Clique-colouring and biclique-colouring unichord-free graphs

Abstract

29 p. : il.The class of unichord-free graphs was recently investigated in the context of vertex-colouring [J. Graph Theory 63 (2010) 31{67], edge-colouring [Theoret. Comput. Sci. 411 (2010) 1221{1234] and total- colouring [Discrete Appl. Math. 159 (2011) 1851{1864]. Unichord-free graphs proved to have a rich structure that can be used to obtain in- teresting results with respect to the study of the complexity of colour- ing problems. In particular, several surprising complexity dichotomies of colouring problems are found in subclasses of unichord-free graphs. In the present work, we investigate clique-colouring and biclique-colouring problems restricted to unichord-free graphs. We show that the clique- chromatic number of a unichord-free graph is at most 3, and that the 2-clique-colourable unichord-free graphs are precisely those that are per- fect. We prove that the biclique-chromatic number of a unichord-free graph is at most its clique-number. We describe an O(nm)-time algo- rithm that returns an optimal clique-colouring, but the complexity to optimal biclique-colour a unichord-free graph is not classi ed yet. Nev- ertheless, we describe an O(n2)-time algorithm that returns an optimal biclique-colouring in a subclass of unichord-free graphs called cactus. Keywords: unichord-free, decomposition, hypergraphs, Petersen graph, Heawood graph, clique-colouring, biclique-colouring, cactus