Metric Dimension of a Diagonal Family of Generalized Hamming Graphs

TL;DR

This paper determines the metric dimension of a specific family of generalized Hamming graphs, providing exact values for the 3-dimensional case and introducing a novel characterization method using auxiliary hypergraphs.

Contribution

It introduces a new approach to compute metric dimension for a family of generalized Hamming graphs using forbidden subgraph characterization.

Findings

Exact metric dimension for 3-dimensional generalized Hamming graphs

Characterization of resolving sets via forbidden subgraphs

Constructive approach enabling precise calculations

Abstract

Classical Hamming graphs are Cartesian products of complete graphs, and two vertices are adjacent if they differ in exactly one coordinate. Motivated by connections to unitary Cayley graphs, we consider a generalization where two vertices are adjacent if they have no coordinate in common. Metric dimension of classical Hamming graphs is known asymptotically, but, even in the case of hypercubes, few exact values have been found. In contrast, we determine the metric dimension for the entire diagonal family of $3$-dimensional generalized Hamming graphs. Our approach is constructive and made possible by first characterizing resolving sets in terms of forbidden subgraphs of an auxiliary edge-colored hypergraph.