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

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 $K_a \times K_b \times K_c$.