(p,k)-coloring problems in line graphs

The (p,k)-coloring problems generalize the usual coloring problem by replacing stable sets by cliques and stable sets. Complexities of some variations of (p,k)-coloring problems (split-coloring and cocoloring) are studied in line graphs; polynomial algorithms or proofs of NP-completeness are given according to the complexity status. We show that the most general (p,k)-coloring problems are more difficult than the cocoloring and the split-coloring problems while there is no such relation between the last two problems. We also give complexity results for the problem of finding a maximum (p,k)-colorable subgraph in line graphs. Finally, upper bounds on the optimal values are derived in general graphs by sequential algorithms based on Welsh-Powell and Matula orderings.