The edge colorings of $K_{5}$-minor free graphs

TL;DR

This paper extends edge coloring results to $K_5$-minor free graphs, showing they are $ ext{edge } ext{max degree}$-colorable for maximum degree at least seven, generalizing known planar graph results.

Contribution

It proves that all $K_5$-minor free graphs with maximum degree at least seven are edge $ ext{max degree}$-colorable, broadening the class of graphs with this property.

Findings

$K_5$-minor free graphs with $ ext{max degree} ext{ at least } 7$ are edge $ ext{max degree}$-colorable.

Generalization of planar graph edge coloring results to $K_5$-minor free graphs.

Extends classical results by Vizing, Sanders, and Zhao to a larger class of graphs.

Abstract

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta\geq 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta=7$ is edge $\Delta$-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every $K_5$-minor free graph with maximum degree $\Delta$ at least seven is edge $\Delta$-colorable.