Gracefulness of a Cycle with Parallel Chords and Parallel $P_k$-Chords of Different Lengths

Abstract

In this paper, we prove that every $n$-cycle ($n\ge 6$) with parallel chords is graceful for all $n\ge 6$ and every $n$-cycle with parallel $P_k$-chords of increasing lengths is graceful for $n\equiv 2(\mathrm{mod}4)$ with $1\le k\le \lfloor \frac{n}{2}\rfloor –1$.