Generalized Heawood Graphs and Triangulations of Tori

The Heawood graph is a remarkable graph that played a fundamental role in the development of the theory of graph colorings on surfaces in the 19th and 20th centuries. Based on permutahedral tilings, we introduce a generalization of the classical Heawood graph indexed by a sequence of positive integers. We show that the resulting generalized Heawood graphs are toroidal graphs, which are dual to higher dimensional triangulated tori. We also present explicit combinatorial formulas for their $f$-vectors and study their automorphism groups.