3-dynamic coloring and list 3-dynamic coloring of [formula omitted]-free graphs

Abstract

For positive integers k and r, a (k,r)-coloring of a graph G is a proper coloring of the vertices with k colors such that every vertex of degree i will be adjacent to vertices with at least min{i,r} different colors. The r-dynamic chromatic number of G, denoted by χr(G), is the smallest integer k for which G has a (k,r)-coloring. For a k-list assignment L to vertices of G, an (L,r)-coloring of G is a coloring c such that for every vertex v of degree i, c(v)∈L(v) and v is adjacent to vertices with at least min{i,r} different colors. The list r-dynamic chromatic number of G, denoted by χL,r(G), is the smallest integer k such that for every k-list L, G has an (L,r)-coloring.In this paper, the behavior and bounds of 3-dynamic coloring and list 3-dynamic coloring of K1,3-free graphs are investigated. We show that if G is K1,3-free, then χL,3(G)≤max{χL(G)+3,7} and χ3(G)≤max{χ(G)+3,7}. The results are best possible as 7 cannot be reduced.