On t-relaxed chromatic number of r-power paths

Let G be a graph and t a non-negative integer. Suppose f is a mapping from the vertex set of G to {1, 2,…,k}. If, for any vertex u of G, the number of neighbors v of u with f(v) = f(u) is less than or equal to t, then f is called a t-relaxed k-coloring of G. And G is said to be (k,t)-colorable. The t-relaxed chromatic number of G, denote by χt(G), is defined as the minimum integer k such that G is (k,t)-colorable. Let n and r be two positive integers with n ≥ r + 1. Denote by Pn the path on n vertices and by Pnr the rth power of Pn. This paper determines the t-relaxed chromatic number of Pnr the rth power of Pn.