Hamiltonicity of doubly semi-equivelar maps on the torus

TL;DR

This paper proves that all doubly semi-equivelar maps on the torus contain a Hamiltonian cycle, confirming the Nash-Williams conjecture for their associated graphs by demonstrating high connectivity.

Contribution

It establishes the existence of Hamiltonian cycles in all doubly semi-equivelar torus maps and verifies the Nash-Williams conjecture for their graphs.

Findings

All doubly semi-equivelar maps on the torus have Hamiltonian cycles.

Associated graphs are either 3-connected or 4-connected.

Confirmed Nash-Williams conjecture for these graphs.

Abstract

The well-known twenty types of 2-uniform tilings of the plane give rise infinitely many doubly semi-equivelar maps on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle. As a consequence, we establish the Nash-Williams conjecture for the graphs associated with these doubly semi-equivelar maps by showing that these graphs are either 3-connected or 4-connected.