3,886 research outputs found

    Chromatic thresholds in dense random graphs

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    The chromatic threshold δχ(H,p)\delta_\chi(H,p) of a graph HH with respect to the random graph G(n,p)G(n,p) is the infimum over d>0d > 0 such that the following holds with high probability: the family of HH-free graphs GG(n,p)G \subset G(n,p) with minimum degree δ(G)dpn\delta(G) \ge dpn has bounded chromatic number. The study of the parameter δχ(H):=δχ(H,1)\delta_\chi(H) := \delta_\chi(H,1) was initiated in 1973 by Erd\H{o}s and Simonovits, and was recently determined for all graphs HH. In this paper we show that δχ(H,p)=δχ(H)\delta_\chi(H,p) = \delta_\chi(H) for all fixed p(0,1)p \in (0,1), but that typically δχ(H,p)δχ(H)\delta_\chi(H,p) \ne \delta_\chi(H) if p=o(1)p = o(1). We also make significant progress towards determining δχ(H,p)\delta_\chi(H,p) for all graphs HH in the range p=no(1)p = n^{-o(1)}. In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with minor modifications from arXiv:1108.1746 for a self-contained proof of a technical lemma; accepted to Random Structures and Algorithm

    Bipartite induced density in triangle-free graphs

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    We prove that any triangle-free graph on nn vertices with minimum degree at least dd contains a bipartite induced subgraph of minimum degree at least d2/(2n)d^2/(2n). This is sharp up to a logarithmic factor in nn. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of n/dn/d and (2+o(1))n/logn(2+o(1))\sqrt{n/\log n} as nn\to\infty. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most O(min{n,(nlogn)/d})O(\min\{\sqrt{n},(n\log n)/d\}) as nn\to\infty. Relatedly, we also make two conjectures. First, any triangle-free graph on nn vertices has fractional chromatic number at most (2+o(1))n/logn(\sqrt{2}+o(1))\sqrt{n/\log n} as nn\to\infty. Second, any triangle-free graph on nn vertices has list chromatic number at most O(n/logn)O(\sqrt{n/\log n}) as nn\to\infty.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring

    Strong chromatic index of sparse graphs

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    A coloring of the edges of a graph GG is strong if each color class is an induced matching of GG. The strong chromatic index of GG, denoted by χs(G)\chi_{s}^{\prime}(G), is the least number of colors in a strong edge coloring of GG. In this note we prove that χs(G)(4k1)Δ(G)k(2k+1)+1\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1 for every kk-degenerate graph GG. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that % \chi_{s}^{\prime}(G)\leq 4\Delta -3 for any chordless graph GG. Both bounds remain valid for the list version of the strong edge coloring of these graphs

    Distance-generalized Core Decomposition

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    The kk-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least kk other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of kk-core, which we refer to as the (k,h)(k,h)-core, i.e., the maximal subgraph in which every vertex has at least kk other vertices at distance h\leq h within that subgraph. We study the properties of the (k,h)(k,h)-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic number) and it preserves its usefulness to speed-up or approximate distance-generalized notions of dense structures, such as hh-club. Computing the distance-generalized core decomposition over large networks is intrinsically complex. However, by exploiting clever upper and lower bounds we can partition the computation in a set of totally independent subcomputations, opening the door to top-down exploration and to multithreading, and thus achieving an efficient algorithm

    Density theorems for bipartite graphs and related Ramsey-type results

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    In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements

    The t-improper chromatic number of random graphs

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    We consider the tt-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph G(n,p)G(n,p). The t-improper chromatic number χt(G)\chi^t(G) of GG is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most tt. If t=0t = 0, then this is the usual notion of proper colouring. When the edge probability pp is constant, we provide a detailed description of the asymptotic behaviour of χt(G(n,p))\chi^t(G(n,p)) over the range of choices for the growth of t=t(n)t = t(n).Comment: 12 page
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