On The Edge-graceful Trees Conjecture

Abstract

A $(p,q)$-graph $G$ is said to be \textbf{edge-graceful} if the edges can be labeled by $1,2,\dots ,q$ so that the vertex sums are distinct, mod $p$. It is shown that if a tree $T$ is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. The conjecture is still unsettled. In this paper, we give the state of the progress toward this tantalizing conjecture.