Precoloring co-Meyniel graphs

TL;DR

This paper proves that the pre-coloring extension problem is polynomial-time solvable for complements of Meyniel graphs and characterizes co-Meyniel graphs as exactly the PrExt perfect graphs, advancing understanding of graph coloring.

Contribution

It establishes the polynomiality of pre-coloring extension for co-Meyniel graphs and characterizes PrExt perfect graphs as co-Meyniel graphs, answering a key open question.

Findings

Pre-coloring extension is polynomial for complements of Meyniel graphs.

Co-Meyniel graphs are exactly the PrExt perfect graphs.

Contracted graphs of co-Meyniel graphs are in a restricted class of perfect graphs.

Abstract

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.