Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c   is a coloring such that for every edge [xi,xj][xi,xj], c(xi)≠c(xj)c(xi)≠c(xj) and for every arc (xp,xq)(xp,xq), c(xp)