List [formula omitted]-dynamic coloring of graphs with small maximum average degree

Abstract

An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least min{d(v),r} different classes. The r-dynamic chromatic number ofG, written χr(G), is the minimum k such that G has such a coloring. The list r-dynamic chromatic number of G is denoted chr(G). In this paper, we prove that every graph G with one of the following conditions has a list (r+1)-dynamic coloring: (1) mad(G)<187 and r≥8; (2) mad(G)<52 and r≥6; (3) mad(G)<125 and r≥5; (4) mad(G)<2.8−ϵ and r≥f(ϵ)=165ϵ−2, where 0<ϵ≤110. Also we prove that every graph G with mad(G)<4−ϵ and r≥f(ϵ)=5M−11 has a list (r+f(ϵ))-dynamic coloring, where 0<ϵ<1, M=⌈8ϵ⌉−2.