247 research outputs found

    Scott's induced subdivision conjecture for maximal triangle-free graphs

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    Scott conjectured that the class of graphs with no induced subdivision of a given graph is χ\chi-bounded. We verify his conjecture for maximal triangle-free graphs

    Disproving the normal graph conjecture

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    A graph GG is called normal if there exist two coverings, C\mathbb{C} and S\mathbb{S} of its vertex set such that every member of C\mathbb{C} induces a clique in GG, every member of S\mathbb{S} induces an independent set in GG and CSC \cap S \neq \emptyset for every CCC \in \mathbb{C} and SSS \in \mathbb{S}. It has been conjectured by De Simone and K\"orner in 1999 that a graph GG is normal if GG does not contain C5C_5, C7C_7 and C7\overline{C_7} as an induced subgraph. We disprove this conjecture

    The Erd\H{o}s-Hajnal Conjecture for Paths and Antipaths

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    We prove that for every k, there exists ck>0c_k>0 such that every graph G on n vertices not inducing a path PkP_k and its complement contains a clique or a stable set of size nckn^{c_k}

    Clique versus Independent Set

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    Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O(nlogn)O(n^{\log n}), and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant cHc_H for which we find a O(ncH)O(n^{c_H}) CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a O(nck)O(n^{c_k}) CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, cHc_H is of order O(HlogH)O(|H| \log |H|) resulting from Vapnik-Chervonenkis dimension, and on the other side, ckc_k is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by O(nlogn)O(n^{\log n}) instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator

    Isolating highly connected induced subgraphs

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    We prove that any graph GG of minimum degree greater than 2k212k^2-1 has a (k+1)(k+1)-connected induced subgraph HH such that the number of vertices of HH that have neighbors outside of HH is at most 2k212k^2-1. This generalizes a classical result of Mader, which states that a high minimum degree implies the existence of a highly connected subgraph. We give several variants of our result, and for each of these variants, we give asymptotics for the bounds. We also we compute optimal values for the case when k=2k=2. Alon, Kleitman, Saks, Seymour, and Thomassen proved that in a graph of high chromatic number, there exists an induced subgraph of high connectivity and high chromatic number. We give a new proof of this theorem with a better bound
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