ABSTRACT
An acyclic colouring of a graph is a proper colouring of the graph, for which every cycle uses at least three colours. In this paper, we describe a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertices. Unlike previous operations, our method successfully establishes a connection with (acyclic) 4-colourings. Moreover, we discuss a special class of graphs, called diamond triangulations, and give a necessary and sufficient condition for a diamond triangulation to be acyclically 4-colourable.
Acknowledgements
The authors are very much indebted to the referees for their valuable comments and suggestions which greatly improved the original manuscript of this paper.
Disclosure statement
No potential conflict of interest was reported by the authors.