On k-graceful, locally finite graphs

AbstractWhile a finite, bipartite graph G can be k-graceful for every k ≥ 1, a finite nonbipartite G can be k-graceful for only finitely many values of k. An improved bound on such possible values of k is presented. Definitions are extended to infinite graphs, and it is shown that if G is locally finite and vertex set V(G) and edge set E(G) are countably infinite, then for each k ≥ 1 the graph G has a k-graceful numbering h mapping V(G) onto the set of nonnegative integers