On the super edge-magic deficiency of some graphs

Vira Hari Krisnawati,Anak Agung Gede Ngurah,Noor Hidayat,Abdul Rouf Alghofari

doi:10.1016/j.heliyon.2020.e05561

Vira Hari Krisnawati, Anak Agung Gede Ngurah + Show 2 more

Open Access

https://doi.org/10.1016/j.heliyon.2020.e05561

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Abstract

A graph G is called super edge-magic if there exists a bijection f:V(G)∪E(G)⟶{1,2,⋯,|V(G)|+|E(G)|}, where f(V(G))={1,2,⋯,|V(G)|}, such that f(u)+f(uv)+f(v) is a constant for every edge uv∈E(G). Such a case, f is called a super edge magic labeling of G. A bipartite graph G with partite sets A and B is called consecutively super edge-magic if there exists a super edge-magic labeling f with the property that f(A)={1,2,⋯,|A|} and f(B)={|A|+1,|A|+2,⋯,|V(G)|}. The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G∪nK1 is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a bipartite graph G, denoted by μc(G), is either the minimum nonnegative integer n such that G∪nK1 is consecutively super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph 2C3∪Cn and join product of K1,n∪Pm with an isolated vertex.

Highlights

Let G be a finite and simple graph having vertex set V (G) and edge set E(G), where p = |V (G)| and q = |E(G)|
We study the super edge-magic deficiency (SEMD) of some graphs
We find the exact value or upper bound of the SEMD of forests with two components

Summary

Introduction

[5] A graph G is SEM if and only if there exists a bijection f ∶ V (G) → {1, 2, ⋯ , p} such that the set of all edge-sums S = {f (x) + f (y) ∶ xy ∈ E(G)} consists of q consecutive integers In this case, f extends to be a SEM labeling of G with magic constant k = p + q + min(S). The super edge-magic deficiency (SEMD) of a graph G, μs(G), is defined as either the minimum nonnegative n such that G ∪ nK1 is a SEM graph or +∞ if there exists no such n. The consecutively super edge-magic deficiency (consecutively SEMD) of a bipartite graph G, μc (G), is defined to be either the smallest nonnegative integer n with the property that G ∪ nK1 is consecutively SEM or +∞ if there exists no such n. We study the SEMD of graph (K1,n ∪ Pm) + K1 for any integer n ≥ 1 and m ≥ 3

The SEMD of forests with two components

Conclusion