### On the super edge-magic deficiency of join product and chain graphs

#### Abstract

A graph *G* of order ∣*V*(*G*)∣ = *p* and size ∣*E*(*G*)∣ = *q* is called *super edge-magic if there exists a bijection f : V(G) ∪ E(G) → {1, 2, 3, ⋯, p + q} such that f(x) + f(xy) + f(y) is a constant for every edge xy ∈ E(G) and f(V(G)) = {1, 2, 3, ⋯, p}. Furthermore, the super edge-magic deficiency of a graph G, μ_{s}(G), is either the minimum nonnegative integer n such that G ∪ nK_{1} is super edge-magic or + ∞ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product of a graph which has certain properties with an isolated vertex and the super edge-magic deficiency of chain graphs.*

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.12

#### References

A. Ahmad, M.F. Nadeem, M. Javaid, and R. Hasni, On the super edge-magic deficiency of some families related to ladder graphs, Australas. J. Combin. 51 (2011), 201–208.

A. Ahmad, I. Javaid and M.F. Nadeem, Further results on super edge-magic deficiency of unicyclic graphs, Ars Combin. 99 (2011), 129–138.

C. Barrientos, Graceful labeling of chain and corona graphs, Bull. Inst. Combin. Appl. 34 (2002), 17–26.

E.T. Baskoro, I.W. Sudarsana, and Y.M. Cholily, How to construct new super edge-magic graphs from some old ones, J. Indones. Math. Soc. 2 (2005), 155–162.

Z. Chen, On super edge magic graphs, J.Combin, Math. Combin. Comput. 38 (2001), 55–64.

H. Enomoto, A. Llado, T. Nakamigawa, and G. Ringel, Super edge magic graphs, SUT J. Math. 34 (1998), 105–109.

R.M. Figueroa-Centeno, R. Ichishima, and F.A. Muntaner-Batle, The place of super edge-magic labelings among other classes of labelings, Discrete Math. 231 (2001), 153–168.

R.M. Figueroa-Centeno, R. Ichishima, and F.A. Muntaner-Batle, On the super edge-magic deficiency of graphs, Ars Combin. 78 (2006), 33–45.

R.M. Figueroa-Centeno, R. Ichishima, and F.A. Muntaner-Batle, Some new results on the super edge-magic deficiency of graphs, J. Combin. Math. Combin. Comput., 55 (2005), 17– 31.

J.A. Gallian, A dinamic survey of graph labelings, Electron. J. Combin. 16 (2017) # DS6.

R. Ichishima and A. Oshima, On the super edge-magic deficiency and α-valuation of graphs, J. Indones. Math. Soc., Special Edition (2011), 59–69.

A. Kotzig and A. Rosa, Magic valuation of finite graphs, Canad. Math. Bull. 13 (4) (1970), 451–461.

S.M. Lee and J. Wang, On super edge-magicness of chain graphs whose blocks are complete graphs, Cong. Numer. 162 (2003), 147–160.

A.A.G. Ngurah and Adiwijaya, New results on the (super) edge-magic deficiency of chain graphs, International Journal of Mathematics and Mathematical Sciences, Vol. 2017, Article ID 5156974, 6 pages, https://doi.org/10.1155/2017/5156974.

A.A.G. Ngurah, E.T. Baskoro, and R. Simanjuntak, On super edge-magic deficiency of graphs, Australas. J. Combin. 40 (2008), 3–14.

A.A.G. Ngurah and R. Simanjuntak, Super edge-magic labelings: deficiency and maximality, Electron. J. Graph Theory Appl. 5 (2) (2017), 212–220.

A.A.G. Ngurah and R. Simanjuntak, Super edge-magic deficiency of join-product graphs, Util. Math. 105 (2017), 279–289.

A.A.G. Ngurah, R. Simanjuntak, E.T. Baskoro, and S. Uttunggadewa, On super edge-magic strength and deficiency of graphs, Computational Geometry and Graph Theory, LNCS 4535 (2008), 144–154.

Slamin, M. Baca, Y. Lin, M. Miller and R. Simanjuntak, Edge-magic total labelings of wheels, fans and friendship graphs, Bull. Inst. Combin. Appl. 35 (2002), 89–98.

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