia State University least 2k +1 andorder k contains k vertex-disjoint theta Dept of Mathematics and Statistics subgraphs. In this talk, we consider a minimum degree [email protected] condition for a graph to contain k vertex-disjoint isomorphic theta subgraphs. The result implies the existence of Zhicheng Gao k vertex-disjoint cycles of the same even length. School of Mathematics and Statistics Carleton University Shinya Fujita [email protected] International College of Arts and Science Yokohama City University Songling Shan [email protected] Department of Mathematics and Statistics Georgia State University Katsuhiro Ota [email protected] Department of Mathematics Keio University Xingxing Yu [email protected] School of Mathematics Georgia Tech Tadashi Sakuma [email protected] Faculty of Education, Art and Science Yamagata University Wenan Zang [email protected] Department of Mathematics University of Hong Kong MS16 [email protected] Halin Graphs and Generalized Halin Graphs

MS16 A Halin graph is a plane graph H = T ∪ C such that T is a spanning tree of H with no vertices of degree 2 where Minimum Degree and Disjoint Cycles in General- | |≥ ized Claw-free Graphs T 4andC is a cycle whose vertex set is the set of leaves of T . It is known that Halin graphs satisfy a lot of nice

For s ≥ 3agraphisK1,s-free, if it does not contain an properties. So, it is a natural question whether relaxations induced subgraph isomorphic to K1,s.Fors =3,such of Halin graphs preserve such properties or not. Now we graphs are called claw-free graphs. Results on disjoint cy- define a generalized Halin graph which is a graph H = T ∪C cles in claw-free graphs satisfying certain minimum degree such that T is a spanning tree of H with no vertices of conditions will be be discussed, such as if G is claw-free degree 2 where |T |≥4andC is a cycle whose vertex set of sufficiently large order n =3k with δ(G) ≥ n/2, then is the set of leaves of T (note that some generalized Halin G contains k disjoint triangles. Also, the extension of re- graphs may not be plane graphs). In this talk, we introduce sults on disjoint cycles in claw-free graphs satisfying certain some known results on Halin graphs and generalized Halin minimum degree conditions to K1,s-free graphs for s>3 graphs. After that, we investigate difference between Halin will be presented. These results will be used to prove the graphs and generalized Halin graphs. existence of minimum degree conditions that imply the existence of powers of Hamiltonian cycles in generalized claw- Shoichi Tsuchiya free graphs. Tokyo University of Science [email protected] Ralph Faudree University of Memphis Department of Math Sciences MS17 [email protected] Large Forbidden Configurations Define a matrix A to be simple if it is a (0,1)-matrix with no MS16 repeated columns. Given a matrix F ,wesayA has no con- On Chorded Cycles figuration F if there is no submatrix of A which is a row and column permutation of F . Our extremal problem, given m I will introduce results that have been shown over the last and F , is to determine an upper bound forb(m, F )onthe 62 DM14 Abstracts

number of columns in an m-rowed simple matrix which has σ(H) is the smallest size of a set of vertices that meets each no configuration F . For this talk we consider forbidding a edge of H in exactly one vertex. We also determine the ex- configuration F (m), a configuration that grows with m. act values of exr(n, H) for all large n for certain so-called Some design theory questions arise. σ-critical H. Our results generalize many recent asymptotic or exact results on the topic. The work is joint with Richard P. Anstee Z. Füredi. We also survey related results and pose some Mathematics Department questions. University of British Columbia [email protected] Tao Jiang Department of Mathematics and Statistic Attila Sali Miami University Renyi Institute [email protected] [email protected] MS17 MS17 Turán Numbers for Bipartite Graphs Plus Odd Cy- The Manickam-Miklós-Singhi Conjectures for Sets cles and Vector Spaces In this talk, I will discuss a general approach to a class of More than twenty-five years ago, Manickam, Miklós, and extremal problems for bipartite graphs. Specifically, Erd˝os Singhi conjectured that for positive integers n, k with n ≥ and Simonovits made three important conjectures concern- 4k,everysetof n real numbers with nonnegative sum has ing the asymptotic behavior of the extremal number for a n−1 bipartite graph containing a cycle. One of the conjectures at least k−1 k-element subsets whose sum is also non- states that if Ck is the set of odd cycles of length less than negative. We verify this conjecture when n ≥ 8k2,which k, then for every bipartite graph F containing a cycle, the simultaneously improves and simplifies a bound of Alon, extremal Ck-free F -free graphs should have asymptotically Huang, and Sudakov and also a bound of Pokrovskiy when as many edges as an extremal F -free bipartite graph. We k<1045. Moreover, our arguments resolve the vector prove this conjecture for a wide class of graphs F ,inad- space analogue of this conjecture. dition exhibiting a “near-bipartite’ structure of extremal Ameera Chowdhury graphs. The proof depends strongly on the existence of Carnegie-Mellon University an exponent for a bipartite graph F , which constitutes an- [email protected] other of the three conjectures of Erd˝os and Simonovits. Jacques Verstraete Ghassan Sarkis, Shahriar Shahriari University of California-San Diego Pomona College [email protected] [email protected], [email protected]

MS18 MS17 Sudoku, Latin Squares, and Deﬁning Sets in Graph Maximizing the Number of Nonnegative Subsets Coloring

Given a set of n real numbers, if the sum of elements of Over the last decade, Sudoku, a combinatorial number- every subset of size larger than k is negative, what is the placement puzzle, has become a favorite pastimes of many maximum number of subsets of nonnegative sum? In this n−1 n−1 all around the world. In this puzzle, the task is to complete talk we will show that the answer is k−1 + ... + 0 +1 a partially ﬁlled 9 by 9 square with numbers 1 through 9, by establishing and applying a weighted version of Hall’s subject to the constraint that each number must appear Theorem. This settles a problem of Tsukerman. Joint work once in each row, each column, and each of the nine 3 with Noga Alon. by 3 blocks. As it turns out, this is very similar to the notion of critical sets for Latin squares, or more generally, Hao Huang deﬁning sets for graph colorings. In this talk, we discuss DIMACS, Rutgers University this connection and present a number of new results and Institute for Advanced Study open problems for Sudoku squares. [email protected] Mohammed Mahdian Google MS17 [email protected] Turan Problem for Hypergraph Forests

The Turan number exr(n, H)ofanr-uniform hypergraph MS18 H is the largest size of an r-uniform hypergraph on n ver- Deﬁning Sets in Graph Coloring ticesthatdoesnotcontainH as a subgraph. A hypergraph is a hypergraph forest if its edges can be ordered as In a given graph G,asetofverticesS with an assignment E1,...,Em such that for any i>1thereexistsa(i)

c set. The following conjecture (1995), is still open: For any Let Δmon(Kn) denote the maximum number of edges of the 2 c n, d(Kn × Kn,n)=n /4 . Defining sets in graph col- same colour incident with a vertex of Kn. In 1976, Bollobás c c oring are closely related with the idea of “uniquely k-list and Erd˝os conjectured that every Kn with Δmon(Kn) < colorable graphs”. In this talk we mention these concepts n/2 contains a properly coloured Hamilton cycle, that in different areas and introduce some more open problems. is, a spanning cycle in which adjacent edges have distinct colours. In this talk, we show that for any ε>0andforall c n ≥ n0(ε), Δmon(Kn) < (1/2 − ε)n is sufficient. Hence the Ebadollah S. Mahmoodian conjecture of Bollobás and Erd˝os is true asymptotically. Sharif University of Technology [email protected] Allan Lo University of Birmingham [email protected] MS18 Coloring Maps on Surfaces MS19 Albertson conjectured every surface F 2 admits an integer Decomposition of Random Graphs into Complete N(F 2) such that if G ⊂ F 2,thenG admits some S ⊂ V (G) Bipartite Graphs with |S|≤N(F 2) such that G − S is 4-colorable. In my talk, as a variation of it, we prove that every locally pla- For a graph G, the bipartition number τ(G) is the mini- nar graph G on an orientable surface has a 5-coloring such mum number of complete bipartite subgraphs whose edge that the cardinality of the smallest color class is bounded sets partition the edge set of G. In 1971, Graham and by ε|V (G)|. Moreover, we prove similar results for some Pollak proved that τ(Kn)=n − 1. For a graph G with classes of maps on orientable surfaces. n vertices, one can show τ(G) ≤ n − α(G) easily, where α(G) is the independence number of G.Erd˝os conjec- Atsuhiro Nakamoto tured that almost all graphs G with n vertices satisfy Yokohama National University τ(G)=n − α(G). In this talk, we present upper and Japan lower bounds for τ(G(n, p)) which gives support to Erd˝os’ [email protected] conjecture.

Xing Peng MS18 University of South Carolina Distinguishing Colorings of 3-regular Maps on [email protected] Closed Surfaces Fan Chung A k-coloring of a map on a closed surface is said to be dis- University of California, San Diego tinguishing if there is no automorphism of the map other [email protected] than the identity map which preserves the colors. In particular, if a map has a k-coloring which uses color k at most once, then it is said to be nearly distinguishing (k − 1)- MS19 colorable. We shall show that any 3-regular map on a closed A Blow-up Lemma for Sparse Pseudorandom surface is nearly distinguishing 3-colorable unless it is one Graphs of three exceptions. We present a new blow-up lemma for spanning graphs with Seiya Negami bounded maximum degree in sparse pseudorandom graphs Yokohama National Univeristy and discuss its possible applications. [email protected] Peter Allen, Julia Boettcher London School of Economics MS18 [email protected], [email protected] Small Snarks and 6-chromatic Triangulations on the Klein Bottle Hiep Han Emory University We take a dual approach to the embedding of snarks on the Universidade de Sao Paulo Klein bottle, and investigate edge-colorings of 6-chromatic [email protected] triangulations of the Klein bottle. In the process, we discover the smallest snarks that embed polyhedrally on the Klein bottle. Additionally, we ﬁnd that every triangula- Yoshiharu Kohayakawa tion containing certain 6-critical graphs on the Klein bottle Instituto de Matematica e Estatistica must have a Gr¨unbaum coloring and thus cannot admit a Universidade de So Paulo, Brazil dual embedded snark. [email protected] sarah-marie belcastro Yury Person Smith College Universitaet Frankfurt Hampshire College Summer Studies in Mathematics [email protected] [email protected]

