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

Abstract

An odd graceful labeling of a graph ( , ) G V E = is a function : ( ) {0,1,2, . . .2 ( ) 1} f V G E G → − such that | ( ) ( )| f u f v − is odd value less than or equal to 2 ( ) 1 E G − for any , ( ) u v 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 n P and m P denoted n m P P ∧ . Firstly, we describe an algorithm to label the vertices and the edges of the vertex set ( ) n m V P P ∧ and the edge set ( ) n m E P P ∧ respectively. Finally, we prove that the graph n m P P ∧ is odd graceful for all integers n and . m