SET COLORINGS IN PERFECT GRAPHS

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 is an element of V(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) not equal 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 xs(G). We show that the decision variant of determining xs(G) is NP-complete in the general case, and show that xs(G) can be efficiently calculated when G is a threshold graph. We study the difference x(G) xs(G), presenting new bounds that are sharp for all graphs G satisfying x(G) = omega(G). We finally present results of the Nordhaus-Gaddum type, giving sharp bounds on the sum and product of xs(G) and xs((G) over bar).