MS19 MS19 The Typical Structure of Sparse H-free Graphs Properly Coloured Hamilton Cycles in Edge- coloured Complete Graphs Two central topics of study in combinatorics are the so- called evolution of random graphs, introduced by the sem- c Let Kn be an edge-coloured complete graph on n vertices. inal work of Erd˝os and R´enyi, and the properties of H-free 64 DM14 Abstracts

graphs, that is, graphs which do not contain a subgraph [email protected] isomorphic to a given (usually small) graph H.Awidely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical (ran- MS20 dom) H-free graph with n vertices and m edges changes as A Refinement of the Grid Theorem m grows from 0 to ex(n, H). We first show how a recent result of Balogh, Morris, and the speaker, proved inde- A k-connected set of vertices in a graph G is a set X of pendently by Saxton and Thomason, allows one to derive vertices in G such that if X1,X2 ⊆ X then G contains an approximate structural description of a typical H-free min{|X1|, |X2|,k} vertex-disjoint paths between X1 and graph for each non-bipartite H. Then, extending classi- X2. The unavoidable minors for graphs containing a large cal results of Kolaitis, Osthus, Prömel, Rothschild, Steger, k-connected set will be presented. In particular, for graphs and Taraz, we show that in several interesting cases, such containing a large, highly connected set, all of the unavoid- as when H is a clique, these approximate structural de- able minors contain a large grid-minor, which implies the scriptions may be made precise. grid theorem of Robertson and Seymour. This is joint work with Jim Geelen. Jozsef Balogh University of Illinois Jim Geelen, Benson Joeris [email protected] University of Waterloo [email protected], [email protected] Robert Morris IMPA, Rio de Janeiro MS20 [email protected] Packing Edge-Disjoint Odd S-Cycles in 4-Edge- Connected Graphs Wojciech Samotij Tel Aviv University It has been known that for a 4-edge-connected graph G, School of Mathematical Sciences the Erd˝os-Pósa property holds for edge-disjoint odd cycles, [email protected] and there is a simple polynomial-time algorithm to test whether or not G has k edge-disjoint odd cycles for fixed Lutz Warnke k. We generalize these results to the problem with parity University of Oxford constraints. [email protected] Yusuke Kobayashi, Naonori Kakimura University of Tokyo MS19 [email protected], [email protected] The Structure of Digraphs not Containing a Fixed Subgraph Ken-ichi Kawarabayashi National Institute of Informatics, Japan We count the number of digraphs and oriented graphs not k [email protected] containing a fixed digraph H. The corresponding question has been studied extensively for graphs and hypergraphs. Our proof uses the hypergraph container method developed MS20 by Saxton and Thomason and independently by Balogh, Unavoidable Vertex-minors in Large Prime Graphs Morris, and Samotij. Moreover, we describe the structure of typical H-free digraphs and oriented graphs for certain Agraphisprime (with respect to the split decomposition) H. This answers two questions of Cherlin. if its vertex set does not admit a partition (A, B) (called a split)with|A|, |B|≥2 such that the set of edges joining A Daniela Kuhn, Deryk Osthus, Timothy Townsend and B induces a complete bipartite graph. We prove that University of Birmingham for each n,thereexistsN such that every prime graph [email protected], [email protected], on at least N vertices contains a vertex-minor isomorphic [email protected] to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching. Yi Zhao Georgia State University O-Joung Kwon [email protected] Department of Mathematical Sciences, KAIST [email protected]

MS20 Sang-Il Oum KAIST Size Conditions for Rainbow Matchings [email protected]

Let E =(Ei | 1 ≤ i ≤ k) be a collection of graphs (or hypergraphs). A choice of disjoint edges, one from each Ei, MS20 E is called a rainbow matching for . In this talk I will discuss Packing A-paths with Speciﬁed Endpoints some recent results pertaining to rainbow matchings. In particular, I will focus on size conditions for the Ei’s such Consider a graph G and a speciﬁed subset A of vertices. that there exists a rainbow matching for E. An A-path is a path with both ends in A and no internal vertex in A. Given a list of {(si,ti):1≤ i ≤ k} of pairs of David M. Howard vertices in A, consider the question of whether there exist Colgate University many pairwise disjoint A-paths P1,...,Pt such that for all DM14 Abstracts 65

j, the ends of Pj are equal to si and ti for some value [email protected] i. This generalizes the disjoint paths problem and is NP- hard if k is not ﬁxed. We further restrict the question, and ask if there either exist t pairwise disjoint such A-paths or MS21 alternatively, a bounded set of f(t) vertices intersecting all How to Squash a 6-Cycle System into a Steiner such paths. In general, there exist examples where no such Triple System function f(t) exists; we present an exact characterization of when such a function exists. The spectra for Steiner triple systems and 6-cycle systems agree when n =1or9(mod 12). Let (X, C) be a 6-cycle Daniel Marx system of order n =1or9(mod 12). Let T be a collection Computer and Automation Research Institute, Hungarian of bowties obtained by squashing each 6-cycle of C into Academy of Sciences (MTA SZTAKI) a bowtie (i.e. identifying two ’opposite’ vertices of the 6- [email protected] cycle). If (X, T) is a Steiner triple system we say that the 6-cycle system (X, C) is squashed into the Steiner triple Paul Wollan system (X, T). In this talk we construct, for every n =1 University of Rome ”La Sapienza” or 9 (mod 12) a 6-cycle system that can be squashed into [email protected] a Steiner triple system. (This is joint work with Mariusz Meszka and Alex Rosa.)

MS21 Charles C. Lindner Resolvable Or Near Resolvable Designs in Multiple Auburn University Rooms [email protected]

I will discuss the existence of RBIBDs or NRBs which are MS21 L-colorable in the following sense. Let D =(V,B, R)bea v v−1 On Large Sets of Combinatorial Objects (v, k, λ)-RBIBD (or NRB), let t be a divisor of k (or k ) and let L be a list of t non-negative integers summing up − to λ.IsaythatD is L-colorable if it is possible to color Resolutions of t designs were studied as early as 1847 its blocks with t colors 1,...,t in such a way that: by Reverend T. P. Kirkman who proposed the famous 15 schoolgirls problem. In this talk we discuss large sets • v every parallel (or near parallel) class has exactly kt LS[N](t, k, v)ofsimplet − (v, k, λ) designs, large sets v− 1 ≤ ≤ q − (or kt ) blocks colored i for 1 i t; LS [N](t, k, n)ofgeometrict designs in linear spaces of dimension n over GF (q), orthogonal large sets and large • if ci(x, y) is the number of blocks colored i through x { } sets of large sets. We present some recent results which and y, then the list c1(x, y),...,ct(x, y) is equal to use groups to extend previous results, and pose open prob- L for every pair of distinct points x and y. lems.

Marco Buratti Spyros S. Magliveras Dipartimento di Matematica e Informatica Florida Atlantic University Universit`a di Perugia, Italy Department of Mathematical Sciences [email protected] [email protected]

MS21 MS22 A Three-Factor Product Construction for Mutually Finitely Forcible Graphons Orthogonal Latin Squares In extremal graph theory, we often consider large graphs It is well known that mutually orthogonal latin squares, that are in the limit uniquely determined by finitely many or MOLS, admit a (Kronecker) product construction. I densities of their subgraphs. The corresponding limits (so- will discuss how, under mild conditions, ‘triple products’ of called graphons) are called finitely forcible. Motivated by MOLS can result in a gain of one square. This observation classical results in extremal combinatorics as well as by re- leads to a few improvements to the MOLS table and a cent developments in the study of finitely forcible graphons, slight strengthening of the famous theorem of MacNeish. Lovasz and Szegedy made some conjectures about the This is joint work with Alan C.H. Ling and hinges on an structure of such graphons. In particular, they conjectured important construction of Rolf Rees. that the topological space of typical points of every finitely forcible graphon is compact and finitely dimensional. In Peter Dukes this talk we give a brief overview about the corresponding University of Victoria, Canada results, and in particular show that these two conjectures [email protected] are false.

Roman Glebov MS21 ETH Zurich Vertex-transitive Graphs of Prime-squared Order [email protected] are Hamilton-decomposable

We prove that all connected vertex-transitive graphs of or- MS22 der p2, p a prime, can be decomposed into Hamilton cycles. Estimating the Distance from Testable Properties (Joint work with Brian Alspach and Darryn Bryant.) Fischer and Newman showed that it is possible to test the Donald L. Kreher distance of a graph from every testable graph property. In Department of Mathematical Sciences this talk we simplify their test, and furthermore general- Michigan Technological University ize it the algebraic setting of aﬃne-invariant properties on 66 DM14 Abstracts

finite fields. This is based on a joint work with Shachar holds for sufficiently large n. Wealsoprovetheconjec- Lovett. ture if n =4k for all k ≥ 0. These results imply that F (n) lim n =0.4, and determine the unique limit object. In Hamed Hatami (3) Department of Mathematics the proof we use flag algebras combined with stability ar- Princeton Univ. guments. [email protected] Jozsef Balogh University of Illinois MS22 [email protected] The Inducibility of Directed Graphs Ping Hu In modern extremal combinatorics, a substantial number University of Illinois at Urbana-Champaign of problems study the asymptotic relations between densi- [email protected] ties of subgraphs. One particularly interesting topic called inducibility is to find the maximum possible induced den- Bernard Lidicky sity of a given subgraph. We study this problem in the University of Illinois, Urbana, IL digraph setting, and solve the inducibilies for all the com- [email protected] plete bipartite digraphs. Our result confirms a conjecture by Falgas-Ravry and Vaughan, and provides the first Florian Pfender known explicit instance of density problems for which one Dept. of Math & Stat Sciences can prove the extremality of an iterated blow-up construc- University of Colorado Denver tion. fl[email protected] Hao Huang DIMACS, Rutgers University Jan Volec Institute for Advanced Study IUUK Charles University [email protected] [email protected]

Michael Young MS22 Iowa State University First Order Limits [email protected]

The notion of ﬁrst order convergence and the corresponding limit object, modelling, were proposed to bridge the realms MS23 of dense and sparse graph limits. In this talk, we recall the Extending Partial Representations of Circle notions and we will present positive and negative results Graphs we have obtained on the existence of modelings for some classes of graphs and matroids. AcirclegraphG is an intersection graph of chords of a circle. We study the algorithmic question of whether one can Daniel Kral extend a given representation R of an induced subgraph University of Warwick of G toarepresentationR of the entire G.Wegivea [email protected] polynomial-time algorithm for this problem. To show this, we describe the structure of all representations of a circle Martin Kupec graph based on split decomposition. Charles University in Prague [email protected]ﬀ.cuni.cz Steven Chaplick Department of Computer Science, Anita Liebenau University of Toronto University of Warwick [email protected] [email protected] Radoslav Fulek Lukas Mach Industrial Engineering and Operations Research Computer Science Institute of Charles University (IUUK) Department Lesser Town Square, no. 25, Prague Columbia University [email protected] [email protected]

Vojtech Tuma Pavel Klavik Charles University in Prague Computer Science Institute [email protected]ﬀ.cuni.cz Charles University in Prague [email protected]ﬀ.cuni.cz

MS22 Rainbow Triangles in Three-colored Graphs MS23 Contact Representations of Planar Graph: Re- Erd˝os and S´os proposed a problem of maximizing the num- building Is Hard ber F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n)= Planar graphs have geometric representations of various F (a)+F (b)+F (c)+F (d)+abc + abd + acd + bcd,where types, e.g. as contacts of disks, triangles or - in the bi- a + b + c + d = n and a, b, c, d are as equal as possible partite case - vertical and horizontal segments. We show and F (0) = 0. We prove that the conjectured recurrence that in each of these cases, the problem of completing a DM14 Abstracts 67

partial representation becomes NP-hard. We also give two [email protected] polynomial time algorithms for the grid contact case. The ﬁrst one when all vertical segments are pre-represented and the second is when the vertical segments have only their MS23 x-coordinates speciﬁed. Picking Planar Edges; Or, Drawing a Graph with a Planar Subgraph

Jan Kratochvil Given a graph G and a subset F ⊆ E(G)ofitsedges,is Charles University, Czech Republic there a drawing of G in which all edges of F are free of [email protected]ﬀ.cuni.cz crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If Steven Chaplick we require the drawing of G to be straight-line, and allow Department of Computer Science, at most one crossing along each edge in F , the problem University of Toronto turns out to be as hard as the existential theory of the real [email protected] numbers.

