Proof of a conjecture of Plummer and Zha

TL;DR

This paper proves that all pentagraphs, graphs where every cycle has length at least five and all induced odd cycles are length five, are three-colorable, confirming a conjecture related to their structure.

Contribution

The paper establishes that every pentagraph is three-colorable, resolving a conjecture about their colorability and structural properties.

Findings

All pentagraphs are three-colorable.

Disproves the uniqueness of the Petersen graph as a special pentagraph.

Confirms the conjecture that 3-connected, internally 4-connected pentagraphs are three-colorable.

Abstract

Say a graph $G$ 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.