ON COMPLETE BIPARTITE DECOMPOSITION OF COMPLETE MULTIGRAPHS

Let K(n \ t) denote the complete multigraph containing n vertices and exactly t edges between every pair of distinct vertices, and let f(n; t) be the minimum number of complete bipartite subgraphs into which the edges of K(n \ t) can be decomposed. Pritikin [3] proved that f(n; t) greater-than-or-equal-to max {n - 1,t}, and that f(n; 2) = n if n = 2, 3, 5, and f(n; 2) = n - 1, otherwise. In this paper, for t greater-than-or-equal-to 3 using Hadamard designs, skew-Hadamard matrices and symmetric conference matrices [6], we give some complete multigraph families K(n \ t) with f(n; t) = n - 1.