5,881 research outputs found

    Monochromatic Clique Decompositions of Graphs

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    Let GG be a graph whose edges are coloured with kk colours, and H=(H1,,Hk)\mathcal H=(H_1,\dots , H_k) be a kk-tuple of graphs. A monochromatic H\mathcal H-decomposition of GG is a partition of the edge set of GG such that each part is either a single edge or forms a monochromatic copy of HiH_i in colour ii, for some 1ik1\le i\le k. Let ϕk(n,H)\phi_{k}(n,\mathcal H) be the smallest number ϕ\phi, such that, for every order-nn graph and every kk-edge-colouring, there is a monochromatic H\mathcal H-decomposition with at most ϕ\phi elements. Extending the previous results of Liu and Sousa ["Monochromatic KrK_r-decompositions of graphs", Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in H\mathcal H is a clique and nn0(H)n\ge n_0(\mathcal H) is sufficiently large.Comment: 14 pages; to appear in J Graph Theor

    On the Chromatic Thresholds of Hypergraphs

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    Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c(V(H)r1)c \binom{|V(H)|}{r-1} has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure

    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
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