Hamiltonian paths in m x n projective checkerboards

TL;DR

This paper investigates the existence of Hamiltonian paths on m x n projective checkerboards, extending previous work by analyzing routes that visit each square exactly once with specific movement constraints.

Contribution

It generalizes prior results by characterizing Hamiltonian paths on projective checkerboards of arbitrary dimensions, incorporating edge-twisting to model projective plane properties.

Findings

Determines conditions for Hamiltonian paths between any two squares.

Extends known results from square to rectangular projective boards.

Provides a comprehensive framework for path existence in this setting.

Abstract

For any two squares A and B of an m x n checkerboard, we determine whether it is possible to move a checker through a route that starts at A, ends at B, and visits each square of the board exactly once. Each step of the route moves to an adjacent square, either to the east or to the north, and may step off the edge of the board in a manner corresponding to the usual construction of a projective plane by applying a twist when gluing opposite sides of a rectangle. This generalizes work of M.H.Forbush et al. for the special case where m = n.