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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 $\cn(G)\leq \la(G)+1$. Inthis 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 andonly if $G$ contains a block which can be obtained from copies of $K_{k+1}$ byrepeated applications of the Haj\'os join.

A Brooks type theorem for the maximum local edge connectivity