Bounds for the (m,n)-mixed chromatic number and the oriented chromatic number

A bound for the (n,m)-mixed chromatic number in terms of the chromatic number of the square of the underlying undirected graph is given. A similar bound holds when the chromatic number of the square is replaced by the injective chromatic number. When restricted to n = 1 andm = 0 (i.e., oriented graphs) this gives a new bound for the oriented chromatic number. In this case, a slightly improved bound is obtained if the chromatic number of the square is replaced the 2-dipath chromatic number (defined in Section 4). In all cases, the method of proof generalizes an argument that has been used to obtain Brooks-type theorems for injective oriented colorings [1, 2, 3, 4, 5, 6, 7, 8]. Similar, though not identical, arguments have appeared in the work of Sopena [9], and Nešetřil and Raspaud [10, 11]. Colorings of mixed graphs were first studied by Nešetřil and Raspaud [11]. They gave a bound on the (n,m)-mixed chromatic number in terms of the acyclic chromatic number of the underlying undirected graph. Bounds for the (0, 2)-mixed chromatic number of planar graphs and other graph families have been found [12]. Some of these have been extended to arbitrary n and m [13].