Paul Dorbec Marcus Schaefer Univ. Bordeaux, CNRS, France DePaul University [email protected] [email protected]

Mickael Montassier MS23 LIRMM, France [email protected] Coloring Geometric Intersection Graphs via on-line Games

Juraj Stacho Game graphs are graphs encoding presentation strategies DIMAP and University of Warwick in on-line graph coloring games. The chromatic number of [email protected] a game graph is equal to the number of colors that any online coloring algorithm is forced to use playing against the corresponding strategy. I will present how game graphs of MS23 appropriately defined games can be used in proving lower and upper bounds on the maximum chromatic number in classes of intersection graphs of geometric objects. The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial Tomasz Krawczyk Jagiellonian University Intersection graphs of geometric objects have been exten- [email protected] sively studied, both due to their interesting structure and their numerous applications; prominent examples include Bartosz Walczak interval graphs and permutation graphs. In this paper we Theoretical Computer Science Department study a natural graph class that generalizes both interval Jagiellonian University and permutation graphs, namely simple-triangle graphs. [email protected] Simple-triangle graphs - also known as PI graphs (for Point-Interval) - are the intersection graphs of triangles MS24 that are defined by a point on a line L1 andanintervalon a parallel line L2. They lie naturally between permutation Pulling Self-avoiding Walks from an Adsorbing and trapezoid graphs, which are the intersection graphs Plane of line segments between L1 and L2 and of trapezoids between L1 and L2, respectively. Although various effi- In this talk a self-avoiding walk model of a polymer pulled cient recognition algorithms for permutation and trapezoid from an adsorbing surface by a vertically applied force is graphs are well known to exist, the recognition of simple- considered. The self-avoiding walk is fixed in the surface triangle graphs has remained an open problem since their at its one endpoint, and pulled at the other endpoint. I introduction by Corneil and Kamula three decades ago. In will show that the appropriate thermodynamic limit in this this paper we resolve this problem by proving that simple- model exists and that the free energy is convex. An ex- triangle graphs can be recognized in polynomial time. As pression for the free energy in terms of the free energies of a consequence, our algorithm also solves a longstanding pulled and adsorbing self-avoiding walks will be given, and open problem in the area of partial orders, namely the it will be shown that there is a phase boundary between an recognition of linear-interval orders, i.e. of partial orders adsorbed phase and a ballistic phase in the model. Some P = P1 ∩P2,whereP1 is a linear order and P2 is an interval qualitative properties of this phase boundary will be given, order. This is one of the first results on recognizing par- and some bounds on the location of the boundary will be tial orders P that are the intersection of orders from two shown. This work was done in collaboration with SG Whit- different classes P1 and P2. In complete contrast to this, tington. partial orders P which are the intersection of orders from thesameclassP have been extensively investigated, and EJ Janse van Rensburg in most cases the complexity status of these recognition York University problems has been already established. [email protected]

George B. Mertzios MS24 School of Engineering and Computing Sciences The Systematic Application of Analytic Combina- Durham University torics of Several Variables to Lattice Path Enumer- 68 DM14 Abstracts

ation Problems ysed previously.

This work combines recent work on lattice path enumera- Thomas Wong tion using the orbit sum method and the analysis of mul- University of British Columbia tivariate generating functions and determines new asymp- [email protected] totic formulas for d-dimensional lattice models with particular symmetries. We determine the dominant asymptotics for the number of walks restricted to the positive orthant MS25 taking unit steps which move positively in each coordinate and are symmetric with respect to each axis. These Symmetric Chain Decompositions of Quotients of expressions are derived by analyzing the singular variety Products of a multivariate rational function whose diagonal counts the lattice path models in question. (Collaboration with Given a subgroup G of the symmetric group Sn on [n] Stephen Melczer (SFU)) and the Boolean lattice of all subsets of [n], the quotient Bn/G has the ordering induced on the orbits under G.It Marni Mishna is conjectured that Bn/G has a symmetric chain decompo- Simon Fraser University sition [SCD]. Earlier results are extended to G constructed [email protected] from wreath products and applied to powers P n of any poset with an SCD. Several questions involving abelian subgroups of G are raised. MS24 On the Number of Walks in a Triangular Domain Dwight Duffus Emory University We consider walks on a triangular domain that is a sub- Mathematics and Computer Science Department set of the triangular lattice. We then specialize this by [email protected] dividing the lattice into two directed sub-lattices with different weights. Our central result is an explicit formula for Kyle Thayer the generating function of walks starting at a fixed point Fort Collins, CO in this domain and ending anywhere within the domain. [email protected] Intriguingly, the vspecialization of this formula to walks starting in a fixed corner of the triangle shows that these are equinumerous to two-colored Motzkin paths, and two- coloured three-candidate Ballot paths, in a strip of finite MS25 height. On the Dimension of Posets with Cover Graphs of Treewidth 2 Thomas Prellberg Queen Mary, University of London [email protected] Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well- Paul Mortimer known construction of Kelly shows that there are posets Queen Mary University of London of arbitrarily large dimension whose cover graphs have [email protected] treewidth 3. The focus of this talk is the boundary case of treewidth 2. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, MS24 and Wiechert) or if it has pathwidth 2 (Biró, Keller, and Random Knots and Polymer Models Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover This talk will survey some of the results on properties of graph has treewidth 2. We show that it is indeed the case: random knots in 3-space and in confined volumes, with ap- Every such poset has dimension at most 2554. plications to enzyme action on duplex DNA and the structure and dynamics of duplex DNA confined to viral capsids. Gwenael Joret Universite Libre de Bruxelles De Witt L. Sumners [email protected] Florida State University Department of Mathematics [email protected] Piotr Micek Jagiellonian University [email protected] MS24 Numerical Approach of Two Friendly Walks in a William T. Trotter Sticky Slab School of Mathematics Georgia Institute of Technology In previous work we derived an exact solution of two [email protected] friendly directed walks above a sticky wall with single and double interactions. This models the behaviour of a pair Ruidong Wang of attracting polymers above an adsorbing surface. In this Georgia Institute of Technology talk, we extend this to two friendly directed walks in a [email protected] sticky slab. I’ll discuss some difficulties encountered with the analytic approach, the results from numerical analysis, Veit Wiechert and then compare these results with similar models anal- TU - Berlin DM14 Abstracts 69

[email protected] random vertices. At each step we insert a vertex chosen uniformly at random from the collection of vertices v for which I ∪{v} is independent. This process terminates at MS25 a maximal independent set. We give a nontrivial lower Forbidden Structures in the Boolean Lattice bound on the size of the final indpendent set, assuming H satisfies certain degree and codegree conditions. We show The Lubell function measures a set of points F in the that the algorithm produces objects of interest in additive n-dimensional Boolean lattice by the expected number of combinatorics, and gives new lower bounds on the Turán times that a random, full chain meets F. WeusetheLubell numbers of certain hypergraphs. function to prove Turán-type results on the maximum size of a set F that avoids certain structures, such as Boolean Patrick Bennett algebras and induced posets of height 2. Joint with J. T. University of Toronto Johnston and Linyuan Lu. [email protected]

J. T. Johnston, Linyuan Lu Thomas Bohman University of South Carolina Carnegie Mellon University [email protected], [email protected] [email protected]

Kevin Milans Dept. of Mathematics MS26 Univ. of West Virginia Juggling Card Sequences [email protected] One of the many ways to represent juggling patterns is through the use of so-called juggling cards. These are tem- MS25 plates which indicate at each time step which balls are to Some Recent Results about Cross Intersecting be thrown to which heights. In this talk, we describe a Families number of new combinatorial and probabilistic results (as well as some unsolved problems) arising in the study of A family of subsets is called intersecting if any two mem- these objects. bers of the family have non-empty intersection. Suppose that each subsets has a given weight, and we want to ﬁnd Ron Graham an intersecting family with the maximum total weight. UCSD The Erd˝os–Ko–Rado theorem is a typical example. I will [email protected] present some new results such as the exact bound for the product sizes of cross t-intersecting k-uniform families. I will also present two very diﬀerent tools: one uses a random MS26 walk and the other uses graph spectra. Extremal Graphs for Connectedness

Norihide Tokushige It is known that the topological connectedness of the inde- Ryukyu University pendence complex of a line graph L(G) is bounded below Japan by ν(G)/2 − 2, where ν(G) denotes the matching number [email protected] of G. We study graphs G for which this parameter is close to ν(G)/2 − 2, and describe the consequences of our work for some long-standing conjectures about hypergraphs. MS25 Planar Posets and Minimal Elements Penny Haxell University of Waterloo Streib and Trotter showed that the dimension of a planar [email protected] poset is bounded in terms of its height. Answering a question posed by R. Stanley, we show that the dimension of Lothar Narins, Tibor Szabo aplanarposetwitht minimal elements is at most 2t +1. Freie Universitaet Berlin This is tight for t =1andt =2.Forlargert,wehavea [email protected], [email protected] lower bound of t +3.

William T. Trotter MS26 School of Mathematics Hamiltonian Increasing Paths in Random Edge Or- Georgia Institute of Technology derings [email protected] If the edges of Kn are totally ordered, a simple path whose Ruidong Wang edges are in ascending order is called increasing.The Georgia Institute of Technology worst-case length of the longest increasing path has re- [email protected] mained an open problem√ for several decades, with asymptotic bounds between n (Graham and Kleitman, 1973) 1 and 2 n (Calderbank, Chung, and Sturtevant, 1984). We MS26 consider the average case. Surprisingly, if a uniformly ran- The Random Greedy Algorithm for Independent dom ordering is chosen, a Hamiltonian increasing path ex- 1 Sets in Hypergraphs ists with probability at least e . Let r be a ﬁxed constant and let H be an r-uniform, D- Mikhail Lavrov regular hypergraph on N vertices. Consider an algorithm Carnegie Mellon University which forms an independent set I by iteratively inserting [email protected] 70 DM14 Abstracts

Po-Shen Loh [email protected] Department of Mathematical Sciences Carnegie Mellon University Stefan van Zwam [email protected] Princeton University [email protected] MS26 Extremal Problems on Diameter 2-critical Graphs MS27 Well-quasi-ordering Graphs by the Topological Mi- A graph is called diameter 2-critical if its diameter is 2 nor Relation and the deletion of any edge would strictly increase the diameter. In 1979, Murty and Simon conjectured that if G Robertson and Seymour proved that graphs are well-qausi- is a diameter 2-critical graph with n vertices and m edges, 2 ordered by the minor relation and the weak immersion then m is no more than n /4, with equality if and only if G relation in the prominent Graph Minors series. That is, is the complete bipartite graph Kn/2,n/2.Inthesameyear, given infinitely many graphs, one graph contains another Caccetta and Haggkvist made the following conjecture for as a minor (or a weak immersion, respectively). However, diameter 2-critical graphs, which, if true, would imply the the topological minor relation does not well-quasi-order Murty-Simon conjecture: the sum of squares of all vertex graphs. An old conjecture of Robertson states that for ev- degrees is at most nm. We present some results on these ery integer k, graphs with no topological minor isomorphic two conjectures, as well as their related problems. Joint to the graph obtained from a path of length k by doubling work with Po-Shen Loh. every edge are well-quasi-ordered by the topological minor relation. We will sketch our recent proof of this conjecture Po-Shen Loh in this talk. This work is joint with Robin Thomas. Department of Mathematical Sciences Carnegie Mellon University Chun-hung Liu [email protected] Georgia Institute of Technology [email protected] Jie Ma Carnegie Mellon University Robin Thomas [email protected] Georgia Tech [email protected] MS27 Treewidth-destroying Partitions of Graphs MS27 Non-planar Extensions of Subdivisions of Planar Planar graphs (and more generally, graphs avoiding a fixed Graphs minor) have the following property, useful in design of approximation algorithms: For every k>0, there exists d>0 Almost 4-connectivity is a weakening of 4-connectivity such that each such graph can be partitioned to k parts so which allows for vertices of degree three. In this talk I will that removal of any of the parts reduces the tree-width to describe generalizations and applications of the following at most d. We study the properties of graph classes with theorem. Let G be an almost 4-connected triangle-free pla- this property, as well as its variations. nar graph, and let H be an almost 4-connected non-planar graph such that H has a subgraph isomorphic to a subdi- Zdenek Dvorak vision of G. Then there exists a graph G such that G is Computer Science Institute isomorphic to a minor of H, and either Charles University, Prague [email protected]ff.cuni.cz (i) G = G + uv for some vertices u, v ∈ V (G) such that no facial cycle of G contains both u and v,or MS27 (ii) G = G + u1v1 + u2v2 for some distinct vertices u ,u ,v ,v ∈ V (G) such that u ,u ,v ,v appear Inequivalent Representations of Highly Connected 1 2 1 2 1 2 1 2 on some facial cycle of G in the order listed. Matroids

