Even-hole-free graphs still have bisimplicial vertices
Combinatorics
2020-05-18 v2
Abstract
A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques. In an earlier paper \cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using a different method.
Keywords
Cite
@article{arxiv.1909.10967,
title = {Even-hole-free graphs still have bisimplicial vertices},
author = {Maria Chudnovsky and Paul Seymour},
journal= {arXiv preprint arXiv:1909.10967},
year = {2020}
}