Face-degree bounds for planar critical graphs

TL;DR

This paper investigates face-degree bounds in planar critical graphs to understand the structure of potential counterexamples to Vizing's conjecture, providing partial characterizations and bounds for such graphs.

Contribution

It introduces face-degree parameters for planar graphs and establishes upper bounds for critical graphs related to Vizing's conjecture, advancing understanding of their structure.

Findings

Upper bounds on face-degree parameters for critical graphs with small maximum degree

Partial characterization of potential counterexamples to Vizing's conjecture

Insights into the structure of planar edge-chromatic critical graphs for k ≤ 5

Abstract

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist. For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.