On a conjecture of McNeil

TL;DR

This paper investigates the maximum length of a Hamiltonian walk on grid graphs with Manhattan distance, confirming McNeil's conjecture for the specific case of square grids and extending the analysis to rectangular grids.

Contribution

The paper generalizes McNeil's conjecture from square grids to rectangular grid graphs, providing bounds on the maximum walk length with an additive factor of one.

Findings

Confirmed McNeil's conjecture for square grids up to an additive factor of one.

Extended the analysis to rectangular grid graphs, establishing bounds on maximum walk length.

Provided a unified approach to understanding Hamiltonian walks in grid graphs.

Abstract

Suppose that the $n^2$ vertices of the grid graph $P_n^2$ are labeled, such that the set of their labels is $\{1,2,\ldots,n^2\}$. The labeling induces a walk on $P_n^2$, beginning with the vertex whose label is $1$, proceeding to the vertex whose label is $2$, etc., until all vertices are visited. The question of the maximal possible length of such a walk, denoted by $M(P_n^2)$, when the distance between consecutive vertices is the Manhattan distance, was studied by McNeil, who, based on empirical evidence, conjectured that $M(P_n^2)=n^3-3$, if $n$ is even, and $n^3-n-1$, otherwise. In this work we study the more general case of $P_m\times P_n$ and capture $M(P_m\times P_n)$, up to an additive factor of $1$. This holds, in particular, for the values conjectured by McNeil.