It is known that the problem of settling the existence of an n X n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph K n n such that Tcontains at least one circuit of each copy of K 23 in K n ". Here we investigate the case where n = 2 (mod 4), and we show that the problem of finding the cardinality of T is equivalent to that of settling the existence of a certain kind of n X n matrix.

more »

... reover, we show that the case where n m 2 (mod 4) differs from that where n s 0 (mod 4) in that the problem of finding the cardinality of T is not equivalent to that of maximising the determinant of an n X n (l,-l)-matrix. 1980 Mathematics subject classification (Amer. Math. Soc.): 05 C 50 Thus the Hadamard conjecture is equivalent to a problem about the 4-circuits of K n ", where n = 0 (mod 4). It is also well-known to be equivalent to the