Proof of a conjecture of Plummer and Zha
Combinatorics
2022-02-01 v2
Abstract
Say a graph is a {\em pentagraph} if every cycle has length at least five, and every induced cycle of odd length has length five. N. Robertson proposed the conjecture that the Petersen graph is the only pentagraph that is three-connected and internally 4-connected, but this was disproved by M. Plummer and X. Zha in 2014. Plummer and Zha conjectured that every 3-connected, internally 4-connected pentagraph is three-colourable. We prove this: indeed, we will prove that every pentagraph is three-colourable.
Keywords
Cite
@article{arxiv.2201.11505,
title = {Proof of a conjecture of Plummer and Zha},
author = {Maria Chudnovsky and Paul Seymour},
journal= {arXiv preprint arXiv:2201.11505},
year = {2022}
}