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

TL;DR

This work determines the metric dimension $f(a,b,c)$ of the cross product $K_a \times K_b \times K_c$ for all $a\le b\le c$ by recasting the problem as a Static Black-Peg Mastermind variant with per-peg color sets. It develops a suite of feasibility lemmas, a projection framework, and explicit strategy constructions to yield tight bounds across four major regimes: $3a<b+c$, $3a=b+c$, $3a> b+c$, with subcases based on parity and mod-3 considerations. Key results include $f(a,b,c)=c-1$ when $3a<b+c$ and $2b\le c$; $f(a,b,c)=\left\lfloor \dfrac{2}{3}(b+c-1)\right\rfloor$ when $3a<b+c$ and $2b> c$; $f(a,b,c)=\left\lfloor \dfrac{a+b+c}{2} \right\rfloor -1$ for $3a=b+c$ (except $(4,6,6)$); and $f(a,b,c)\le \left\lfloor \dfrac{a+b+c}{2} \right\rfloor$ with many cases achieving equality when $3a\ge b+c$, plus special $f(a,a-1,2a)$-type results. The paper closes many cases with explicit, provably optimal strategies, and it conjectures a tight formula for the remaining open regime $3a>b+c$, supported by exhaustive checks up to $a+b+c\le 21$. This Mastermind‑driven framework thus unifies and extends prior results on $K_a\times K_b$ and $K_a\times K_a\times K_a$ in a single approach. The work provides explicit constructions, lower/upper bounds, and a clear roadmap for resolving the final open case.

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 Cácere et al., who computed the metric dimension of $ K_a \times K_b$, and of Drewes and Jäger, 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 $K_a \times K_b \times K_c$.

Figures (1)

• Figure 1: The questions and the colors of the secrets $s_1$ and $s_2$ at the start of each subcase of Lemma \ref{['lem:super_lemma']}

Theorems & Definitions (51)

• Definition 1
• Definition 2
• Lemma 3
• Corollary 4
• proof
• Lemma 5
• proof
• Remark 6
• Definition 7
• Lemma 8