Generalized Heawood Graphs and Triangulations of Tori

TL;DR

This paper introduces a family of generalized Heawood graphs derived from permutahedral tilings, revealing their toroidal nature, duality with higher-dimensional triangulated tori, and analyzing their combinatorial and symmetry properties.

Contribution

It generalizes classical Heawood graphs using permutahedral tilings, establishing their toroidal structure and providing explicit formulas for their combinatorial invariants.

Findings

Generalized Heawood graphs are toroidal.

They are dual to higher-dimensional triangulated tori.

Explicit formulas for their f-vectors and automorphism groups.

Abstract

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.