On r-hued coloring of product graphs

Abstract

A (k,r)-coloring of a graph G is a proper coloring with k colors such that for every vertex v with degree d(v) in G, the color number of the neighbors of v is at least min{d(v),r}. The smallest integer k such that G has a (k,r)-coloring is called the r-hued chromatic number and denoted by χr(G). In Kaliraj et al. [Taibah Univ. Sci. 14 (2020) 168–171], it is determined the 2-hued chromatic numbers of Cartesian product of complete graph and star graph. In this paper, we extend its result and determine the r-hued chromatic number of Cartesian product of complete graph and star graph.