A unit-time scheduling problem with makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of colors is minimal. We consider an optimal coloring which defines a schedule that minimizes the makespan for a unit-time job-shop problem, present complexity results for some special cases and improve a geometrical algorithm. For the general case, we develop three branch-and-bound algorithms and test them on randomly generated mixed graphs of order $n \leq 200$ for the exact solution and of order $n \leq 900$ for the approximate solution, where $n$ is the number of vertices.