Isometric embeddings in Hamming graphs

The results presented in this paper are parts of my doctoral thesis [ 193; the algorithm and some of the characterizations in Section 7 have been introduced in [ 181. Motivation for isometric embeddings into Hamming graphs has come from communication theory (Graham and Pollak [ 121) an-d linguistics (Firsov 1181). Isometric subgraphs of Hamming graphs also appear in biology as “quasi-species” (Eigen and Winkler-Oswatitsch [7]). Garey and Graham [9] mention a relation to coding theory. Isometric embeddings into Cartesian products of arbitrary graphs are studied in [ 111. A nice survey about isometric embeddings, factorization, and related problems is c221. The vertices of a Hamming graph are labelled by s-tuples of nonnegative integers (see Fig. 2), such that the distance between vertices is just the number of different coordinates of the corresponding s-tuples, known as the Humming distance from coding theory. Thus isometric embedding of a graph G into a Hamming graph just means labelling the vertices of G by s-tuples such that their Hamming distances represent the lengths of shortest paths in G. Figure 1 shows how the algorithm described in Section 5 finds such a labelling by partitioning the vertex-set of G. Vertices which lie in the same part of the k th partition get the same k th coordinate. A graph is bipartite if and only if we always get a partition into two parts. Djokovic [S] proved that a bipartite graph is isometrically embed-