Abstract:

A(k, r)-coloring of a graph G is a proper k-coloring of graph G such that the neighbors of any vertex receive at least min{r, d(v)} different colors. The smallest positive integral k such that graph G has a(k, r)-coloring is defined as the r-hued chromatic number and denoted by χ_r(G). The Cartesian product of two graphs G and H, denoted by G□H, has vertex set V(G)×V(H), where(u₁, v₁) and(u₂, v₂) are adjacent if and only if either u₁=u₂ and v₁v₂∈E(G), or v₁=v₂ and u₁u₂∈E(G). In this paper, the r-hued chromatic number of W_n□P_m is determined.

Key words: (k, r)-coloring, r-hued chromatic number, Cartesian product of graphs