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Forbidden Subgraph Characterization of Quasi-Line Graphs Medha Dhurandhar [email protected]
Forbidden Subgraph Characterization of Quasi-line Graphs Medha Dhurandhar [email protected] Abstract: Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques. 1. Introduction: A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. In [2] Chudnovsky and Seymour gave a constructive characterization of quasi-line graphs. An alternative characterization of quasi-line graphs is given in [3] stating that a graph has a fuzzy reconstruction iff it is a quasi-line graph and also in [4] using the concept of sums of Hoffman graphs. Here we characterize quasi-line graphs in terms of the forbidden induced subgraphs like line graphs. We consider in this paper only finite, simple, connected, undirected graphs. The vertex set of G is denoted by V(G), the edge set by E(G), the maximum degree of vertices in G by Δ(G), the maximum clique size by (G) and the chromatic number by G). N(u) denotes the neighbourhood of u and N(u) = N(u) + u. For further notation please refer to Harary [3]. 2. Main Result: Before proving the main result we prove some lemmas, which will be used later. Lemma 1: If G is {3K1, C5}-free, then either 1) G ~ K|V(G)| or 2) If v, w V(G) are s.t. -
Clay Mathematics Institute 2005 James A
Contents Clay Mathematics Institute 2005 James A. Carlson Letter from the President 2 The Prize Problems The Millennium Prize Problems 3 Recognizing Achievement 2005 Clay Research Awards 4 CMI Researchers Summary of 2005 6 Workshops & Conferences CMI Programs & Research Activities Student Programs Collected Works James Arthur Archive 9 Raoul Bott Library CMI Profile Interview with Research Fellow 10 Maria Chudnovsky Feature Article Can Biology Lead to New Theorems? 13 by Bernd Sturmfels CMI Summer Schools Summary 14 Ricci Flow, 3–Manifolds, and Geometry 15 at MSRI Program Overview CMI Senior Scholars Program 17 Institute News Euclid and His Heritage Meeting 18 Appointments & Honors 20 CMI Publications Selected Articles by Research Fellows 27 Books & Videos 28 About CMI Profile of Bow Street Staff 30 CMI Activities 2006 Institute Calendar 32 2005 Euclid: www.claymath.org/euclid James Arthur Collected Works: www.claymath.org/cw/arthur Hanoi Institute of Mathematics: www.math.ac.vn Ramanujan Society: www.ramanujanmathsociety.org $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ In addition to major,0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT ongoing activities such as JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE the summer schools,UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM the Institute undertakes a 5IF.JMMFOOJVN1SJ[F1SPCMFNT SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF number of smaller'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB special projects -
Some Remarks on Even-Hole-Free Graphs
Some remarks on even-hole-free graphs Zi-Xia Song∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Abstract A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger’s conjecture and the Erd˝os-Lov´asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies the Erd˝os-Lov´asz Tihany conjecture provided that t ≥ s>χ(G)/3. Furthermore, we prove that every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs. 1 Introduction All graphs in this paper are finite and simple. For a graph G, we use V (G) to denote the vertex set, E(G) the edge set, |G| the number of vertices, e(G) the number of edges, δ(G) the minimum degree, ∆(G) the maximum degree, α(G) the independence number, ω(G) the clique number and χ(G) the chromatic number. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. -
The Structure of Claw-Free Graphs
