A local epsilon version of Reed's Conjecture
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In 1998, Reed conjectured that every graph G satisfies χ(G) ≤1/2(Δ(G) + 1 + ω(G)), where χ(G) is the chromatic number of G, Δ(G) is the maximum degree of G, and ω(G) is the clique number of G. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some ε > 0 such that χ(G) ≤ (1 - ε)(Δ(G) + 1) + εω(G). It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version of Reed's conjecture. Namely, we conjecture that if G is a graph with list-assignment L such that for each vertex v of G, |L(v)| ≥1/2(d(v) + 1 + ω(v)), where d(v) is the degree of v and ω(v) is the size of the largest clique containing v, then G is L-colorable. Our main result is that an "epsilon version" of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of k-critical graphs with clique number less than k/2, as follows. For every α > 0, if ε≤α^2/1350, then if G is an L-critical graph for some k-list-assignment L such that ω(G) < (1/2 - α)k and k is sufficiently large, then G has average degree at least (1 + ε)k. This implies that for every α > 0, there exists ε > 0 such that if G is a graph with ω(G)≤ (1/2 - α)mad(G), where mad(G) is the maximum average degree of G, then χ_ℓ(G) ≤ (1 - ε)(mad(G) + 1) + εω(G).
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