A Note on Strongly Graceful Trees

Abstract

A tree $T$ with $n$ vertices and a perfect matching $M$ is strongly graceful if $T$ admits a graceful labeling $f$ such that $f(u)+f(v)=n-1$ for every edge $uv\in M$. Broersma and Hoede $[5]$ conjectured that every tree containing a perfect matching is strongly graceful in $1999$. We prove that a tree $T$ with diameter $D(T)\le 5$ supports the strongly graceful conjecture on trees. We show several classes of basic seeds and some constructive methods for constructing large scales of strongly graceful trees.