2,495 research outputs found

    Colouring Diamond-free Graphs

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    The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8

    Colouring diamond-free graphs

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    The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8

    Colouring graphs with no induced six-vertex path or diamond

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    The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6P_6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6P_6, diamond)-free graph GG is no larger than the maximum of 6 and the clique number of GG. We do this by reducing the problem to imperfect (P6P_6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P6P_6, diamond, K6K_6)-free graph. Together with the Lov\'asz theta function, this gives a polynomial time algorithm to compute the chromatic number of (P6P_6, diamond)-free graphs.Comment: 29 page

    Bounding clique-width via perfect graphs

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    Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1,H2)-free graphs and present three new classes of (H1,H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to (H1,H2)-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs

    A note on 2--bisections of claw--free cubic graphs

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    A \emph{kk--bisection} of a bridgeless cubic graph GG is a 22--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most kk. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a 22--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs

    Homogeneous sets, clique-separators, critical graphs, and optimal χ\chi-binding functions

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    Given a set H\mathcal{H} of graphs, let fH ⁣:N>0N>0f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0} be the optimal χ\chi-binding function of the class of H\mathcal{H}-free graphs, that is, fH(ω)=max{χ(G):G is H-free, ω(G)=ω}.f_\mathcal{H}^\star(\omega)=\max\{\chi(G): G\text{ is } \mathcal{H}\text{-free, } \omega(G)=\omega\}. In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal χ\chi-binding functions for subclasses of P5P_5-free graphs and of (C5,C7,)(C_5,C_7,\ldots)-free graphs. In particular, we prove the following for each ω1\omega\geq 1: (i)  f{P5,banner}(ω)=f3K1(ω)Θ(ω2/log(ω)),\ f_{\{P_5,banner\}}^\star(\omega)=f_{3K_1}^\star(\omega)\in \Theta(\omega^2/\log(\omega)), (ii) $\ f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),(iii) (iii) \ f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin \mathcal{O}(\omega),and(iv) and (iv) \ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.Wealsocharacterise,foreachofourconsideredgraphclasses,allgraphs We also characterise, for each of our considered graph classes, all graphs Gwith with \chi(G)>\chi(G-u)foreach for each u\in V(G).Fromthesestructuralresults,wecanproveReedsconjecturerelatingchromaticnumber,cliquenumber,andmaximumdegreeofagraphfor. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for (P_5,banner)$-free graphs
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