The structure of claw-free graphs Maria Chudnovsky and Paul Seymour Abstract A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. At first sight, there seem to be a great variety of types of claw-free graphs. For instance, there are line graphs, the graph of the icosahedron, complements of triangle-free graphs, and the Schl¨afli graph (an amazingly highly-symmetric graph with 27 vertices), and more; for instance, if we arrange vertices in a circle, choose some intervals from the circle, and make the vertices in each interval adjacent to each other, the graph we produce is claw-free. There are several other such examples, which we regard as \basic" claw-free graphs. Nevertheless, it is possible to prove a complete structure theorem for claw- free graphs. We have shown that every connected claw-free graph can be ob- tained from one of the basic claw-free graphs by simple expansion operations. In this paper we explain the precise statement of the theorem, sketch the proof, and give a few applications. 1 Introduction A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. (Graphs in this paper are finite and simple.) Line graphs are claw-free, and it has long been recognized that claw-free graphs are an interesting generalization of line graphs, sharing some of the same properties. For instance, Minty [16] showed in 1980 that there is a polynomial-time algorithm to find a stable set of maximum weight in a claw-free graph, generalizing the algorithm of Edmonds [9, 10] to find a maximum weight matching in a graph. -
Strong Perfect Graph Theorem a Graph G Is Perfect If and Only If Neither G Nor Its Complement G¯ Contains an Induced Odd Circuit of Length ≥ 5
THEOREM OF THE DAY The Strong Perfect Graph Theorem A graph G is perfect if and only if neither G nor its complement G¯ contains an induced odd circuit of length ≥ 5. An induced odd circuit is an odd-length, circular sequence of edges having no ‘short-circuit’ edges across it, while G¯ is the graph obtained by replacing edges in G by non-edges and vice-versa. A perfect graph G is one in which, for every induced subgraph H, the size of a largest clique (that is, maximal complete subgraph) is equal to the chromatic number of H (the least number of vertex colours guaranteeing no identically coloured adjacent vertices). This deep and subtle property is confirmed by today’s theorem to have a surprisingly simple characterisation, whereby the railway above is clearly perfect. The railway scenario illustrates just one way in which perfect graphs are important. We wish to dispatch goods every day from depots v1, v2,..., choosing the best-stocked depots but subject to the constraint that we nominate at most one depot per network clique, so as to avoid head-on collisions. The depot-clique incidence relationship is modelled as a 0-1 matrix and we attempt to replicate our constraint numerically from this as a set of inequalities (far right, bottom). Now we may optimise dispatch as a standard linear programming problem unless... the optimum allocates a fractional amount to each depot, failing to respect the one-depot-per-clique constraint. A 1975 theorem of V. Chv´atal asserts: if a clique incidence matrix is the constraint matrix for a linear programme then an integer optimal solution is guaranteed if and only if the underlying network is a perfect graph. -
Mathematics People
Mathematics People information, sometimes even one incorrect bit can de- Chudnovsky and Spielman stroy the integrity of the data stream; adding verification Selected as MacArthur Fellows ‘codes’ to the data helps to test its accuracy. Spielman and collaborators developed several families of codes Maria Chudnovsky of Columbia University and Dan- that optimize speed and reliability, some approaching the iel Spielman of Yale University have been named Mac- theoretical limit of performance. One, an enhanced version Arthur Fellows for 2012. Each Fellow receives a grant of of low-density parity checking, is now used to broadcast US$500,000 in support, with no strings attached, over the and decode high-definition television transmissions. In a next five years. separate line of research, Spielman resolved an enduring Chudnovsky was honored for her work on the classifica- mystery of computer science: why a venerable algorithm tions and properties of graphs. The prize citation reads, for optimization (e.g., to compute the fastest route to the “Maria Chudnovsky is a mathematician who investigates airport, picking up three friends along the way) usually the fundamental principles of graph theory. In mathemat- works better in practice than theory would predict. He and ics, a graph is an abstraction that represents a set of similar a collaborator proved that small amounts of randomness things and the connections between them—e.g., cities and convert worst-case conditions into problems that can be the roads connecting them, networks of friendship among solved much faster. This finding holds enormous practi- people, websites and their links to other sites. When used cal implications for a myriad of calculations in science, to solve real-world problems, like efficient scheduling for engineering, social science, business, and statistics that an airline or package delivery service, graphs are usually depend on the simplex algorithm or its derivatives. -
