Even pairs in square-free Berge graphs with no odd prism

TL;DR

This paper proves that square-free Berge graphs with no odd prism are either cliques or contain an even pair, enabling polynomial-time optimal coloring algorithms for this class.

Contribution

It establishes that such graphs are either cliques or contain an even pair, confirming a conjecture and facilitating efficient coloring.

Findings

Graphs are either cliques or have an even pair.

Polynomial-time algorithm for optimal coloring.

Supports the conjecture by Everett and Reed.

Abstract

We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as conjectured by Everett and Reed. This result is used to devise a polynomial-time algorithm to color optimally every graph in this class.