Analogues of Cliques for (m, n)-Colored Mixed Graphs

Abstract

An (m, n)-colored mixed graph is a mixed graph with arcs assigned one of m different colors and edges one of n different colors. A homomorphism of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m, n)-colored mixed chromatic number, denoted χm,n(G) , of an (m, n)-colored mixed graph G is the order of a smallest homomorphic image of G. An (m, n)-clique is an (m, n)-colored mixed graph C with χm,n(C) = | V(C) |. Here we study the structure of (m, n)-cliques. We show that almost all (m, n)-colored mixed graphs are (m, n)-cliques, prove bounds for the order of a largest outerplanar and planar (m, n)-clique and resolve an open question concerning the computational complexity of a decision problem related to (0, 2)-cliques. Additionally, we explore the relationship between χ1 , 0 and χ0 , 2.