Subcubic edge chromatic critical graphs have many edges

TL;DR

This paper investigates the structure of subcubic critical graphs with edge chromatic number 4, establishing a new lower bound on their average degree, which is tight and characterizes such graphs beyond the Petersen graph with a vertex removed.

Contribution

The paper improves the lower bound on the average degree of subcubic critical graphs, showing it is at least 46/17, and characterizes the extremal graphs achieving this bound.

Findings

Critical graphs with $ riangle=3$ and $ ext{chromatic index}=4$ have average degree at least 46/17.

The Petersen graph with a vertex removed ($P^*$) uniquely attains the minimum average degree of 8/3.

The bound of 46/17 is sharp, exemplified by the Hajos join of two copies of $P^*$.

Abstract

We consider graphs $G$ with $\Delta=3$ such that $\chi'(G)=4$ and $\chi'(G-e)=3$ for every edge $e$, so-called \emph{critical} graphs. Jakobsen noted that the Petersen graph with a vertex deleted, $P^*$, is such a graph and has average degree only $\frac83$. He showed that every critical graph has average degree at least $\frac83$, and asked if $P^*$ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if $G$ is a subcubic critical graph other than $P^*$, then $G$ has average degree at least $\frac{46}{17}\approx2.706$. This bound is best possible, as shown by the Hajos join of two copies of $P^*$.