A remark on Petersen coloring conjecture of Jaeger

TL;DR

This paper explores a coloring relation between cubic graphs, introduces the Sylvester coloring conjecture, and proves that certain graphs are uniquely characterized by this relation, specifically the Petersen and Sylvester graphs.

Contribution

It introduces the Sylvester coloring conjecture and proves uniqueness results for Petersen and Sylvester graphs under the coloring relation.

Findings

If G is a connected bridgeless cubic graph with G prec P, then G is P.

If G is a connected cubic graph with G prec S, then G is S.

The paper defines and investigates the Sylvester coloring conjecture.

Abstract

If $G$ and $H$ are two cubic graphs, then we write $H\prec G$, if $G$ admits a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. Let $P$ and $S$ be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if $G$ is a connected bridgeless cubic graph with $G\prec P$, then $G=P$. Finally, if $G$ is a connected cubic graph with $G\prec S$, then $G=S$.