A Brooks type theorem for the maximum local edge connectivity

TL;DR

This paper characterizes graphs where the chromatic number equals the maximum local edge connectivity plus one, extending Dirac's theorem and building on previous work for the case when local edge connectivity is 3.

Contribution

It provides a complete characterization of graphs with chromatic number equal to maximum local edge connectivity plus one for all cases where the connectivity is at least 4.

Findings

Graphs with $ u(G)=k+1$ contain blocks formed from $K_{k+1}$ via Hajós joins.

The result generalizes Dirac's theorem for all local edge connectivity values.

The case $ u(G)=k+1$ is characterized by the presence of specific block structures.

Abstract

For a graph $G$, let $\cn(G)$ and $\la(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \cite{Dirac53} implies that every graph $G$ satisfies $\cn(G)\leq \la(G)+1$. In this paper we characterize the graphs $G$ for which $\cn(G)=\la(G)+1$. The case $\la(G)=3$ was already solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}. We show that a graph $G$ with $\la(G)=k\geq 4$ satisfies $\cn(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\'os join.