List r-dynamic coloring of sparse graphs

Abstract

The r-dynamic coloring is a generalization of the L(1,1)-labeling. An r-dynamic k-coloring of a graph G 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 of G, 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 show that chr(G)≤r+5 for planar graphs G with g(G)≥5 and r≥15, chr(G)≤r+10 for graphs G with mad(G)<103 and chr(G)≤r+1 for graphs G with mad(G)<83 and r≥14.