On the Metric Dimension of $K_a \times K_b \times K_c$

TL;DR

This paper determines the metric dimension of the Cartesian product of three complete graphs for all parameter ranges, extending previous results and introducing a new variant of the Mastermind game to establish optimal strategies.

Contribution

It provides a complete characterization of the metric dimension of $K_a imes K_b imes K_c$ for all positive integers, including a novel game-based approach for proof.

Findings

Exact metric dimension formulas for all parameter cases.

Introduction of a new variant of Static Black-Peg Mastermind.

Optimal strategies for determining the metric dimension.

Abstract

In this work we determine the metric dimension of $ K_a \times K_b \times K_c$ for all $a,b,c\in \mathbb N$ with $ a \le b \le c $ as follows. For $3a<b+c$ and $2b \le c$, this value is $c-1$, for $3a<b+c$ and $2b > c$, it is $\left \lfloor \frac{2}{3}(b+c-1) \right \rfloor$, and for $3a=b+c$, it is $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $. The only open case is $3a>b+c$, where two values are possible, namely $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $ and $\left \lfloor \frac{a+b+c}{2} \right \rfloor $. This result extends previous results of C\'acere et al., who computed the metric dimension of $ K_a \times K_b$, and of Drewes and J\"ager, who computed the metric dimension of $ K_a \times K_a \times K_a$. We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows, as the number of questions of these strategies is equal to the metric dimension of $K_a \times K_b \times K_c$.