All Trees of Odd Order with Three Even Vertices are Super Edge-Graceful

Abstract

A $(p,q)$-graph $G$ is said to be 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. In [7], we establish that every tree of odd order with one even vertex is edge-graceful. Mitchem and Simoson [19] introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for trees. We show that every tree of odd order with three even vertices is super edge-graceful.