Precoloring co-Meyniel graphs
Vincent Jost (Leibniz - IMAG), Benjamin L\'ev\^eque (Leibniz - IMAG),, Fr\'ed\'eric Maffray (Leibniz - IMAG)

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 and a subset of nodes to which some colors are already assigned, in finding a coloring of 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…
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