6,572 research outputs found

    Perfect packings with complete graphs minus an edge

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    Let K_r^- denote the graph obtained from K_r by deleting one edge. We show that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a collection of disjoint copies of K_r^- which covers all vertices of G. Here chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n

    Approximating Vizing's independence number conjecture

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    In 1965, Vizing conjectured that the independence ratio of edge-chromatic critical graphs is at most 12\frac{1}{2}. We prove that for every ϵ>0\epsilon > 0 this conjecture is equivalent to its restriction on a specific set of edge-chromatic critical graphs with independence ratio smaller than 12+ϵ\frac{1}{2} + \epsilon.Comment: Revised version: The title is change

    Packing chromatic vertex-critical graphs

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    The packing chromatic number χρ(G)\chi_{\rho}(G) of a graph GG is the smallest integer kk such that the vertex set of GG can be partitioned into sets ViV_i, i[k]i\in [k], where vertices in ViV_i are pairwise at distance at least i+1i+1. Packing chromatic vertex-critical graphs, χρ\chi_{\rho}-critical for short, are introduced as the graphs GG for which χρ(Gx)<χρ(G)\chi_{\rho}(G-x) < \chi_{\rho}(G) holds for every vertex xx of GG. If χρ(G)=k\chi_{\rho}(G) = k, then GG is kk-χρ\chi_{\rho}-critical. It is shown that if GG is χρ\chi_{\rho}-critical, then the set {χρ(G)χρ(Gx): xV(G)}\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\} can be almost arbitrary. The 33-χρ\chi_{\rho}-critical graphs are characterized, and 44-χρ\chi_{\rho}-critical graphs are characterized in the case when they contain a cycle of length at least 55 which is not congruent to 00 modulo 44. It is shown that for every integer k2k\ge 2 there exists a kk-χρ\chi_{\rho}-critical tree and that a kk-χρ\chi_{\rho}-critical caterpillar exists if and only if k7k\le 7. Cartesian products are also considered and in particular it is proved that if GG and HH are vertex-transitive graphs and diam(G)+diam(H)χρ(G){\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G), then GHG\,\square\, H is χρ\chi_{\rho}-critical

    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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