Characterizing subgraphs of Hamming graphs

Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) incident edges receive the same label if and only if they lie on a common triangle; (ii) for any vertices u and v at distance at least two, there exist two labels which both appear on any induced u, v-path.