On the cyclic decomposition of complete graphs into almost-bipartite graphs Original Research Article

Techniques of labeling the vertices of a bipartite graph G with n edges to yield cyclic G-decompositions of the complete graph K2nx+1 have received much attention in the literature. Up until recently, these techniques have been used mostly with bipartite graphs. An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. Examples of such graphs include the odd cycles. Here we introduce the concept of a γ-labeling of an almost-bipartite graph and show that if an almost-bipartite graph G with n edges has a γ-labeling then there is a cyclic G-decomposition of K2nx+1 for all positive integers x. We also show that odd cycles as well as certain other almost-bipartite 2-regular graphs have γ-labelings.