192 research outputs found

    On bounding the difference between the maximum degree and the chromatic number by a constant

    Full text link
    We provide a finite forbidden induced subgraph characterization for the graph class Υk\varUpsilon_k, for all kN0k \in \mathbb{N}_0, which is defined as follows. A graph is in Υk\varUpsilon_k if for any induced subgraph, Δχ1+k\Delta \leq \chi -1 + k holds, where Δ\Delta is the maximum degree and χ\chi is the chromatic number of the subgraph. We compare these results with those given in [O. Schaudt, V. Weil, On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI: 10.1007/s00373-014-1468-3], where we studied the graph class Ωk\varOmega_k, for kN0k \in \mathbb{N}_0, whose graphs are such that for any induced subgraph, Δω1+k\Delta \leq \omega -1 + k holds, where ω\omega denotes the clique number of a graph. In particular, we give a characterization in terms of Ωk\varOmega_k and Υk\varUpsilon_k of those graphs where the neighborhood of every vertex is perfect.Comment: 10 pages, 4 figure

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

    Full text link
    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
    corecore