We prove that for each fixed finite field F, every sufficiently Sergey Norin connected matroid has only a bounded number of inequiv- McGill University alent F-representations. We obtain this result via a chain [email protected] theorem for highly connected matroids, which as far as we know is new even when restricted to graphs. This theo- Robin Thomas rem will be used in the recently announced proof of Rota’s Georgia Tech conjecture by Geelen, Gerards and Whittle. [email protected] Tony Huynh KAIST, Korea MS27 [email protected] Spanning Trees with Vertices having Large Degrees

Jim Geelen Let G be a connected graph, and let f be a mapping from University of Waterloo V (G)toZ>0,whereZ>0 is the set of positive integers. [email protected] The purpose of this talk is to consider the existence of a spanning tree T such that for each vertex v, Bert Gerards CWI Amsterdam f(v)degT (v), DM14 Abstracts 71

where degT (v) is the degree of v in T . Note that when ferred to as the Small-Set Expansion problem has proven f(v) = 2 for all but two vertices v, a spanning tree sat- to be intimately connected to the Unique Games Conjec- isfying the above condition corresponds to a Hamiltonian turea conjecture. In this talk, we survey the following re- path with prescribed ends in a given graph. In this talk, cent developments in connection to the small set expansion we will concentrate on necessary conditions that are almost problem: necessary conditions. • Higher order Cheeger inequalities and their use in approximating small-set expansion. Kenta Ozeki National Institute of Informatics, Japan • Efficacy of SDP hierarchies for combinatorial opti- [email protected] mization problems on small-set expanders. • Complexity connections between small set expansion Yoshimi Egawa and unique games conjecture. Tokyo University of Science nothing Prasad Raghavendra U C Berkeley [email protected] MS28 Introduction to Hardness of Approximation MS28 Many computational problems of interest are NP-hard. An Analytical Approach to Parallel Repetition extensively studied approach to cope with NP-hardness is to design efficient algorithms that compute approximately We propose an analytical framework for studying parallel optimal solutions, with a guarantee on the quality of ap- repetition, a basic product operation for one-round two- proximation so achieved. Surprisingly, different problems player games. In this framework, we consider a relaxation behave very differently in terms of how well they can be of the value of projection games. We show that this relax- approximated. The discovery of the PCP Theorem in early ation is multiplicative with respect to parallel repetition nineties showed that there are inherent reasons for this var- and that it provides a good approximation to the game ied behavior and the apparent hardness of approximation value. Based on this relaxation, we prove the following for some problems. Subsequent developments have led to parallel repetition bound: For every projection game G a much better understanding of this topic, with several with value at most ρ,thek-fold parallel repetition G⊗k challenges ahead. This talk will give an overview of these has value at most developments, as a preparation towards the follow-up talks √ k/2 ⊗k 2 ρ in the mini-symposium. val(G ) ≤ . 1+ρ Subhash Khot New York University This statement implies a parallel repetition bound for pro- [email protected] jection games with low value ρ. Previously, it was not known whether parallel repetition decreases the value of such games. This result implies stronger inapproxima- MS28 bility bounds for set cover and label cover.Inthis Toward PCPs with Minimal Error framework, we also show improved bounds for few parallel repetitions of projection games, showing that Raz’s coun- The Sliding Scale Conjecture in PCP states that there terexample is tight even for a small number of repetitions. are PCP verifiers with a constant number of queries and Finally, we also give a short proof for the NP-hardness of soundness error that is exponentially small in the ran- label cover(1,δ) for all δ>0, starting from the basic domness of the verifier and the length of the prover’s an- PCP theorem. swers. The Sliding Scale Conjecture is one of the oldest open problems in PCP, and it implies hardness of approx- Irit Dinur imation up to polynomial factors for problems like Max- Weizmann Institute CSP (with polynomial-sized alphabet), Directed-Sparsest- [email protected] Cut and Directed-Multi-Cut. A strengthening of the conjecture to projection games has further applications, e.g., David Steurer to hardness of approximation up to polynomial factors for Cornell University the Closest-Vector-Problem in lattices and to exploring the [email protected] threshold behavior of CSPs. In this talk we discuss the conjecture and approaches to proving it. MS28 Dana Moshkovitz A Characterization of Strong Approximation Re- MIT sistance [email protected] −1 − k → |f (1)| For a predicate f : 1, 1 0, 1withρ(f)= 2k ,we call the predicate strongly approximation resistant if given MS28 a near-satisfiable instance of CSP(f), it is computationally Unique Games Conjecture and Small Set Expan- hard to find an assignment such that the fraction of con- sion straints satisfied is outside the range [ρ(f) − Ω(1),ρ(f)+ Ω(1)]. We present a characterization of strongly approxi- A small set expander is a graph where every set of suffi- mation resistant predicates under the Unique Games Con- ciently small size has near perfect edge expansion. This jecture. We also present characterizations in the mixed talk concerns the computational problem of distinguish- linear and semi-definite programming hierarchy and the ing a small set-expander, from a graph containing a small Sherali-Adams linear programming hierarchy. In the for- non-expanding set of vertices. This problem henceforth re- mer case, the characterization coincides with the one based 72 DM14 Abstracts

on UGC. Each of the characterizations is in terms of exis- [email protected] tence of a probability measure on a natural convex polytope associated with the predicate. Based on joint work with Subhash Khot and Pratik Worah. MS29 Connected Hypergraphs with Small Spectral Ra- Madhur Tulsiani dius TTI Chicago [email protected] In 1970 Smith classified all connected graphs with the spectral radius at most 2. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Re- MS29 cently, the definition of spectral radius has been extend to r-uniform hypergraphs. In this paper, we generalize the Eigenvalue Stability under Hypermatrix Perturba- Smith’s theorem to r-uniform hyper graphs. We show that tion and Random Hypergraphs the smallest limit point of the spectral√ radii of connected k r-uniform hypergraphs is ρk =(k − 1)! 4. We classify all There has been a recent flurry of interest in the spectral connected k-uniform hypergraphs with spectral radius at theory of tensors and hypergraphs as new ideas success- most ρk. fully analogize spectral graph theory to hypergraphs. For one natural incarnation – the homogeneous adjacency spec- Linyuan Lu trum – a substantial number of seemingly basic questions University of South Carolina about hypergraph spectra remain out of reach. One of [email protected] the problems that has yet to be resolved is the (asymptotically almost sure) spectrum of a random hypergraph in the Erd˝os-Rényi sense; another closely related one is MS29 that of describing the spectrum of complete hypergraphs Nonlinear Eigenvalues in Random Regular Graphs (other than a kind of implicit description for the 3-uniform case). We present the requisite background and discuss Given a graph G and a metric space (X, d), one can define some progress in this area that involves tools from alge- the spectral gap γ = γ(G, X)ofG with respect to X via a braic geometry, perturbation theory, and large deviations. generalization of the Rayleigh quotient under the metric d. In this way, one can call a graph family Gn with spectral gaps γn = γ(Gn,X)anexpander family with respect to X Joshua Cooper →∞ University of South Carolina if γn is bounded away from 0 as n .Wediscussthe expansion properties of Gn,d with respect to Gm,d under Department of Mathemeatics [email protected] the standard distance metric, where m n,andsome applications of these concepts.

Mary Radcliffe MS29 UC San Diego Minimum Rank, Maximum Nullity, and Zero Forc- radcliff[email protected] ing Number MS30 This talk will survey recent developments in the problem of determining the minimum rank/maximum nullity of sym- Hamiltonicity of 3-connected Planar Graphs with metric matrices described by a graph, including equality No K2,5 Minor of maximum nullity and zero forcing number for complete subdivision graphs, and Nordhaus-Gaddum type problems Tutte showed that 4-connected planar graphs are hamil- for minimum rank and related parameters. tonian. But examples, such as the Herschel graph, show that this cannot be extended to 3-connected planar graphs without extra conditions. We show that K , -minor-free Leslie Hogben 2 5 3-connected planar graphs are hamiltonian. Results on Iowa State University, hamilton paths in outerplanar graphs form an important American Institute of Mathematics tool for our arguments. Computational evidence provided [email protected] by Gordon Royle suggests that it may be feasible to use our methods to characterize K2,6-minor-free 3-connected nonhamiltonian planar graphs. MS29 Numerical Methods for Estimating the Minimum Mark Ellingham Rank and Zero Forcing Number of Large Graphs Department of Mathematics Vanderbilt University [email protected] Given a graph G, the minimum rank of a graph, mr(G), is the minimum rank over all symmetric matrices with a sparsity pattern determined by the graph. Calculating the Emily Marshall minimum rank and the related zero forcing number of a Vanderbilt University graph are computationally difficult. We offer a iterative al- [email protected] gorithm to estimate these parameters effectively. We show this algorithm is effective for small graphs and apply the al- Kenta Ozeki gorithm to numerically verify related conjectures for larger National Institute of Informatics, Japan graphs. [email protected]

Franklin Kenter Shoichi Tsuchiya Rice University Tokyo University of Science DM14 Abstracts 73

[email protected] [email protected]

MS30 MS30 Forbidden Pairs for the Existence of a Spanning Combining Degree and Connectivity Bounds for Halin Subgraph Graph Linkages

{H1,H2} is called a forbidden pair of G if G contains no For k ≥ 2, it is easy to construct (2k −1)-connected graphs induced subgraph isomorphic to H1 or H2. A Halin graph with minimum degree δ>n/2 which are not k-linked. On is a plane graph H = T ∪ C consisting of a spanning tree T the other hand, all 2k-connected graphs with minimum de- without vertices of degree 2 and a cycle C induced by all gree δ>10k are k-linked, independently of n.Inthistalk leaves of the tree T . In this paper, we will talk some results we will study similar threshold behaviors for other graph about the forbidden pairs for existence of a spanning Halin linkages, and discover that this threshold is not always this subgraph in G. sharp. Ping Yang Florian Pfender Geogia State University Dept. of Math & Stat Sciences [email protected] University of Colorado Denver fl[email protected] Zdenek Ryjácek Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic MS30 [email protected] A Spanning Tree Homeomorphic to a Small Tree MS31 In this talk, we discuss the existence of a spanning tree Modeling Polymer Adsorption at an Inhomoge- which is a subdivision of a small tree. A hamiltonian path neous Surface is a spanning tree which is also a subdivision of K2.In- spired by this observation, for a given integer k,wegive We consider several models of polymers interacting with a degree sum condition for a connected graph to have a an impenetrable surface. We use both directed and self- spanning tree which is a subdivision of a tree of order at avoiding walks to model the polymer configurations and most k. introduce a regular inhomogeneity in the polymer, the surface or both. In each case, we compute the phase diagram Akira Saito and consider the effect of the inhomogeneity on the phase Department of Computer Science and System Analysis boundary. We give qualitative arguments about the form Nihon University of the boundary and consider how it relates to the phe- [email protected] nomenon of pattern recognition. Gary K. Iliev, Stuart Whittington Kazuki Sano University of Toronto Department of Information Science [email protected], [email protected] Nihon University [email protected] Enzo Orlandini University of Padova Department of Physics and Astronomy MS30 [email protected] From Hists to Halin Graphs MS31 As counterpart notions to path, cycle, and hamiltonicity, homeorphically irreducible trees (HIT) and homeomorphi- Exact Solution of a Simple Adsorption Model of cally irreducible spanning trees (HIST) are trees with no De-naturating DNA vertex of degree 2. HITs and HISTs are closely related to We consider a DNA strand in a solvent near an attrac- a family of graphs called Halin graphs. A Halin graph is tive surface modelled as two interacting (friendly) directed obtained from a plane embedding of a HIT of at least 4 walks along the square lattice. We establish a functional vertices by connecting the leaves into a cycle following the equation for the corresponding generating function, which cyclic order on the leaves determined by the embedding. is further refined by means of the obstinate kernel method. In this talk, we will briefly look into some works have been Specifically, we utilise the kernel method in a novel way done on HISTs and Halin graphs, and some other intrigu- to express the exact-solution of the model in terms of two ing research problems related to HISTs and Halin graphs. simpler generating functions for the same underlying combinatorial class. The Zeilberger-Gosper algorithm is then utilised to computationally determine linear homogeneous Songling Shan differential equations solved by these simpler generating Department of Mathematics and Statistics functions, thereby allowing us to analyse the singularity Georgia State University structure and thus critical behaviour of the model. We [email protected] deduce the phase diagram for this model, finding that the system exhibits four phases with a special point where they Guantao Chen all meet. Georgia State University Dept of Mathematics and Statistics Aleks Owczarek 74 DM14 Abstracts

