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 extension of Graham and Pollak’s result to coverings of a graph G where each edge of G is allowed to be covered a specified number of times and its generalization to complete uniform hypergraphs. We also discuss the recently disproved Alon-Saks-Seymour Conjecture (which can be regarded as 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.