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 K k+1 by repeated applications of the Hajós join.
|Journal||The Electronic Journal of Combinatorics|
|Number of pages||11|
|Publication status||Published - 2018|
- Graph coloring
- Critical graphs
- Brooks’ theorem