Dynamic choosability of triangle-free graphs and sparse random graphs
Abstract
The r-dynamic choosability of a graph G, written chr(G), is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least \dG(v),r\ neighbors of distinct colors. Let ch(G) denote the choice number of G. In this paper, we prove chr(G)≤ (1+o(1)) ch(G) when (G)δ(G) is bounded. We also show that there exists a constant C such that for the random graph G=G(n,p) with 2n<p≤ 12, it holds that ch2(G)≤ ch(G) + C, asymptotically almost surely. Also if G is triangle-free regualr graph, then ch2(G)≤ ch(G)+86 holds.
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