A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree
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A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if G_1, …, G_m are nearly disjoint graphs of maximum degree at most D, then the following holds. For every fixed C, if each vertex v ∈⋃_i=1^m V(G_i) is contained in at most C of the graphs G_1, …, G_m, then the (list) chromatic number of ⋃_i=1^m G_i is at most D + o(D). This result confirms a special case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy, and we derive this result from a more general list coloring result in the setting of `color degrees' that also implies a result of Reed and Sudakov.
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