An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

TL;DR

This paper investigates odd graceful labelings of unions of paths and cycles, proving conditions for odd gracefulness and providing an algorithm for labeling such graphs.

Contribution

It extends odd graceful labeling results to unions of cycles and paths, and introduces a specific labeling algorithm for these graphs.

Findings

C_m ∪ P_n is odd graceful if m is even

Provided an explicit labeling algorithm for these graphs

Extended known results on odd graceful labelings

Abstract

In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1 edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph C_m $\cup$ P_m when m = 4, 6,8,10 and then we proved that the graph C_ $\cup$ P_n is odd graceful if m is even. Finally, we described an algorithm to label the vertices and the edges of the vertex set V(C_m $\cup$ P_n) and the edge set E(C_m $\cup$ P_n).