Odd harmonious labeling of grid graphs

P Jeyanthi,S Philo,Maged Z Youssef

doi:10.22199/issn.0717-6279-2019-03-0027

Abstract

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f* : E(G) → {1, 3, · · · , 2q − 1} defined by f∗ (uv) = f (u) + f (v) is a bijection. In this paper we prove that path union of t copies of Pm×Pn, path union of t different copies of Pmᵢ×Pnᵢ where 1 ≤ i ≤ t, vertex union of t copies of Pm×Pn, vertex union of t different copies of Pmᵢ×Pnᵢ where 1 ≤ i ≤ t, one point union of path of Ptn (t.n.Pm×Pm), t super subdivision of grid graph Pm×Pn are odd harmonious graphs.

Highlights

Throughout this paper by a graph we mean a finite, simple and undirected one
The graph labeling is an assignment of integers to the set of vertices or edges or both, subject to certain conditions
A graph G is said to be harmonious if there exists an injection f : V (G) → Zq such that the induced function f ∗ : E(G) → Zq defined by f ∗(uv) = (f (u) + f (v)) is a bijection and f is called harmonious labeling of G

Summary

Introduction

Throughout this paper by a graph we mean a finite, simple and undirected one. For standard terminology and notation we follow Harary [3]. The Cartesian product of two paths Pm and Pn denoted by Pm × Pn is known as a grid graph on mn vertices and 2mn − (m + n) edges. Let G be a graph and G1, G2, · · · , Gn, n ≥ 2 be n copies of graph G. the graph obtained by adding an edge from Gi to Gi+1 (i = 1, 2, · · · , n − 1) is called path union of graph G.

Results

Conclusion