# Remarks on planar edge-chromatic critical graphs

## TL;DR

This paper proves new structural properties of planar edge-chromatic critical graphs, providing insights that support Vizing's conjecture for planar graphs with maximum degree 6.

## Contribution

It establishes the non-existence of certain 6-critical plane graphs with limited face incidences and characterizes face adjacencies in 5-critical plane graphs, advancing understanding of planar graph colorings.

## Key findings

- No 6-critical plane graph with vertices incident to at most three 3-faces exists.
- Every 5-critical plane graph has a 3-face adjacent to a 3- or 4-face.
- Results support Vizing's conjecture for planar graphs with maximum degree 6.

## Abstract

The only open case of Vizing's conjecture that every planar graph with $\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $\Delta = 5$ our result gives insights into the structure of planar $5$-critical graphs, and the result for $\Delta=6$ gives support for the truth of Vizing's planar graph conjecture.