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 (\(n \geq 6\)) with parallel chords is graceful for all \(n \geq 6\) and every \(n\)-cycle with parallel \(P_k\)-chords of increasing lengths is graceful for \(n \equiv 2 \pmod{4}\) with \(1 \leq k \leq \left\lfloor \frac{n}{2} \right\rfloor – 1\).