Circular flow number of Goldberg snarks

TL;DR

This paper determines the exact circular flow number of Goldberg snarks, confirming a conjecture and advancing understanding of flow properties in specific graph classes.

Contribution

It proves that the circular flow number of Goldberg snarks $G_{2k+1}$ is exactly $4 + 1/(k+1)$, settling a previously conjectured value.

Findings

Circular flow number of Goldberg snarks is $4 + 1/(k+1)$

Confirmed a conjecture by Goedgebeur, Mattiolo, and Mazzuoccolo

Advances understanding of flow properties in snarks

Abstract

A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number of $G$ is the infimum over all values $r$ such that $G$ admits a nowhere-zero $r$-flow. We prove that the circular glow number of Goldberg snark $G_{2k+1}$ is $4+1/(k+1)$, proving a conjecture of Goedgebeur, Mattiolo, and Mazzuoccolo.