Hamilton cycles in 4-connected troidal triangulations

Grünbaum [B. Grünbaum, Polytopes, graphs, and complexes, Bull. Amer. Math. Soc. 76 (1970) 1131-1201] and independently Nash-Williams [C.St.J.A. Nash-Williams, Unexplored and semi-explored territories in graph theory, in “New directions in the theory of graphs” 149-186, Academic Press, New York, 1973] conjectured that every 4-connected graphs on the torus has a hamilton cycle. In this paper, we show that the conjecture is true for triangulations of the torus.