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Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written ch r(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 min{dG(v),r} neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove ch r(G)≤(1+o(1)) ch (G) when Δ(G)δ(G) is bounded. We also show that there exists a constant C such that the random graph G=G(n,p) with 6log(n)n