Hamiltonian Cycles in Polyhedral Maps

TL;DR

This paper establishes a precise condition for the existence of specific Hamiltonian cycles in the edge graphs of polyhedral maps on surfaces, including an algorithm for their construction in certain cases.

Contribution

It provides a necessary and sufficient condition for various types of Hamiltonian cycles in polyhedral maps and introduces an algorithm to construct them when they exist.

Findings

Characterization of Hamiltonian cycles in polyhedral maps

Existence of contractible Hamiltonian cycles in equivelar triangulated maps

Algorithm for constructing such cycles

Abstract

We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an algorithm to construct such cycles whenever it exists.