On Characterizing Game-Perfect Graphs by Forbidden Induced Subgraphs
Volume 7, Number 1, Pages 21{34 ISSN 1715-0868 ON CHARACTERIZING GAME-PERFECT GRAPHS BY FORBIDDEN INDUCED SUBGRAPHS STEPHAN DOMINIQUE ANDRES Abstract. A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we study similar notions for variants of the game chromatic number, namely B-perfect and [A; B]-perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs. 1. Introduction A well-known maker-breaker game is one of Bodlaender's graph coloring games [9]. We are given an initially uncolored graph G and a color set C. Two players, Alice and Bob, move alternately with Alice beginning. A move consists in coloring an uncolored vertex with a color from C in such a way that adjacent vertices receive distinct colors. The game ends if no move is possible any more. The maker Alice wins if the vertices of the graph are completely colored, otherwise, i.e. if there is an uncolored vertex surrounded by colored vertices of each color, the breaker Bob wins. For a graph G, the game chromatic number χg(G) of G is the smallest cardinality of a color set C such that Alice has a winning strategy in the game described above. -
Tournaments and the Strong Erd\H {O} S-Hajnal Property
TOURNAMENTS AND THE STRONG ERDOS-HAJNAL} PROPERTY ELI BERGER, KRZYSZTOF CHOROMANSKI, MARIA CHUDNOVSKY, AND SHIRA ZERBIB Abstract. A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd}os-Hajnal Conjecture, states that for every tournament S there exists "(S) > 0 such that if T is an n-vertex tournament that does not contains S as a subtour- nament, then T contains a transitive subtournament on at least "(S) n vertices. Let C5 be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon- Pach-Solymosi conjecture is known to be true for the case when S = C5. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C5 there exist disjoint vertex subsets A and B, each containing a linear proportion of the vertices of T , and such that every vertex of A is adjacent to every vertex of B. 1. introduction A tournament is a complete graph with directions on edges. A tour- nament is transitive if it has no directed triangles. For tournaments S; T we say that T is S-free if no subtournament of T is isomorphic to S. In [1] a conjecture was made concerning tournaments with a fixed forbidden subtournament: Conjecture 1.1. For every tournament S there exists " > 0 such that every S-free n-vertex tournament contains a transitive subtournament on at least n" vertices. arXiv:2002.07248v2 [math.CO] 14 Sep 2021 It was shown in [1] that Conjecture 1.1 is equivalent to the Erd}os- Hajnal Conjecture [7, 8]. -
Forbidden Induced Subgraphs
Forbidden Induced Subgraphs Thomas Zaslavsky 1 Department of Mathematical Sciences Binghamton University (State University of New York) Binghamton, NY 13902-6000, U.S.A. Abstract In descending generality I survey: five partial orderings of graphs, the induced- subgraph ordering, and examples like perfect, threshold, and mock threshold graphs. The emphasis is on how the induced subgraph ordering differs from other popular orderings and leads to different basic questions. Keywords: Partial ordering of graphs, hereditary class, induced subgraph ordering, perfect graph, mock threshold graph. 1 Preparation This is a very small survey of partial orderings of graphs, hereditary graph classes (mainly in terms of induced subgraphs), and characterizations by for- bidden containments and by vertex orderings, leading up to a new graph class and a new theorem. Our graphs, written G = (V; E), G0 = (V 0;E0), etc., will be finite, sim- ple (with some exceptions), and unlabelled; thus we are talking about iso- morphism types (isomorphism classes) rather than individual labelled graphs. 1 Email: [email protected] Graphs can be partially ordered in many ways, of which five make an inter- esting comparison. We list some notation of which we make frequent use: N(G; v) = the neighborhood of v in G. d(G; v) = the degree of vertex v in G. G:X = the subgraph of G induced by X ⊆ V . 2 Five Kinds of Containment We say a graph G contains H if H ≤ G, where ≤ denotes some partial ordering on graphs. There are many kinds of containment; each yields a different characterization of each interesting hereditary graph class. -
