Graceful Paths to Edge-Graceful Trees

Graceful and edge-graceful graph labelings are dual notions of each other in the sense that a graceful labeling of the vertices of a graph \(G\) induces a labeling of its edges, whereas an edge-graceful labeling of the edges of \(G\) induces a labeling of its vertices. In this paper we show a connection between these two notions, namely, that the graceful labeling of paths enables particular trees to be labeled edge-gracefully. Of primary concern in this enterprise are two conjectures: that a path can be labeled gracefully starting at any vertex label, and that all trees of odd order are edge-graceful. We give partial results for the first conjecture and extend the domain of trees known to be edge-graceful for the second conjecture.