Near α-labelings of bipartite graphs

An a-labeling of a bipartite graph G with n edges easily yields both a cyclic G-decomposition of Kn,n and of K2nx+1 for all positive integers x. A ,B-Iabeling (or graceful labeling) of G yields a cyclic decomposition of K2n+1 only. It is well-known that certain classes of trees do not have a-Iabelings. In this article, we introduce the concept of a near a-labeling of a bipartite graph, and prove that if a graph G with n edges has a near a-labeling, then there is a cyclic G-decomposition of both Kn,n and K2nx+1 for all positive integers x. We conjecture that all trees have a near a-labeling and show that certain classes of trees which are known not to have an a-labeling have a near a-labeling.