Erratum : MCColor is not optimal on Meyniel graphs

TL;DR

This paper clarifies that the previously claimed linear-time optimal coloring algorithm for Meyniel graphs is incorrect, discusses alternative methods, and highlights that the open problem of linear-time coloring remains unsolved.

Contribution

It corrects a previous claim about the optimality of MCColor for Meyniel graphs and discusses the current best algorithms and open problems.

Findings

MCColor is not optimal for Meyniel graphs.

Alternative algorithms can find stable sets and colorings in O(nm) time.

The open problem of linear-time coloring for Meyniel graphs remains unsolved.

Abstract

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. In the manuscript "Coloring Meyniel graphs in linear time" we claimed that our algorithm MCColor produces an optimal coloring for every Meyniel graph. But later we found a mistake in the proof and a couterexample to the optimality, which we present here. MCColor can still be used to find a stable set that intersects all maximal cliques of a Meyniel graph in linear time. Consequently it can be used to find an optimal coloring in time O(nm), and the same holds for Algorithm MCS+Color. This is explained in the manuscript "A linear algorithm to find a strong stable set in a Meyniel graph" but this is equivalent to Hertz's algorithm. The current best algorithm for coloring Meyniel graphs is the O(n^2) algorithm LexColor due to Roussel and Rusu. The question of finding a linear-time algorithm to color Meyniel graphs is still open.