Large Homogeneous Subgraphs in Bipartite Graphs with Forbidden Induced Subgraphs
Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs Maria Axenovich∗ Casey Tompkinsy Lea Weberz Abstract ~ For a bipartite graph G, let h(G) be the largest t such that either G contains Kt;t, a complete bipartite subgraph with parts of size t, or a bipartite complement of G contains Kt;t. For a class of graphs F, let h~(F) = minfh~(G): G 2 Fg. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contain a cycle. By Forb(n; H) we denote the set of bipartite graphs with parts of size n, which do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h~(Forb(n; H)) = O(n1−) for a positive if H is not strongly acyclic. Here, we prove that h~(Forb(n; H)) is linear in n for all strongly acyclic graphs except for four graphs. Introduction A conjecture of Erd}osand Hajnal [3] states that for any graph H, there is a constant > 0 such that any n-vertex graph that does not contain H as an induced subgraph has either a clique or a coclique on at least n vertices. While this conjecture remains open, see for example a survey by Chudnovsky [2], we address a bipartite variation of the problem. Let G be a bipartite graph with parts U and V of size n each, we write G = ((U; V );E), E ⊆ U × V . We further write E = E(G), and for an edge (u; v) 2 E, we simply write uv. -
Detecting a Long Even Hole
Detecting a long even hole Linda Cook Princeton University, Princeton, NJ 08544, USA Paul Seymour1 Princeton University, Princeton, NJ 08544, USA August 28, 2019; revised September 15, 2020 arXiv:2009.05691v1 [math.CO] 12 Sep 2020 1Partially supported by AFOSR grant A9550-19-1-0187 and NSF grant DMS-1800053. Abstract For each integer ℓ ≥ 4, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least ℓ and even. 1 Introduction All graphs in this paper are finite and have no loops or parallel edges. A hole in a graph is an induced subgraph which is a cycle of length at least four. The length of a path or cycle A is the number of edges in A and the parity of A is the parity of its length. For graphs G, H we will say that G contains H if some induced subgraph of G is isomorphic to H. We say G is even-hole-free if G does not contain an even hole. We denote by |G| the number of vertices of a graph G. A graph algorithm is polynomial-time if its running time is at most polynomial in |G|. This paper concerns detecting holes in a graph with length satisfying certain conditions. It is of course trivial to test for the existence of a hole of length at least ℓ, in polynomial time for each constant ℓ ≥ 4, as follows. We enumerate all induced paths P of length ℓ − 2. For each choice of P , let its ends be x and y, let P ∗ = V (P )\{x,y}, and let N be the set of vertices different from x,y that belong to or have a neighbour in P ∗. -
A Survey of Χ-Boundedness
A survey of χ-boundedness Alex Scott1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Paul Seymour2 Princeton University, Princeton, NJ 08544 March 14, 2017; revised May 28, 2020 1Supported by a Leverhulme Trust Research Fellowship 2Supported by ONR grant N00014-14-1-0084 and NSF grants DMS-1265563 and DMS-1800053. Abstract If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr´asGy´arf´asmade a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now. 1 Introduction Let G be a graph. (All graphs in this paper are finite and simple.) We denote the chromatic number of G by χ(G), and its clique number (the cardinality of its largest clique) by !(G). If X ⊆ V (G), we denote the subgraph induced on X by G[X], and write χ(X) for χ(G[X]) when there is no danger of ambiguity. A hole in G is an induced cycle of length at least four, and an odd hole is one with odd length; an antihole is an induced subgraph whose complement graph is a hole of the complement graph G of G. Certainly χ(G) ≥ !(G), and if we are told that χ(G) > !(G), we can deduce something about the induced subgraphs of G: 1.1 If χ(G) > !(G) then some induced subgraph of G is an odd hole or an odd antihole.