Precoloring co-Meyniel graphs

Vincent Jost (1) , Benjamin Lévêque (1, 2, 3) , Frédéric Maffray (3)

1 Leibniz - IMAG - Laboratoire Leibniz
2 LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
3 G-SCOP_OC - Optimisation Combinatoire

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ''PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (''co-Artemis'' graphs, which are ''co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs.

Mots clés: Perfect graphs; Meyniel graphs; PreColoring

Domaines: Mathématique discrète [cs.DM]

Vincent Jost, Benjamin Lévêque, Frédéric Maffray. Precoloring co-Meyniel graphs. Graphs and Combinatorics, 2007, 23 (3), pp.291-301. ⟨10.1007/s00373-007-0724-1⟩. ⟨hal-00008354⟩