B-chromatic number of cacti

Abstract

Abstract 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 χ b ( 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 χ b ( 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 .