When Two-Holed Torus Graphs are Hamiltonian

TL;DR

This paper investigates Hamiltonian cycles in two-holed torus grid graphs, providing algorithms for determining Hamiltonicity, analyzing periodicity patterns, and classifying specific cases with respect to diagonals and grid dimensions.

Contribution

It introduces algorithms for Hamiltonicity detection and diagonal counting in two-holed torus graphs, along with a complete classification of certain graph configurations.

Findings

An $ ext{O}(n^4)$ algorithm for Hamiltonicity determination.

An $ ext{O}( ext{log}(n))$ algorithm for counting diagonals.

Periodic patterns in Hamiltonicity based on fixed grid sides.

Abstract

Trotter and Erd\"os found conditions for when a directed $m \times n$ grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an $\mathcal{O}(n^4)$ algorithm to determine the Hamiltonicity of one of these graphs and an $\mathcal{O}(\log(n))$ algorithm to find the number of diagonals, which are sets of vertices that force the directions of edges in any Hamiltonian cycle. We also show that there is a periodicity pattern in the graphs' Hamiltonicities if one of the sides of the grid is fixed; and we completely classify which graphs are Hamiltonian in the cases where $n=m$, $n=2$, the $m \times n$ graph has $1$ diagonal, or the $\frac{m}{2} \times \frac{n}{2}$ graph has $1$ diagonal.