Using Skolem and Hooked-Skolem Sequences to Generate Graceful Trees

Abstract

In this paper, it will be shown that a Skolem sequence of order $n\equiv 0,1(\mathrm{mod}4)$ implies the existence of a graceful tree on $2n$ vertices which exhibits a perfect matching or a matching on $2n-2$ vertices. It will also be shown that a Hooked-Skolem sequence of order $n\equiv 2,3(\mathrm{mod}4)$ implies the existence of a graceful tree on $2n+1$ vertices which exhibits a matching on either $2n$ or $2n-2$ vertices. These results will be established using an algorithmic approach.