Triangulations and the Hajos Conjecture

In his recent work Thomassen [J Combin Theory Ser B 93 (2005), 95-105] discussed various refinements of Hajos conjecture. Shortly after Mohar [Electr J Combin 12 (2005), N15] provided an answer to Thomassen's Conjecture 6.5, and proposed a possible extension. The aim of this article is to address Mohar's suggestion. In particular, we prove that, for infinitely many integers m, there exists a graph on m vertices forming a triangulation of an orientable surface so that it does not contain a subdivision of a clique of size O(m(1/2)), and its chromatic number is at least m(2/3+o(1)). The main part of the proof is to show that the random graph can be almost covered by oriented triangles which do not contain certain forbidden configurations. We use a technique similar to the ones of Archdeacon and Grable [Discrete Math 142(1-3) (1995), 21-37] and Thomas and the first author [Random Struct Algorithms 6(1) (1995), 1-12]. We obtain a strengthening by replacing the "nibble" method by "random bites" used by Alon et al. [Israel J Math 100 (1997), 171-187]. (C) 2008 Wiley Periodicals Inc. J Graph Theory 59: 293-325, 2008