University of Melbourne of type h. It allows us to prove several results concerning [email protected] behaviour of large non-orientable maps as well as enumera- tive properties of non-orientable maps. This is a joint work with Guillaume Chapuy. MS31 The Shape of a Hopf Link Maciej Dolega Universite Paris 7 We examine the geometry of random embeddings of the [email protected] hopf link. The geometry is characterised by descriptors derived from the radius of gyration tensor, and is compared against similar statistics for a single self-avoiding polygon. MS32 The Brownian Continuum Random Tree Is the Andrew Rechnitzer Scaling Limit of Random Dissections University of British Columbia [email protected] We are interested in the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissec- MS31 tions or uniform p-angulations. As their number of ver- Random Knots and Confinement Considerations tices n goes to infinity, we show that these random graphs, −1/2 Chromosomal DNA is tightly packed in all organisms. rescaled by n , converge in the Gromov–Hausdorff sense When the chromosomes are circular, or divided into looped towards a multiple of Aldous’ Brownian tree when the domains, the packing geometry has a direct impact on the weights decrease sufficiently fast. This gives in particular topology of the chains. Problems of packing motivate our interesting combinatorial consequences concerning the ge- research on minimal step lattice polygons that are uncon- ometry of these random dissections. The scaling constant fined, or confined to slabs and tubes. depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Mariel Vazquez Markov chain which compares the length of geodesics in San Francisco State University dissections with the length of geodesics in their dual trees. [email protected] Joint work with Nicolas Curien and Bndicte Haas. Nicolas Curien MS31 CNRS & LPMA Counting Knotted 2-Spheres in Tubes in Z4 [email protected]

There are many rigorous and numerical results about knot- Igor Kortchemski ting of simple closed curves in three dimensional lattices Ecole Normale Superieure but for higher dimensional cases very little is known. If [email protected] we consider 2-spheres confined to a tube in Z4,sothat the system is quasi-one dimensional, we show how trans- Bénédicte Haas fer matrix techniques can be used to prove that all except Ceremade & DMA exponentially few such 2-spheres are knotted. This is joint [email protected] work with Chris Soteros and De Witt Sumners.

Stuart Whittington MS32 University of Toronto [email protected] Exploring Some Non-constructive Map Bijections Maps play a sometimes hidden role in many different MS32 branches of mathematics. As a result generating series for related classes of maps can often be obtained by several The KP Hierarchy and Generating Series for Maps different unrelated techniques. Algebraic relationships be- In this talk we will introduce and describe the KP hierarchy tween such generating series can guarantee the existence of of partial differential equations and their formal power se- bijections without providing any constructive way to iden- ries solutions. We will then discuss how this can be applied tify their actions. This phenomenon is particularly promi- to problems in map enumeration. nent when maps are used to study the combinatorics of Jack symmetric functions. In this setting, a shifted version Sean Carrel of the Jack parameter can be related to a quantified depar- University of Waterloo ture from orientability. Generating series in this context [email protected] continue to exhibit symmetries that in the restriction to orientable or non-orientable maps are explained by simple bijections, but these bijections cease to provide an expla- MS32 nation in the perturbed setting. A Bijection for Rooted Maps on Non-orientable Surfaces MichaelA.LaCroix MIT One of the major bijective results in the area of enumera- [email protected] tion of maps and in the area of studying uniform random maps is the approach initiated by Cori and Vauquelin and extended by Chapuy, Marcus and Shaeffer which treats MS32 rooted orientable maps of genus g. We extend their bi- A Graphical Proof of a Generalization of Boccara’s jection for rooted maps drawn on a non-orientable surface Theorem Concerning the Multiplication of Long DM14 Abstracts 75

Cycles from “local stability” results, which has other applications.

In 1980, Boccara showed that the number of ways of writ- ing a fixed odd permutation as a product of an n-cycle and Sergey Norin, Liana Yepremyan n − 1-cycle (in the symmetric group on n symbols) is inde- McGill University pendent of the permutation chosen. We give a substantial [email protected], [email protected] generalization of this result, and summaries of an algebraic and combinatorial proof of it. This is joint work with V. Féray. MS33 Partitioning Random Hypergraphs Amarpreet Rattan University of London I will discuss the algorithmic problem of finding a planted [email protected] partition in a random k-uniform hypergraph, a generalization of both the stochastic block model (k = 2) and planted random k-SAT. For k ≥ 3, the problem exhibits an algo- MS33 rithmic gap: for a large range of densities unique solutions Randomly Coloring Random Graphs exist but no known efficient algorithms can find them. I will present a conjecture on the location of the algorith- We discuss some problems related to colorings of random mic threshold and a proof of it for the class of statistical graphs: (i) Rainbow matchings and hamilton cycles; (ii) algorithms. Based on joint work with Vitaly Feldman and Game chromatic number of random graphs; (iii) Rainbow Santosh Vempala. connection of random graphs; (iv) zebraic colorings. Will Perkins Alan Frieze Georgia Tech Carnegie Mellon University [email protected] Department of Mathematical Sciences [email protected] MS34 Phase Transitions in the Laplacian Determinant MS33 A Short Proof of Gowers’ Lower Bound for the We discuss several types of combinatorial objects enumer- Regularity Lemma ated by the Laplacian determinant on a planar graph, and in particular indicate various phase transitions in these ob- A celebrated result of Gowers states that for every >0 jects which occur for infinite graphs. As an example, we there is a graph G such that every -regular partition of compute the growth rate of the set of k-component essen- G (in the sense of Szemerédi’s regularity lemma) has order tial spanning forests on a bi-infinite strip of fixed width. given by a tower of exponents of height polynomial in 1/. In this note we give a new proof of this result that uses a Richard Kenyon construction and proof of correctness that are significantly Brown University simpler and shorter. Joint work with Asaf Shapira. [email protected]

Guy Moshkovitz Tel Aviv University MS34 [email protected] The Structure of the Abelian Sandpile

The Abelian sandpile is a discrete diffusion process on con- MS33 figurations of chips on the integer lattice, introduced in On the Number of Real Roots of Kac Polynomials 1987 by Bak, Bank and Wiesenfeld. Recent developments have provided a mathematical explanation for the fractal Real roots of Kac polynomials have been studied exten- structures produced by this process. We will discuss these sively in the literature under various conditions. With min- results, and some important remaining questions. imal assumptions on the coefficient variables and by using purely combinatorial method, we show that the number of Lionel Levine real roots is (2/pi) log n + O(1) in expected value. This Cornell University bound is almost best possible. (Joint with O. Nguyen and [email protected] V. Vu) Wesley Pegden Hoi Nguyen Courant Institute Ohio State University [email protected] [email protected] Charles Smart Massachusetts Institute of Technology MS33 [email protected] Turán Function for the Generalized Triangle

The generalized triangle Tr is an r-graph with edges MS34 {1, 2,...,r}, {1, 2,...,r−1,r+1} and {r, r +1,...,2r−2}. Slow Mixing for the Hard--Core Model on Z2 Frankl and Füredi computed the Turán density of Tr for r =5, 6. We strengthen their result computing the Turán The hard-core model has attracted much attention across function ex(n, Tr) exactly for large n. The proof is based on several disciplines, representing lattice gases in statistical a technique for deriving exact results for the Turán function physics and independent sets in discrete mathematics and 76 DM14 Abstracts

computer science. On finite graphs, we are given a param- [email protected] eter λ, and an independent set I arises with probability proportional to λ|I|. We are interested in determining the mixing time of local Markov chains that add or remove a MS34 small number of vertices in each step. On finite regions of Local Statistics of the Abelian Sandpile Model Z2 it is conjectured that there is a phase transition at some critical point λc that is approximately 3.79. It is known We show how to compute local statistics of the abelian that local chains are rapidly mixing when λ<2.3882. sandpile model on the square, hexagonal, and triangular We give complementary results showing that local chains lattices. On the square lattice, all local events are rational will mix slowly when λ>5.3646 on regions with periodic polynomials in 1/π, while on the hexagonal√ and triangular (toroidal) boundary conditions and when λ>7.1031 with lattices they are rational polynomials in 3/π. The proofs non-periodic (free) boundary conditions. The proofs use a use the burning bijection between sandpiles and spanning combinatorial characterization of configurations based on trees, and the methods of Kenyon and Wilson for comput- the presence or absence of fault lines and an enumeration ing grove partition functions. of a new class of self-avoiding walks called taxi walks. David B. Wilson Microsoft Corporation Antonio Blanca [email protected] U.C. Berkeley [email protected] MS35 David J. Galvin Recognizing and Colouring Claw-Free Graphs University of Notre Dame Without Even Holes [email protected] An even hole is an induced even cycle. Even-hole-free graphs generalize chordal graphs. We prove that claw-free Dana Randall even-hole-free graphs can be decomposed by clique cutsets Georgia Institute of Technology into, essentially, proper circular-arc graphs. This provides [email protected] the basis for our algorithms for recognizing and colouring these graphs. Our recognition algorithm is more efficient Prasad Tetali (O(n2m2loglogn)) than known algorithms for recognizing School of Mathematics, Georgia Institute of Technology even-hole-free graphs (O(n19)). Colouring claw-free graphs Atlanta, Ga 30332-0160 is NP-hard and the complexity of colouring even-hole-free [email protected] graphs is unknown, but our algorithm colours claw-free even-hole-free graphs in O(n4)time.

Kathie Cameron MS34 Wilfred Laurer University, Canada [email protected] Maximum Independent Sets in Random d-regular Graphs Steven Chaplick Department of Computer Science, University of Toronto Satisfaction and optimization problems subject to random [email protected] constraints are a well-studied area in the theory of computation. These problems also arise naturally in combi- Chinh Hoang natorics, in the study of sparse random graphs. While Wilfrid Laurier University the values of limiting thresholds have been conjectured for Dept of Physics and Computing Science many such models, few have been rigorously established. [email protected] In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d ex- ceeding a constant d0, there exist explicit constants a, c MS35 depending on d such that the maximum size has constant fluctuations around na − c log n establishing the one-step Digraph Analogues of Interval Graphs and of Triv- replica symmetry breaking heuristics developed by statis- ially Perfect Graphs tical physicists. As an application of our method we also Interval graphs are precisely the reflexive graphs for which prove an explicit satisfiability threshold in random regular list homomorphism problems are solvable in polynomial k-NAE-SAT. This is joint work with Jian Ding and Allan time. For general digraphs, this role is played by the class Sly. of digraphs free of structures we call digraph asteroidal triples (DATs). Trivially perfect graphs are precisely the Jian Ding reflexive graphs for which list homomorphism problems are University of Chicago solvable in logarithmic space. For general digraphs, this [email protected] role is taken by the class of digraphs without structures we call circular N’s (cN’s). Both DAT-free and cN-free digraphs exhibit interesting combinatorial properties; in par- Allan Sly ticular, they can recognized in polynomial time. This is University of California, Berkeley joint work with Arash Rafiey, and partly also with Laszlo [email protected] Egri, and Benoit Larose.

