Analogous to cliques for (m,n)-colored mixed graphs

A (m,n)-colored mixed graph is a graph having arcs of m different colors and edges of n different colors. A graph homomorphism of a (m,n)-colored mixed graph G to a (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 χ(m,n)(G) of a (m,n)-colored mixed graph G, introduced by Nešetřil and Raspaud, is the order (number of vertices) of the smallest homomorphic image of G. In this article, we define the analogue of clique for (m,n)-colored mixed graphs. A (m,n)-clique is a (m,n)-colored mixed graph C with χ(m,n)(C) = |C|. We show that almost all (m,n)-colored mixed graphs are (m,n)-cliques. We prove exact bound on the order of the biggest outerplanar (m,n)-clique and provide upper and lower bounds for the order of the biggest planar (m,n)-clique. Furthermore, we address a particular complexity problem related to (0, 2)-colored mixed graphs and compare the parameters χ(1,0) and χ(0,2).