The average degree of edge chromatic critical graphs with maximum degree seven

TL;DR

This paper proves that 7-critical graphs have an average degree of at least 6, supporting Vizing's conjecture for planar graphs with maximum degree 7 and related surface-embeddable graphs.

Contribution

The paper introduces new adjacency lemmas for paths on 4 or 5 vertices, establishing a lower bound on the average degree of 7-critical graphs.

Findings

Average degree of 7-critical graphs is at least 6

Supports Vizing's planar graph conjecture for maximum degree 7

Extends results to graphs embeddable in surfaces with nonnegative Euler characteristic

Abstract

In this paper, by developing several new adjacency lemmas about a path on $4$ or $5$ vertices, we show that the average degree of 7-critical graphs is at least 6. It implies Vizing's planar graph conjecture for planar graphs with maximum degree $7$ and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao (J. Combin. Theory Ser. B 83 (2001) 201-212 and J. Combin. Theory Ser. B 87 (2003) 254-263) and Zhang (Graphs and Combinatorics 16 (2000) 467-495).