Variations on a theme of Graham and Pollak

Graham and Pollak proved that one needs at least n − 1 complete bipartite subgraphs (bicliques) to partition the edge set of the complete graph on n vertices. In this paper, we study the generalizations of their result to coverings of graphs with specified multiplicities and to complete uniform hypergraphs. We also discuss the recently disproved Alon–Saks–Seymour Conjecture (which proposed a generalization of the previous result of Graham and Pollak) and compute the exact values of the ranks of the adjacency matrices of the known counterexamples to the Alon–Saks–Seymour Conjecture. The rank of the adjacency matrix of a graph G is related to important problems in computational complexity and provides a non-trivial lower bound for the minimum number of bicliques that partition the edge set of G .