b-chromatic number of cacti

A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G) is the maximum number of colours used in a b-colouring of G. If m(G) is the largest integer k such that G has at least k vertices with degree at least k−1, then we know that χb(G)⩽m(G). Irving and Manlove [Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, 91 (1999), pages 127–141] prove that, if T is a tree, then the b-chromatic number of T is at least m(T)−1. In this paper, we prove that, if G is a connected cactus and m(G)⩾7, then the b-chromatic number of G is at least m(G)−1.