Edge-Odd Graceful for Cartesian Product of a Wheel With n Vertices and a Path with Two Vertices

Abstract

Abhyankar [1] investigated direct methods of gracefully labeling graphs. Bahl [2] got gracefulness labeling for few families of spiders in few terms of merging such graphs. Barrientos [3] obtained graceful labelings for chain graphs. Edwards and Howard [4] analyzed a survey of some classes of graceful trees. Kaneria [6] obtained graceful labeling for graphs related to cycle. Kaneria [7] made new graceful graphs by merging stars. Kaneria [8] received graceful labeling by attaching cycle to cycles and cycle with a complete bipartite graph. Mishra and Panigrahi [10] investigated new classes of graceful lobsters obtained from diameter four trees. Ramachandran and Sekar [11] got graceful labelling of super subdivision of ladder. $A (p, q)$ connected graph is edge-odd graceful graph if there exists an injective map $f: E(G) \to\{1, 3, \dots, 2q-1\}$ so that induced map $f_{+}: V(G)\to \{0, 1,2, 3, \dots, (2k-1)\}$ defined by $f_{+}(x) \equiv\sum f(xy)\ (mod\ 2k)$, where the vertex x is incident with other vertex y and $k = \max \{p, q\}$ makes all the edges distinct and odd. In this article, the edge-odd gracefulness of cartesian product of $P_{2}$ and $W_{n}$ is obtained.