Nike Sun Pavol Hell Stanford School of Computer Science DM14 Abstracts 77

Simon Fraser University [email protected] [email protected]

MS36 MS35 Shelah’s Grid Ramsey Problem Coloring Graphs Without Induced Paths and Cy- cles Shelah introduced the following problem related to Hales- 2 Jewett numbers. The grid graph Γn has vertex set [n] ,and Let Pt and C denote a path on t vertices and a cycle on any two vertices in the same row or column are adjacent. vertices, respectively. A graph G is (Pt,C)-free if it does Let g(k) be the minimum n such that every k-edge-coloring not contains Pt or C as an induced subgraph. In this of Γn has an axis-parallel rectangle with opposite edges the talk we study the k-COLORING problem for (Pt,C)-free same color. Shelah asked: is g(k) at most polynomial in graphs. We shall show that for most combinations of k, t k? We give a negative answer. and the problem turns out to be NP-complete. On the other hand, we give certifying polynomial time algorithms Jacob Fox for (P6,C4)-free graphs for every k.Fork =3ork =4, MIT we also provide an explicit and complete list of minimal [email protected] non-k-colorable (P6,C4)-free graphs. David Conlon Shenwei Huang University of Oxford Department of Computer Science [email protected] Simon Fraser University [email protected] Choongbum Lee MIT Pavol Hell cb [email protected] School of Computer Science Simon Fraser University Benny Sudakov [email protected] ETH Zurich [email protected] MS35 Non-Planar Extensions of Planar Signed Graphs MS36 Monochromatic Bounded Degree Subgraph Parti- A signed graph is a pair (G, Σ) where G is a graph, and tions Σ ⊆ E(G). Given a planar signed graph (G, Σ), we are interested in determining the minimal non-planar signed Let F = {F1,F2,...} be a sequence of graphs such that Fn graphs (H,Γ) that “contain” (G, Σ). We show (assuming is a graph on n vertices with maximum degree at most 3-connectivity for (G, Σ)) that there are a small number Δ. We show that there exists an absolute constant C of these “minimal non-planar extensions” of (G, Σ), and such that the vertices of any 2-edge-colored complete graph describe them explicitly. This result generalizes work done can be partitioned into at most 2CΔlogΔ vertex disjoint by Robertson, Seymour, and Thomas. monochromatic copies of graphs from F.

Bertrand Guenin Andrey Grinshpun University of Waterloo MIT [email protected] [email protected]

Katie Naismith Gabor Sarkozy Department of Combinatorics and Optimization Worcester Polytechnic University of Waterloo [email protected] [email protected]

MS36 MS35 Restricted Ramsey-Type Theorems On the Contour of Chordal Bipartite Graphs Abstract not available at time of publication. A vertex in a graph is a contour vertex if none of its neighbors has a larger eccentricity than the vertex. We show Vojtech R¨odl that the set of contour vertices of a connected chordal bi- Emory University partite graph, one without chordless cycles on at least six [email protected] vertices, is geodetic; thus every vertex of the graph lies on a shortest path between some pair of contour vertices. A Hiep Han similar result does not hold for bipartite graphs with long Emory University chordless cycles. Universidade de Sao Paulo [email protected] Danilo Artigas Universidade Federal Fluminense Brazil MS36 [email protected] Multipass Random Coloring of Simple Uniform Hypergraphs R. Sritharan University of Dayton Let D∗(n) be the largest number such that every simple 78 DM14 Abstracts

n-uniform hypergraph with maximum edge degree at most Karlsruhe Institute of Technology D∗(n) is two colorable (has Property B). We prove that [email protected] D∗(n)=Ω(n2n). The method generalizes to b-simple hypergraphs (every pair of distinct edges intersects in at most b vertices) and hypergraphs of arithmetic progres- MS37 sions. This way we obtain an improved lower bound for VanderWaerdennumbersandgetW (n)=Ω(2n). List Coloring the Square of Sparse Graphs

Jakub Kozik The square of a graph G is obtained from G by adding all Theoretical Computer Science edges between vertices with a common neighbor. Borodin Jagiellonian University et al. proved that squares of planar graphs of girth g ≥ 7 [email protected] and sufficiently large maximum degree Δ are list (Δ + 1)- colorable, which is optimal with regard to the girth. We Dmitry Shabanov strengthen their result by replacing the girth and planarity Lomonosov Moscow State University constraint by a maximum average degree constraint, which Faculty of Mechanics and Mathematics yields a more refined bound. [email protected] Marthe Bonamy,BenjaminLéveque MS36 LIRMM, Université Montpellier 2 [email protected], [email protected] Hypergraph Ramsey Numbers

In this talk I will discuss an extension of cycle versus com- Alexandre Pinlou plete graph Ramsey numbers to hypergraphs. A general- LIRMM, Université Montpellier 3 ized triangle is a k-uniform hypergraph (or simply k-graph) [email protected] of three edges e, f, g such that |e ∩ f| = |f ∩ g| = |g ∩ e| =1 and e ∩ f ∩ g = ∅. The Ramsey number Rk(3,t)isthe minimum n such that every k-graph on at least n vertices MS37 contains either a generalized triangle or a clique of size t. 2 − The celebrated Ajtai-Komlos-Szemerédi Theorem together Painting Squares in Δ 1 Shades 2 with Kim’s construction show that R2(3,t)=Θ(t / log t). I will outline here the proof that for every k ≥ 3, Rk(3,t)= Cranston and Kim conjectured that if G is a connected Θ∗(t3/2) where the star denotes implicit powers of loga- graph with maximum degree Δ and G is not a Moore 2 2 rithms in t and a constant depending only on k. General- Graph, then χ(G ) ≤ Δ − 1; here χ is the list chro- izations to cycle-complete graph Ramsey numbers will be matic number. (This saves one color over the easy upper 2 discussed, together with some open problems. This is joint bound of Δ given by combining Brooks’ Theorem (the list-coloring version) with the fact that G2 has maximum work with Dhruv Mubayi and Sasha Kostochka. degree at most Δ2.) We prove their conjecture; in fact, this upper bound holds even for online list chromatic number. Jacques Verstraete University of California-San Diego [email protected] Daniel Cranston Virginia Commonwealth University Department of Mathematics and Applied Mathematics MS37 [email protected] Online and Size anti-Ramsey Numbers Landon Rabern A graph is properly edge-colored if no two adjacent edges LBD Software Solutions have the same color. For a graph H, we introduce the size [email protected] anti-Ramsey number ARs(H) to be the smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of H.Inthistalkwe MS37 shall discuss this analogue of a size-Ramsey number. We determine the bounds on ARs(H) for general as well as for Choosability of Graphs With Bounded Order some specific graphs and provide a number of results for the on-line setting of this problem. Several open problems The choice number of a graph G, denoted ch(G), is the will be presented. minimum integer k such that, for any assignment of lists of size k to the vertices of G, there exists a proper colouring Maria Axenovich of G in which every vertex is mapped to a colour in its list. Iowa State University In this talk, we discuss recent results and open problems [email protected] regarding the choice number of graphs for which the number of vertices is bounded in terms of χ. In particular, this Torsten Ueckerdt includes a solution to Ohba’s Conjecture, which says that Karlsruhe Institute of Technology if |V (G)|≤2χ(G)+1,thench(G)=χ(G). We will also [email protected] talk about a recent strengthening of this result, and propose several open problems for future study. The results Kolja Knauer of this talk are joint work with Bruce Reed, Douglas West, Université Montpellier 2 Hehui Wu and Xuding Zhu. [email protected] Jonathan A. Noel Judith Stumpp McGill University DM14 Abstracts 79

[email protected] [email protected]

MS37 MS38 Biembeddings and Graph Decompositions The Potential Technique in Graph Coloring A face 2-colourable embedding of a simple graph in which We discuss recent advancements and applications of the black faces have boundary lengths m1,m2,...,mr and potential technique developed by Kostochka and Yancey white faces have boundary lengths n1,n2,...,nt is a biem- for lower bounding the edge density of k-critical graphs. bedding of the cycle lists M = m1,m2,...,mr and N = In particular, we will discuss 3-coloring graphs of girth ﬁve n1,n2,...,nt. We use Euler’s classic embedding formula on surfaces. to derive necessary and suﬃcient conditions for the existence of such an embedding in the sphere given two arbi- Luke Postle trary lists of cycle lengths. We also discuss a surprising Georgia Tech relationship between these biembeddings and more general [email protected] cycle decomposition problems. (Joint work with Barbara Maenhaut.)

MS38 Ben Smith University of Queensland Triple Systems, Gray Codes and Snarks [email protected]

The 2-block-intersection graph of any simple twofold triple system is necessarily a cubic graph. If it is Hamiltonian MS38 then a 2-intersecting cyclic Gray code for the design can Switchings for 1-Factorizations be found. If it is not Hamiltonian then the graph has the potential to be a somewhat rare type of graph known as a We deﬁne four graphs where the vertices are isomorphism snark. classes of 1-factorisations of a complete graph Kn and edges are produced by simple switchings. These graphs are stud- David A. Pike iedindetailforn ≤ 12. For some switchings the graph will Memorial University of Newfoundland be disconnected for all orders because 1-factorisations have [email protected] a parity that is preserved by the switchings. We also study isolated vertices, including the celebrated case of perfect 1- factorisations, and connections with atomic Latin squares. MS38 On a Problem of Mariusz Meszka Ian Wanless Monash University We consider a problem due to Mariusz Meszka similar to [email protected] the well-known conjecture of Marco Buratti. Does there exist a near-1-factor in the complete graph on Zp, p an odd prime, whose multiset of edge-lengths equals a given MS39 multiset L? This may be viewed as a generalization of the On the Number of Bh-Sets existence problem for extended Skolem sequences. A set of positive integers is a Bh-set if all the sums of h Alexander Rosa elements from the set are distinct. We provide asymptotic Department of Mathematics bounds for the number of Bh-sets of a given cardinality { } McMaster University contained in [n]= 1,...,n . As a consequence of our [email protected] results, better upper bounds for a problem of Cameron and Erd˝os (1990) in the context of Bh-sets are obtained. We use these results to estimate the maximum size of a Bh-set contained in a random subset of [n]withagiven MS38 cardinality. Approaching the Minimum Number of Clues Su- doku Problem Via the Polynomial Method Sang June Lee Korea Advanced Institute of Science and Technology Determining the minimum number of clues that must be [email protected] present in a Sudoku puzzle in order to uniquely complete the puzzle is known as the minimum number of clues prob- Domingos Dellamonica JR. lem. For a 9-by-9 Sudoku board, it has been conjectured Emory University that one needs 17 clues, and this has been conﬁrmed by [email protected] McGuire et al. in a massive computer search. We apply the polynomial method to the analogous problem for the Yoshiharu Kohayakawa 4-by-4 Shidoku board to illustrate how one might approach Instituto de Matematica e Estatistica the more general problem. This is joint work with Aden Universidade de So Paulo, Brazil Forrow. [email protected]

John Schmitt Vojtech RODL Middlebury College Emory University Middlebury, VT [email protected] 80 DM14 Abstracts

