Properties of Hamiltonian Circuits in Rectangular Grids

TL;DR

This paper investigates the properties and invariants of Hamiltonian circuits in rectangular grids, establishing minimum turn and straight counts for various grid sizes, with proofs for specific cases and conjectures for general cases.

Contribution

It provides new bounds and invariants for Hamiltonian circuits in rectangular grids, including proofs for specific grid sizes and conjectures for the general case.

Findings

All circuits on a 2n x 2n grid have at least 4n turns.

Circuits on an n x (n+1) grid have minimum 2n or 2n+2 straights depending on parity.

Results are extended to general n x m grids with partial proofs.

Abstract

We present properties and invariants of Hamiltonian circuits in rectangular grids. It is proved that all circuits on a $2n \times 2n$ chessboard have at least $4n$ turns and at least $2n$ straights if $n$ is even and $2n+2$ straights if $n$ is odd. The minimum number of turns and straights are presented and proved for circuits on an $n \times (n+1)$ chessboard. For the general case of an $n \times m$ chessboard similar results are stated but not all proofs are given.