Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor

TL;DR

This paper proves that all 4- and 5-regular projective planar graphs without Petersen minors can be properly edge-colored with r colors, extending understanding of edge-coloring in specific graph classes.

Contribution

It establishes that such graphs are r-edge colorable for r=4,5, a new result in the study of edge-coloring for these graph classes.

Findings

4- and 5-regular projective planar graphs without Petersen minors are r-edge colorable

The result extends edge-coloring theory to a new class of graphs

No Petersen-minor condition is crucial for the coloring property

Abstract

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. We prove for $r \in \{4,5\}$, every projective planar $r$-graph with no Petersen-minor is $r$-edge colorable.