Total-Chromatic Number and Chromatic Index of Dually Chordal Graphs

A graph is dually chordal if it is the clique graph of a chordal graph. Alternatively, a graph is dually chordal if it admits a maximum neighbourhood order. This class generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs and interval graphs. We prove that Vizing's total-colour conjecture holds for dually chordal graphs. We describe a new heuristic that yields an exact total-colouring algorithm for even maximum degree dually chordal graphs and an exact edge-colouring algorithm for odd maximum degree dually chordal graphs. Key words. graph algorithms, chordal graphs, total colouring, edge colouring, clique graphs, maximum neighbourhood ordering. AMS subject classification. 05C85, 05C15, 68R10, 90C27 1 Introduction We consider the problem of total colouring and edge colouring dually chordal graphs. A total colouring of a graph is a colouring of its vertices and edges such that no adjacent vertices, no adjacent edges, and no incident ver..