Wojciech Samotij [email protected], [email protected] Tel Aviv University School of Mathematical Sciences [email protected] MS40 Degree Sequences and Forced Adjacency Relation- ships MS39 Degree sequences of graphs usually have several labeled re- On Erds Conjecture on the Number of Edges in alizations with differing edge sets. In some cases, however, 5-Cycles vertices with certain degrees are adjacent (or non-adjacent) in every realization. The quintessential example of this Erdös, Faudree, and Rousseau in 1992 showed that a graph occurs with a threshold graph, where every adjacency re- on n vertices and with at least n2/4 + 1 edges comprise lationship is uniquely determined by the degree sequence. at least 2n/2 + 1 edges on triangles and this result is Given a degree sequence, we characterize degree pairs cor- sharp. They also considered a conjecture of Erdös that responding to vertices that are forcibly (non)-adjacenct, re- such a graph have at most n2/36 non-pentagonal edges. vealing connections with the Erd˝os–Gallai inequalities and This was mentioned in other paper of Erdös and also in the majorization order. Fan Chung’s problem book. In this talk we give a graph√ of n2/4 + 1 edges with much more, namely n2/8(2 + 2) + MichaelD.Barrus O(n) pentagonal edges, disproving the original conjecture. Brigham Young University We also show that this coefficient is asymptotically the best [email protected] possible. MS40 Zeinab Maleki Isfahan University of Technology Stability of the Potential Function [email protected] The potential number of a graph H, denoted σ(H,n), is the minimum integer such that any graphic sequence of Zoltan Furedi length n has a realization containing H as a subgraph. University of Illinois, Urbana-Champaign AgraphH has degree-sequence stability if every graphic [email protected] sequence with sum close to σ(H,n) having no realization containing a copy of H can be transformed into an extremal sequence with o(n) additions and subtractions. We discuss MS39 the degree-sequence stability of various families of graphs. Large Subgraphs Without Short Cycles Catherine Erbes, Michael Ferrara We study two extremal problems about subgraphs exclud- University of Colorado Denver ing a family F of fixed graphs. i) Among all graphs with m [email protected], edges, what is the smallest size f(m, F)ofalargestF–free [email protected] subgraph? ii) Among all graphs with minimum degree δ and maximum degree Δ, what is the smallest minimum de- Ryan R. Martin F F gree h(δ, Δ, ) of a spanning –free subgraph with largest Iowa State University F minimum degree? We study the case where is composed [email protected] of all cycles of length at most 2r +1, r ≥ 1. In this case, we give meaningful lower bounds on f(m, F) and deter- Paul Wenger mine the asymptotic value of h(δ, Δ, F). This is joint work Rochester Institute of Technoloy with F. Foucaud and M. Krivelevich, and with B. Reed. [email protected] Guillem Perarnau Universitat Politecnica de Catalunya MS40 [email protected] Degree Sequences of Uniform Hypergraphs: Re- sults and Open Problems

MS39 A sequence of nonnegative integers is k-graphic if it is the degree sequence of a k-uniform hypergraph. We present Asymptotic Distribution of the Numbers of Ver- sharp suﬃcient conditions for k-graphicality based on a tices and Arcs in the Giant Strong Component in sequence’s length and degree sum. Further, Kocay and Li Sparse Random Digraphs gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3- For the random digraph on n vertices, D(n, Prob(arc)= graphic sequence into any other realization. We extend p), Karp (90) andLuczak (90) found that for p = c/n, c > their result to k-graphic sequences for all k ≥ 3andgive 1, with probability tending to 1, there is a giant strong several applications of edge exchanges in hypergraphs. component of size linear in n. We show that in D(n, p = c/n)andD(n, number of arcs = m), m = cn,thejoint Michael Ferrara distribution of the numbers of vertices and arcs in the giant University of Colorado Denver strong component is asymptotically Gaussian with mean [email protected] and covariance matrix linear in n. Behrens Sarah Daniel J. Poole, Boris Pittel University of Nebraska-Lincoln The Ohio State University [email protected] DM14 Abstracts 81

Catherine Erbes Catherine Erbes University of Colorado Denver University of Colorado Denver [email protected] [email protected]

Stephen Hartke Ryan R. Martin University of Nebraska-Lincoln Iowa State University [email protected] [email protected]

Benjamin Reiniger University of Illinois Urbana-Champaign MS41 [email protected] Mitochondrial DNA Organization in Trypanoso- matid Parasite Hannah Spinoza University of Illinois Urbana Champaign Trypanosomes and leishmania, two trypanosomatid para- [email protected] sites, are protozoa which cause fatal diseases such as sleep- ing sickness, Chagas disease and Leishmaniasis. A dis- Charles Tomlinson tinctive feature of trypanosomatid parasites is that their University of Nebraska-Lincoln mitochondrial DNA, known as kinetoplast DNA, is orga- [email protected] nized into thousands of minicircles and a few dozen maxi- circles. Minicircles in these organisms are topologically linked, forming a gigantic chainmail-like network. The MS40 topological structure of the minicircles is of great signif- Minimal Forbidden Sets for Degree Sequence Char- icance both because it is species-specific and because it is acterizations a promising target for the development of drugs. How- ever, the biophysical factors that led to the formation of Given a set F of graphs, a graph G is F-free if G does not the network during evolution and that maintain the net- contain any member of F as an induced subgraph. Barrus, work remain poorly understood. In this talk I will dis- Kumbhat, and Hartke (2008) called F a degree-sequence- cuss a mathematical and computational model that help forcing (DSF) set if, for each graph G in the class C of us quantify possible growth mechanisms of the network. F-free graphs, every realization of the degree sequence of We find that minicircle networks rapidly form, following a G is also in C.ADSFsetisminimal if no proper subset is percolation-saturation pathway, when the density of mini- also DSF. In this talk, we present new properties of minicircles is increased, and that the valence (the number of mal DSF sets, including that every graph is in a minimal minicircles linked to any given minicircle) grows linearly DSF set and that there are only finitely many DSF sets with density of minicircles. Limitations and extensions of of cardinality k. Using these properties and a computer these models and results will also be discussed. search, we characterize the minimal DSF triples. Javier Arsuaga Stephen Hartke San Francisco State University University of Nebraska-Lincoln [email protected] [email protected]

Michael D. Barrus MS41 Brigham Young University Assorted Topics in the Monte Carlo Simulation of [email protected] Polymers

We will touch on various aspects of Monte Carlo simula- MS40 tions of polymers, with the particular goal of motivating The Asymptotic Behavior of the Potential Function the use of moves which are neither local nor global, but which occur at intermediate length scales. We will touch Given a graph H,asequenceπ of nonnegative integers is on applications of Monte Carlo to enumerations of walks, potentially H-graphic if there is a graph G whose vertex work in progress on possible methods to adapt the pivot al- degrees match the entries of π and H is a subgraph of G. gorithm to the study of conﬁned polymers and θ polymers, The potential number of H is the minimum even integer and a method to rapidly calculate the hydrodynamic radius σ(n, H) such that every n-term sequence of nonnegative of a polymer. integers with sum at least σ(n, H) is potentially H-graphic. We present recent results on potential numbers, including Nathan Clisby precise asymptotics and stability for sequences that are not University of Melbourne potentially H-graphic. [email protected]

Paul Wenger Rochester Institute of Technoloy MS41 [email protected] Inhomogeneous Percolation with a Defect Plane

Michael Ferrara Let L be an s-dimensional sublattice of the d-dimensional University of Colorado Denver hypercubic lattice Zd,with2≤ s

and their boundaries. the permanent. The goal of this talk is to present several conjectures generalizing to general stable polynomials new Neal Madras (entropic) lower bounds on the permanent. Time permit- York University ting, we will present also conjectures related to the relative [email protected] approximation of 4D Pascal’s determinant and the permanent of positive semideﬁnite matrices. Gary K. Iliev University of Toronto Leonid Gurvits [email protected] The City College of New York - CUNY [email protected] EJ Janse van Rensburg York University [email protected] MS42 Title Not Available at Time of Publication

MS41 Abstract not available at time of publication. Subknots in Closed Chains Robin Pemantle For a fixed knot configuration, the subknots are the knot University of Pennsylvannia types seen in the open subchains. For nice knot configura- [email protected] tions (like ones minimized with respect to some knot energy), the subknots are typically simpler knot types than thehostknottype.Wecompareandcontrastthesetof MS42 subknots coming from KnotPlot configurations, tight knot Introduction to Hyperbolic and Real Stable Poly- configurations, and random configurations. This is joint nomials work with Ken Millett and Andrzej Stasiak. We will introduce real stable and hyperbolic polynomials Eric Rawdon and discuss their basic properties in preparation for the Department of Mathematics rest of the talks. No background will be assumed. University of St. Thomas [email protected] Nikhil Srivastava,NikhilSrivastava Microsoft Research India [email protected], [email protected] MS41 Enumerating Knots in Lattice Tube Models of Polymers MS42 Self-avoiding polygons on the cubic lattice are the standard Hyperbolic Polynomials, Interlacers, and Sums of statistical mechanics model for ring polymers in dilute so- Squares lution. For this model, the effects of geometrical confinement are studied by constraining the polygons to an infinite Hyperbolic polynomials are real polynomials whose real rectangular lattice tube. For small tube sizes, exact enu- hypersurfaces are nested ovals, the inner most of which is meration and generation of polygons is accomplished using convex. These polynomials appear in many areas of math- transfer-matrices. Using this approach, I will present re- ematics, including optimization, combinatorics and differ- cent results regarding the knot complexity of self-avoiding ential equations. I’ll give an introduction to this topic and polygons in a lattice tube. discuss the special connection between hyperbolic polynomials and their interlacing polynomials (whose real ovals Christine Soteros interlace the those of the original). This will let us related University of Saskatchewan inner oval of a hyperbolic hypersurface to the cone of non- [email protected] negative polynomials and, sometimes, to sums of squares. An important example will be the bases generating polynomial of a matroid. MS42 Stable Multivariate Polynomials and the Lower Cynthia Vinzant Bounds: (old)Results and (new)Conjectures University of Michigan Dept. of Mathematics

Let p(x1, ..., xm) be homogeneous stable polynomial, i.e. [email protected] it does not have zeros with positive real parts. In the last 10 years these were several spectacular applications of this class of polynomials and its natural generalizations MS42 to many problems in various areas: open problems had Stable Multivariate Eulerian Polynomials been solved and very complicated proofs had been drasti- cally simplified. One of the first applications was a radical We discuss a multivariate generalization of the well-known simplification, vast generalization and even some improve- Eulerian polynomials that arose from the solution of the ment of A. Schrijver’s asymptotically optimal lower bound Monotone Column Permanent Conjecture. We give several on the number of perfect matchings in regular bipartite examples how this stable multivariate generalization sim- graphs. The polynomials related to the Schrijver’s lower plifies and extends results in permutation statistics. Fi- bound are products of linear forms. It turned out that for nally, we describe another surprising connection linking those very special polynomials the combination of Schri- these polynomials to stationary distributions of certain Ex- jver’s approach and the Bethe approximation leads to a clusion Processes. This talk is based on collaboration with new very precise lower bound on the mixed derivative, i.e. P. Brändén, J. Haglund, M. Leander, D.G. Wagner, and DM14 Abstracts 83

