Non-Hamiltonian Holes in Grid Graphs

TL;DR

This paper introduces polygonal grid graphs and characterizes non-Hamiltonian holes, providing a necessary and sufficient condition for Hamiltonicity, which could lead to polynomial-time algorithms for the Hamiltonian cycle problem.

Contribution

It extends grid graphs to polygonal tilings, formulates conditions for non-Hamiltonian holes, and offers a new criterion for Hamiltonicity in polygonal grid graphs.

Findings

Defined non-Hamiltonian holes in polygonal grid graphs.

Derived a formula for inside faces using Grinberg theorem.

Established necessary and sufficient conditions for Hamiltonicity.

Abstract

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in the basis, we deduce the formula of inside faces in Grinberg theorem, called Grinberg equation, and derive a kind of cycles whose existence make a polygonal grid graph non-Hamiltonian, called non-Hamiltonian holes, and then we characterize the existence condition of non-Hamiltonian holes and obtain the necessary and sufficient condition of a polygonal grid graph to be Hamiltonian. The result in this paper provides a new starting point for developing a polynomial-time algorithm for Hamilton problem in general grid graphs.