On Chromatic Number of Colored Mixed Graphs

An (m,n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f 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 an arc (edge) of color c in H . The (m,n)-colored mixed chromatic number χ(m,n)(G) of an (m,n)-colored mixed graph G is the order (number of vertices) of the smallest homomorphic image of G. This notion was introduced by Nešetřil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147–155). They showed that χ(m,n)(G) ≤ k(2m+ n) k−1 where G is a k-acyclic colorable graph. We proved the tightness of this bound. We also showed that the acyclic chromatic number of a graph is bounded by k + k(2m+n)(2m+n) if its (m,n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we showed that for graphs with maximum degree ∆ its (m,n)-colored mixed chromatic number is at most 2(∆− 1)(2m+ n). In particular, the last result directly improves the upper bound 2∆2 of oriented chromatic number of graphs with maximum degree ∆, obtained by Kostochka, Sopena and Zhu (1997, J. Graph Theory 24, 331–340) to 2(∆ − 1)2. We also show that there exists a graph with maximum degree ∆ and (m,n)-colored mixed chromatic number at least (2m+ n).