Set vertex colorings and joins of graphs

Abstract

For a nontrivial connected graph G, let c: V (G) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) ≠ NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number x _s (G). A study is made of the set chromatic number of the join G+H of two graphs G and H. Sharp lower and upper bounds are established for x _s (G + H) in terms of x _s (G), x _s (H), and the clique numbers ω(G) and ω(H).