Snake Paths in King and Knight Graphs

TL;DR

This paper investigates the maximum length of snake paths in king and knight graphs, providing exact results for king graphs and estimates for knight graphs, highlighting differences between even and odd cases.

Contribution

It determines the maximum snake path length in n x n king graphs and characterizes the paths, also estimating maximum lengths in m x n knight graphs for all dimensions.

Findings

Maximum snake path length in n x n king graphs is found and characterized.

Different behaviors are observed between even and odd cases.

Estimated maximum lengths for snake paths or cycles in m x n knight graphs.

Abstract

A snake path in a graph $G$ is a path in $G$ which is also an induced subgraph of $G$. For all $n$, we find the greatest length of a snake path in the $n \times n$ king graph and we give a complete description of the paths which attain this greatest length. The even and odd cases behave very differently. We also estimate the greatest length of a snake path or cycle in the $m \times n$ knight graph, for all $m$ and $n$.