On r-hued coloring of planar graphs with girth at least 5

Abstract

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, X(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 5 and 7 > r > 3, then Xr(G) < r +11. © 2018 Utilitas Mathematica Publishing Incorporated. All rights reserved.