Pursuit-evasion games on latin square graphs

TL;DR

This paper explores pursuit-evasion parameters like cop number, metric dimension, and localization number on latin square graphs, providing bounds and analyzing specific cases such as back-circulant latin squares.

Contribution

It establishes new bounds for cop number, metric dimension, and localization number on latin square graphs, including cases involving k-MOLS and back-circulant structures.

Findings

Cop number bounds are given for k-MOLS(n), with a specific value for n > (k+1)^2.

Lower and upper bounds for metric dimension and localization number are established.

The metric dimension of back-circulant latin squares is shown to be close to the lower bound.

Abstract

We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the upper bound of $n+O\left(\frac{\log{n}}{\log{\log{n}}}\right)$ on the localization number of a latin square graph of order $n.$