Parameterized mixed graph coloring

Journal article, 2019

Coloring of mixed graphs that contain both directed arcs and undirected edges
is relevant for scheduling of unit-length jobs with precedence constraints and
conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and
Vitaver) relates graph coloring to longest paths. It can be extended to mixed
graphs. In the present paper we further extend the GHRV theorem to weighted
mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm
(with the number of undirected edges as parameter) that is slightly faster than
the brute-force algorithm. The parameter is natural since the directed version
is polynomial whereas the undirected version is NP-complete. Furthermore we
point out a new polynomial case where the edges form a clique.