An Algorithm for Odd Gracefulness of the Tensor Product of Two Line Graphs

TL;DR

This paper introduces an algorithm for assigning odd graceful labelings to the tensor product of two path graphs, expanding the class of graphs known to admit such labelings and providing a constructive method.

Contribution

It presents the first generalized algorithm for odd graceful labeling of tensor products of two paths, proving their odd gracefulness for all integer parameters.

Findings

The tensor product of two paths is odd graceful for all n, m.

A constructive algorithm for labeling these graphs is provided.

The paper extends the class of graphs known to be odd graceful.

Abstract

An odd graceful labeling of a graph G=(V,E) is a function f:V(G)->[0,1,2,...,2|E(G)|-1} such that |f(u)-f(v)| is odd value less than or equal to 2|E(G)-1| for any u, v in V(G). In spite of the large number of papers published on the subject of graph labeling, there are few algorithms to be used by researchers to gracefully label graphs. This work provides generalized odd graceful solutions to all the vertices and edges for the tensor product of the two paths P_n and P_m denoted P_n^P_m . Firstly, we describe an algorithm to label the vertices and the edges of the vertex set V(P_n^P_m) and the edge set E(P_n^P_m) respectively. Finally, we prove that the graph P_n^P_m is odd graceful for all integers n and m.