A Brooks Type Theorem for the Maximum Local Edge Connectivity

For a graph G, let χ(G) and λ(G) denote the chromatic number of G and the maximum local edge connectivity of G, respectively. A result of Dirac implies that every graph G satisfies χ(G) 6 λ(G) + 1. In this paper we characterize the graphs G for which χ(G) = λ(G) + 1. The case λ(G) = 3 was already solved by Aboulker, Brettell, Havet, Marx, and Trotignon. We show that a graph G with λ(G) = k > 4 satisfies χ(G) = k+ 1 if and only if G contains a block which can be obtained from copies of Kk+1 by repeated applications of the Hajós join.