Decompositions of complete graphs into isomorphic bipartite subgraphs

Abstract

Let |E(G)|=ε andf, a 1-1 mapping ofV(G) into {0,1,...,ε}. Thenf is called a β-valuation ofG if the induced function given by$\bar f(u\upsilon ) = |f(u) - f(\upsilon )|$, for alluv∈E(G) is 1-1. A β-valuationf is called an α-valuation ofG if there exists a nonnegative number λ such that for everyuv∈E(G) withf(u)<f(v),f(u)≤λ<f(v). Let$Q_n (G) = G \times \underbrace {K_2 \times ... \times K_2 }_{(n - 1)times}$ denote the graph of then-dimensionalG-cube. ForG=K _{3, 3},K _{4, 4}, andP _k ,it is shown that for any positive integern, then-dimensionalG-cube has an α-valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.