is connected by an edge to a vertex j whenever there exists a k, 1 k t, such that b^-bj^ = -1.
By (2), it follows that G is the complete graph K9. Considering the edges which are contributed
by any one column of B(F), it follows that K9 has a decomposition into the sum, over all i and
j , of XJJ copies of the complete bipartite graph Ky having i vertices of one class and j vertices
of the other class. Thus, we have the following edge-disjoint decomposition:
(9) K9 = X XyKij .
The Graham-Pollak Theorem  (for simpler proofs, see [11,16]; for extensions, see [13,
23, 6]) implies that
I xy 8 .
The Diophantine system (3), (4), (6), (7) and (8) has 24 possible solutions, given in
Table 1. The idea of the proof is to show that each one of these solutions, which will be called here
"cases", leads to contradictions.