10 research outputs found

    Acyclic and frugal colourings of graphs

    Get PDF
    Given a graph G = (V, E), a proper vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed [14] studied proper t-frugal colourings and Yuster [19] studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. In particular, we consider vertex colourings that are not necessarily proper, and in this case, we find qualitative connections with colourings that are t-improper -colourings in which the colour classes induce subgraphs of maximum degree at most t -for choices of t near to d

    Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

    Get PDF

    Proper Conflict-free Coloring of Graphs with Large Maximum Degree

    Full text link
    A proper coloring of a graph is conflict-free if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph GG has a proper conflict-free coloring with at most 5Δ(G)/25\Delta(G)/2 colors and conjectured that Δ(G)+1\Delta(G)+1 colors suffice for every connected graph GG with Δ(G)≥3\Delta(G)\ge 3. Our first main result is that even for list-coloring, ⌈1.6550826Δ(G)+Δ(G)⌉\left\lceil 1.6550826\Delta(G)+\sqrt{\Delta(G)}\right\rceil colors suffice for every graph GG with Δ(G)≥109\Delta(G)\ge 10^{9}; we also prove slightly weaker bounds for all graphs with Δ(G)≥750\Delta(G)\ge 750. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph GG and a ``conflict'' hypergraph H{\mathcal H}. As another corollary of our results in this general framework, every graph has a proper (30+o(1))Δ(G)1.5(\sqrt{30}+o(1))\Delta(G)^{1.5}-list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lov\'{a}sz Local Lemma or entropy compression. We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph GG has a fractional (1+o(1))Δ(G)(1+o(1))\Delta(G)-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al. holds asymptotically in a strong sense

    Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

    No full text
    International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

    Eight Biennial Report : April 2005 – March 2007

    No full text

    Frugal, acyclic and star colourings of graphs

    Get PDF
    Contains fulltext : 197080pre.pdf (preprint version ) (Open Access) Contains fulltext : 197080pub.pdf (publisher's version ) (Closed access
    corecore