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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 \textit{$(m,n)$-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 $χ_{(m,n)}(G) \leq k(2m+n)^{k-1}$ 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 $k^2 + k^{2 + \lceil log_{(2m+n)} log_{(2m+n)} k \rceil}$ if its $(m,n)$-colored mixed chromatic number is at most $k$. Furthermore, using probabilistic method, we showed that for graphs with maximum degree $Δ$ its $(m,n)$-colored mixed chromatic number is at most $2(Δ-1)^{2m+n} (2m+n)^{Δ-1}$. In particular, the last result directly improves the upper bound $2Δ^2 2^Δ$ of oriented chromatic number of graphs with maximum degree $Δ$, obtained by Kostochka, Sopena and Zhu (1997, J. Graph Theory 24, 331--340) to $2(Δ-1)^2 2^{Δ-1}$. We also show that there exists a graph with maximum degree $Δ$ and $(m,n)$-colored mixed chromatic number at least $(2m+n)^{Δ/ 2}$.