On r-hued coloring of planar graphs with girth at least 6

For integers k,r>0, a (k,r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v),r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χr(G), is the least integer k such that a (k,r)-coloring of G exists. In this paper, we proved that if G is a planar graph with girth at least 6, then χr(G)≤r+5. This extends a former result in Bu and Zhu (2012). It also implies that a conjecture on r-hued coloring of planar graphs is true for planar graphs with girth at least 6.