Using Skolem and Hooked-Skolem Sequences to Generate Graceful Trees

In this paper, it will be shown that a Skolem sequence of order \(n \equiv 0,1 \pmod{4}\) 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 \(n \equiv 2,3 \pmod{4}\) 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.