On r-hued coloring of connected P5-free tripartite graphs

A (k,r)-coloring of a graph G is a proper k-vertex coloring such that any vertex is adjacent to vertices with at least min{r,dG(v)}. The minimum integer k such that G has a (k,r)-coloring is called the r-hued chromatic number of G and denoted by χr(G). A graph does not have an induced subgraph isomorphic to P5 is a P5-free graph. It is not known whether it is valid that χr(G)≤rχ(G) among all P5-free graphs. In [Discrete Math. 342(2019)1904-1911], an affirmative answer is presented for all graphs G with χ(G)≤2. We verify that if G is a P5-free graph with χ(G)≤3, then χr(G)≤rχ(G).