Turns in Hamilton cycles of rectangular grids

TL;DR

This paper investigates the maximum number of turns in Hamilton cycles within rectangular grids, providing exact solutions in specific cases and approximate bounds in others, along with a new proof for the square grid case.

Contribution

It introduces a novel connection between the problems of maximizing and minimizing turns in Hamilton cycles on grids, extending previous results and offering new proofs.

Findings

Exact maximum turns for certain grid sizes

Approximate bounds with additive error of 2 for others

A new proof for the square grid case

Abstract

For a Hamilton cycle in a rectangular $m \times n$ grid, what is the greatest number of turns that can occur? We give the exact answer in several cases and an answer up to an additive error of $2$ in all other cases. In particular, we give a new proof of the result of Beluhov for the case of a square $n \times n$ grid. Our main method is a surprising link between the problem of 'greatest number of turns' and the problem of 'least number of turns'.