[1, 1, t]-Colorings of Complete Graphs

Given non-negative integers and an -coloring of a graph is a mapping from to the color set such that for every two adjacent vertices for every two adjacent edges and for all pairs of incident vertices and edges, respectively. The -chromatic number of is defined to be the minimum such that admits an -coloring. In this note we examine for complete graphs We prove, among others, that is equal to whenever but is strictly larger if is even and sufficiently large with respect to Moreover, as and we asymptotically have if and only if.