New Results on the (Super) Edge-Magic Deficiency of Chain Graphs

Ngurah Anak Agung Gede,Adiwijaya Adiwijaya

doi:10.1155/2017/5156974

Abstract

Let G be a graph of order v and size e. An edge-magic labeling of G is a bijection f:V(G)∪E(G)→{1,2,3,…,v+e} such that f(x)+f(xy)+f(y) is a constant for every edge xy∈E(G). An edge-magic labeling f of G with f(V(G))={1,2,3,…,v} is called a super edge-magic labeling. Furthermore, the edge-magic deficiency of a graph G, μ(G), is defined as the smallest nonnegative integer n such that G∪nK1 has an edge-magic labeling. Similarly, the super edge-magic deficiency of a graph G, μs(G), is either the smallest nonnegative integer n such that G∪nK1 has a super edge-magic labeling or +∞ if there exists no such integer n. In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Referring to these, we propose some open problems.

Highlights

Let G be a finite and simple graph, where V(G) and E(G) are its vertex set and edge set, respectively
We investigate the edge-magic deficiency of chain graphs
In [1], Kotzig and Rosa introduced the concepts of edge-magic labeling and edge-magic graph as follows: an edge-magic labeling of a graph G is a bijection f : V(G) ∪ E(G) → {1, 2, 3, . . . , V + e} such that f(x) + f(xy) + f(y) is a constant, called the magic constant of f, for every edge xy of G

Summary

Introduction

Let G be a finite and simple graph, where V(G) and E(G) are its vertex set and edge set, respectively. V} is called a super edge-magic labeling. The edge-magic deficiency of a graph G, μ(G), is defined as the smallest nonnegative integer n such that G ∪ nK1 has an edge-magic labeling.

Results

Conclusion