105 research outputs found

    Beyond Ohba's Conjecture: A bound on the choice number of kk-chromatic graphs with nn vertices

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    Let ch(G)\text{ch}(G) denote the choice number of a graph GG (also called "list chromatic number" or "choosability" of GG). Noel, Reed, and Wu proved the conjecture of Ohba that ch(G)=χ(G)\text{ch}(G)=\chi(G) when V(G)2χ(G)+1|V(G)|\le 2\chi(G)+1. We extend this to a general upper bound: ch(G)max{χ(G),(V(G)+χ(G)1)/3}\text{ch}(G)\le \max\{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil\}. Our result is sharp for V(G)3χ(G)|V(G)|\le 3\chi(G) using Ohba's examples, and it improves the best-known upper bound for ch(K4,,4)\text{ch}(K_{4,\dots,4}).Comment: 14 page

    Sum list coloring, the sum choice number, and sc-greedy graphs

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    Let G=(V,E) be a graph and let f be a function that assigns list sizes to the vertices of G. It is said that G is f-choosable if for every assignment of lists of colors to the vertices of G for which the list sizes agree with f, there exists a proper coloring of G from the lists. The sum choice number is the minimum of the sum of list sizes for f over all choosable functions f for G. The sum choice number of a graph is always at most the sum |V|+|E|. When the sum choice number of G is equal to this upper bound, G is said to be sc-greedy. In this paper, we determine the sum choice number of all graphs on five vertices, show that trees of cycles are sc-greedy, and present some new general results about sum list coloring.Comment: 14 pages, 11 figure

    Every triangle-free induced subgraph of the triangular lattice is (5m,2m)(5m,2m)-choosable

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    International audienceA graph GG is (a,b)(a,b)-choosable if for any color list of size aa associated with each vertex, one can choose a subset of bb colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer m1m\ge 1, every finite triangle-free induced subgraph of the triangular lattice is (5m,2m)(5m,2m)-choosable
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