P-Labeling (Nearly Graceful Labeling) of Cycles with Parallel Pk Chords

A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. A graph G(V,E), |V (G)| = p, |E(G)| = q is said to be nearly graceful if there is an injection from V (G) to {0, 1, 2, . . . , q + 1} such that when each edge xy is assigned the label |f(x) − f(y)| the resulting edge labels are distinct and are in the set {1, 2, 3, . . . , q − 1, q + 1}. In this paper we prove the nearly gracefulness of cycles with parallel chords and cycles with parallel Pk chords for k = 3, 4.