DECOMPOSITIONS OF COMPLETE GRAPHS INTO ISOMORPHIC BIPARTITE SUBGRAPHS

Let \E(G)\ = epsilon and f, a 1-1 mapping of V(G) into {0,1,...,epsilon}. Then f is called a beta-valuation of G if the induced function given by f(uupsilon)BAR = \f(u) -f(upsilon)\, for all uupsilon is-an-element-of E(G)is 1-1. A beta-valuation f is called an alpha-valuation of G if there exists a nonnegative number lambda such that for every uupsilon is-an-element-of E(G) with f(u) < f(upsilon), f(u) less-than-or-equal-to lambda < f(upsilon). Let [GRAPHICS] denote the graph of the n-dimensional G-cube. For G = K3,3, K4,4 and P(k), it is shown that for any positive integer n, the n-dimensional G-cube has an alpha-valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.