Distance monotonicity and a new characterization of Hamming graphs

Hamming graphs are simply Cartesian products of complete graphs. Several characterizations of these graphs involve the notion of intervals in a graph and related topics. The notion of interval distance monotonicity has already been used in order to obtain a characterization of hypercubes. Here, we revisit this notion and use it to obtain a new characterization of Hamming graphs by performing a combination of the hypercube characterization and the interval hypercube characterization.