N. Williams. mr [email protected]

Mirko Visontai Google MS43 [email protected] On the Location of the Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic poly- MS43 nomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we Graph Polynomials from Graph Homomorphisms examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph. A sequence of graphs (Hn)isstrongly polynomial if for every graph G the number of homomorphisms from G to Johann A. Makowsky Hn is given by a polynomial function of n. For example, the Faculty of Computer Science, Technion - IIT sequence (Kn) is strongly polynomial since the number of [email protected] homomorphisms from G to Kn is the chromatic polynomial evaluated at n. Many important graph polynomials are Elena Ravve determined by strongly polynomial sequences of graphs. Department of Software Engineering We give a wide-ranging construction for such sequences. ORT Braude College, Karmiel, Israel [email protected] Delia Garijo Department of applied mathematics Nicolas Blanchard Uinversity of Sevilla, Spain ENS, Paris [email protected] ENS, Paris [email protected] Andrew J. Goodall Computer Science Institute (IUUK) Charles University in Prague, Czech Republic MS43 [email protected]ff.cuni.cz The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Prob- Jaroslav Nesetril lems department of Applied Mathematics Charles University By a result of Vertigan, it is #P-hard to count vertex κ- [email protected]ff.cuni.cz colorings over planar graphs provided κ ∈{0, 1, 2}.We consider the problem of edge coloring a graph, a special case of vertex coloring. We show that it is #P-hard to MS43 count edge κ-colorings over planar r-regular graphs pro- ≥ ≥ On the Complexity of Various Ising Polynomials vided κ r 3. When this condition fails to hold, the problem is trivially tractable. We note that proving #P- hardness for a special case of a problem gives a stronger We study the complexity of partition functions of the result. Ising model with constant energies and external field. The trivariate Ising polynomial is shown to be #P-hard to eval- Jin-Yi Cai, Heng Guo uate at all points, except those in set of low dimension, even University of Wisconsin, Madison on simple bipartite planar graphs. Under the Counting Ex- [email protected], [email protected] ponential Time Hypothesis, we give a dichotomy theorem stating that the bivariate Ising polynomial either take exponential time to compute, or can be done in polynomial Tyson Williams time. Department of Computer Science University of Wisconsin, Madison [email protected] Tomer Kotek Institut für Informationssysteme Technical University, Vienna, Austria MS44 [email protected] A New Generalisation of a de Bruijn Erdos Theo- rem MS43 Let (V,E) be a hypergraph. Assuming that any two edges On the Largest Real Root of the Independence intersectonatmostk vertices and that any two vertices is Polynomials of Trees containedinatleastt edges, we prove that k.|E|≥t.|V |. This generalizes a Theorem of De Bruijn and Erd˝os who In this talk we investigate about the largest real root of proved it in the case where k = t =1. independence polynomials of trees. We define an ordering on graphs. Using the mentioned ordering, we obtain some Pierre Aboulker results about the largest real root. In particular we show Concordia University that among all trees the paths have the minimum and the Montreal stars have the maximum real roots. [email protected]

Mohammad Reza Oboudi Stephan Thomass´e University of Isfahan ENS Lyon 84 DM14 Abstracts

√ [email protected] tices or at least n/5 distinct lines. We also show that a random graph induces Ω(n2) distinct lines.

MS44 Cathryn Supko Lines in Metric Spaces and Hypergraphs: A Survey Concordia University [email protected] We survey the problems and results related to the points and lines in metric spaces and hypergraphs. We will discuss the definition of the lines, the (unofficially) 10-year MS45 old Chen-Chvátal conjecture, its various special cases, the Lazy Cops and Robbers efforts and results in trying to generalize the classical De Bruijn-Erdös theorem, and many recent variations on the We consider a variant of the game of Cops and Robbers, topic. called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the Xiaomin Chen cop number for this game, which we call the lazy cop num- Shanghai Jianshi LTD ber. In this talk we will discuss results concerning the lazy Shanghai cop number of binomial random graphs, hypercubes, and [email protected] graphs with genus g. Deepak Bal, Anthony Bonato MS44 Ryerson University Points, Lines, and the Manhattan Metric [email protected], [email protected]

It is a well-known fact that every non-collinear set of n Bill Kinnersley points in the plane determines at least n distinct lines. This University of Illinois is a corollary to Sylvester-Gallai theorem, as observed by [email protected] Erd˝os in 1943. Chen and Chvátal suggested a generalization of this statement in the context of finite metric spaces Pawel Pralat with lines defined using the notion of betweenness.WeRyerson University prove this generalization in the plane with the L1 metric, [email protected] assuming additionally that no two points share their x-or y-coordinate. Without this additional condition, we can prove the existence of n/37 lines. MS45 Line-of-Sight Pursuit in Sweepable Polygons Ida Kantor Charles University We consider a pursuit game in a simply polygon Q between [email protected]ff.cuni.cz a pursuer P and an evader E,whereP only has line-of-sight visibility. We provide a winning strategy for P a monotone Balazs Patkos polygon in which every line orthogonal to a fixed line L Renyi Institute of Math, Budapest, Hungary intersects Q at most twice. The capture time is linear in [email protected] the area of Q, and independent of the number of vertices of the polygon. Finally, we also extend our algorithm to more general sweepable polygons. Joint work with Volkan MS44 Isler, Jane Butterfield, Lindsay Berry, Zack Keller, Alana A De Bruijn-Erdos Theorem for Distance- Shine and Junyi Wang. hereditary Graphs Andrew J. Beveridge A theorem of De Bruijn and Erd˝osimplies that n non- Department of Mathematics, Statistics and Computer collinear points in the plane determine at least n lines. Science Chen and Chvátal conjectured a generalization to finite Macalester College metric spaces. An interesting case is that of metric spaces [email protected] induced by graphs. In this talk I present a proof of the conjecture for metric spaces induced by distance-hereditary graphs. These are graphs where the distance between two MS45 vertices is the same in any connected induced subgraph. The Computational Complexity of Cops and Rob- bers Rohan Kapadia, Pierre Aboulker Concordia University In the classic Cops and Robbers game, a team of cops at- Montreal tempts to capture a robber on a graph. The game has [email protected], [email protected] been extensively studied, with applications ranging from artificial intelligence to counterterrorism. In this talk, we discuss the computational complexity of deciding whether MS44 k cops can capture a robber on a graph G:howfast(or Lines in Metric Spaces Induced by General Graphs how slow) are the best possible computer algorithms for determining who wins? In 1995, Goldstein and Reingold A theorem of De Bruijn and Erd˝osimplies that n non- conjectured that the problem is EXPTIME-complete – in collinear points in the plane determine at least n lines. other words, that Cops and Robbers is among the “hardest’ Chen and Chvátal conjectured a generalization to finite problems that can be solved in time exponential in the size metric spaces. An interesting case is that of metric spaces of the input. We sketch a recent proof of this conjecture. induced by graphs. We show that every metric space induced by a graph has either a line containing all the ver- Bill Kinnersley DM14 Abstracts 85

University of Illinois [email protected] [email protected]

MS46 MS45 Kekulean Benzenoids Cop Versus Gambler A Kekulé structure forabenzenoidorafullereneΓisaset of edges K such that each vertex of Γ is incident with ex- We consider a variation of Cops and Robbers in which the actly one edge in K, i.e. a perfect matching. All fullerenes robber is not restricted by the graph edges and instead admit a Kekulé structure; however, this is not true for picks a time-independent probability distribution on V (G) benzenoids. In this paper, we develop methods for de- and moves according to this fixed distribution. The cop ciding whether or not a given benzenoid admits a Kekulé moves from vertex to adjacent vertex with the goal of min- structure by constructing Kekulé structures that have a imizing expected capture time. Players move simultane- high density of benzene rings. Our method of construct- ously. We show that when the distribution is known, the ing Kekulé structures gives good estimates for the Clar expected capture time (with best play) on any connected and Fries numbers for benzenoids. Exact values for certain n-vertex graph is exactly n. Time permitting, we will also families of benzenoids will be given. discuss what is known about the case of an unknown distribution. Elizabeth Hartung Massachusetts College of Liberal Arts Natasha Komarov [email protected] Carnegie Mellon University Pittsburgh, PA [email protected] Jack Graver Syracuse University [email protected] MS45 The Eternal Dominating Set Problem on Grids MS46 Split Graphs and Nordhaus-Gaddum Graphs In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is at- AgraphG is an NG-graph if it satisfies the Nordhaus- tacked. After each attack, if the guards can ‘move’ so that Gaddum inequality, χ(G)+χ(G) ≤|V (G)| +1 withequal- they form a dominating set containing the attacked vertex, ity. In recent work, Collins and Trenk gave a new charac- then the guards have defended against the attack.Wewish terization of this class which resulted in a polynomial-time to determine the minimum number of guards required to recognition algorithm. In this talk, we explore connections defend against any sequence of attacks, the eternal domi- between NG-graphs and split graphs and discuss an im- nation number. As the domination number for grid graphs proved algorithm for recognizing NG-graphs based solely on degree sequences. This topic is continued in the talk was recently determined (2011), grid graphs are a natural Counting Split Graphs and Nordhaus-Gaddum Graphs by class of graphs to consider for the eternal dominating set Karen Collins. problem. Though the eternal domination number has been determined for 2 x n grids and 4 x n grids, it has remained Ann N. Trenk only loosely bounded for the 3 x n grid. We provide major Wellesley College improvements to both the upper and lower bounds for the [email protected] 3 x n grid. Joint work with S. Finbow, M. van Bommel, A.Z. Delaney Karen Collins Wesleyan University Margaret-Ellen Messinger [email protected] Dalhousie University [email protected] MS46 On r-dynamic Coloring of Graphs MS46 The r-dynamic chromatic number of a graph G, written Counting Split Graphs and Nordhaus-Gaddum χr(G), is the least k such that G has a proper k-coloring in Graphs which every vertex has neighbors of at least min{d(v),r} different colors. We study upper bounds on χr(G). The AgraphG is an NG-graph if it satisfies the Nordhaus- greedy upper bound χr(G) ≤ rΔ(G)+1 holds with equality Gaddum inequality, χ(G)+χ(G) ≤|V (G)| +1 withequal- if and only if G is r-regular with diameter 2 and girth 5. For ity. In this talk, which is a continuation of Split Graphs and n-vertex graphs, the bound improves to χr(G) ≤ Δ(G)+2r Nordhaus-Gaddum Graphs by Ann Trenk, we continue to when δ(G) ≥ 2r ln n and to χr(G) ≤ Δ(G)+r when δ(G) ≥ explore connections between NG-graphs and split graphs r(r +1)lnn. In terms of the chromatic number, χr(G) ≤ and count the number of NG-graphs on n vertices. This is joint work with Ann N. Trenk (Wellesley College). rχ(G)whenG is k-regular with k ≥ (3 + o(1))r ln r, but 1.377 χr(G) may exceed r χ(G)whenk = r. In contrast, Karen Collins χ2(G) ≤ χ(G)+2when G has diameter 2, with equality Wesleyan University only for complete bipartite graphs and C5.Also,χ2(G) ≤ [email protected] 3χ(G)whenG has diameter 3, which is sharp. However, χ2 is unbounded on bipartite graphs with diameter 4, and Ann N. Trenk χ3 is unbounded on bipartite graphs with diameter 3 or Wellesley College 3-colorable graphs with diameter 2. We also study χr on 86 DM14 Abstracts

cartesian products of two paths or two cycles.

Sogol Jahanbekam University of Colorado Denver [email protected]

Jaehoon Kim University of Illinois [email protected]

Suil O University of Illinois, Urbana [email protected]

Douglas B. West University of Illinois, Urbana Department of Mathematics [email protected]

MS46 Dichromatic Number and Fractional Chromatic Number and Given an undirected graph G, the chromatic number χ(G) is the minimum order of partitions of V (G) into independent sets. Given a directed graph D, a vertex set is acyclic if it does not contain a directed cycle. The chromatic number χ(D) is the minimum order of partitions of V (G)into acyclic sets. The dichromatic number of an undirected −→ graph G, denoted by χ (G), is the maximum chromatic number over all its orientations. Erd¨os and Neumann-Lara n −→ n ≤ n ≤ proved that C1 log n χ (K ) C2 log n for some constants C1,C2. They conjectured that if the dichromatic number of a graph is bounded, so does its chromatic number. The fractional chromatic number is, over all the possible weight functions on the independent sets, the minimum total weight of all the indepdent sets, such that for each vertex the independent sets including it has total weight at least 1. We prove that for any graph G with fractional −→ t n ≥ chromatic number t,wehave χ (K ) 18 log t . Hehui Wu,BojanMohar Simon Fraser University [email protected], [email protected]