3-dynamic coloring and list 3-dynamic coloring of K1, 3-free graphs

For positive integers k and r , a (k, r)-coloring of a graphG 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 verticeswith at leastmin{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. © 2017 Elsevier B.V. All rights reserved.