On dynamic coloring for planar graphs and graphs of higher genus

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper coloring on the vertices of G by k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different colors. The dynamic chromatic number, denoted by χ2(G), is the smallest integer k for which a graph G has a (k, 2)-coloring. A list assignment L of G is a function that assigns to every vertex v of G a set L(v) of positive integers. For a given list assignment L of G, an (L, r)-coloring of G is a proper coloring c of the vertices such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different colors and c(v) ∈ L(v). The dynamic choice number of G, ch2(G), is the least integer k such that every list assignment L with |L(v)| = k, ∀v ∈ V (G), permits an (L, 2)-coloring. It is known that for any graph G, χr (G) ≤ chr (G). Using Euler distributions in this paper, we prove the following results, where (2) and (3) are best possible. (1) If G is planar, then ch2(G) ≤ 6. Moreover, ch2(G) ≤ 5 when ∆(G) ≤ 4. (2) If G is planar, then χ2(G) ≤ 5. (3) If G is a graph with genus g(G) ≥ 1, then ch2(G) ≤ 2 (7 + √ 1 + 48g(G)). © 2012 Elsevier B.V. All rights reserved.