Add time:08/12/2019         Source:sciencedirect.com

For integers k,r>0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v),r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k,r)-coloring. We prove the following:(i) If G is a P4-free graph, then χr(G)≤χ(G)+2(r−1), and this bound is best possible.(ii) If G is a P5-free bipartite graph, then χr(G)≤rχ(G), and this bound is best possible.(iii) If G is a P5-free graph, then χ2(G)≤2χ(G), and this bound is best possible.

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