3-colouring for dually chordal graphs and generalisations

In this paper, we investigate the Colourability problem for dually chordal graphs and some of its generalisations. We show that the problem remains NP-complete if limited to four colours. For the case of three colours, we present a simple linear time algorithm for dismantlable graphs (which include dually chordal graphs) and present an O(n3m) time algorithm for graphs with tree-breadth 1. Additionally, we show that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four.