On the Super Edge-Magic Deficiency of Graphs

An edge-magic total labeling of an .n; m/-graph G D .V; E/ is a one to one map from V .G/ [ E.G/ onto the integers f1; 2; : : : ; n C mg with the property that there exists an integer constant c such that .x/ C .y/ C .xy/ D c for any xy 2 E.G/. It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edgemagic total labeling, called super edge-magic deficiency of a graph G, is denoted by s .G/ [4]. If such vertices do not exist, then deficiency of G will be C1. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by .

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“…The super edge-magic deficiency of a graph G, denoted by s .G/ [4], is mathematically expressed as if…”

Section: Conjecture 11 ([1]) "Every Tree Is Super Edge-magic Total"mentioning

confidence: 99%

Deficiency of forests

et al. 2017

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“…It is not hard to verify that the following labeling is a super edge-magic labeling of G(k). 1 , x 2,1 , y 1,1 , y 2,1 , y 3,1 ), (7,9,8,10),…”

Section: Constructions Of Maximal Graphsmentioning

confidence: 99%

“…However, it is not the case. As an example P n + K 1 , which satisfies the maximal condition, is super edge-magic if and only if 1 ≤ n ≤ 6 (see [7]). …”

Section: Constructions Of Maximal Graphsmentioning

confidence: 99%

“…The edge-magic deficiency of a graph G, µ(G), is the smallest nonnegative integer n such that G ∪ nK 1 is an edge-magic graph. Motivated by Kotzig and Rosa's concept of edge-magic deficiency, Figueroa-Centeno et al [7] introduced the concept of super edge-magic deficiency of a graph. The super edge-magic deficiency of a graph G, µ s (G), is either the smallest nonnegative integer n such that G ∪ nK 1 is a super edge-magic graph or +∞ if there exists no such n.…”

Section: Introductionmentioning

confidence: 99%

“…The super edge-magic deficiency of cycles, complete graphs, complete bipartite graphs K 2,m , disjoin union of cycles, and some forest with two components can be found in [7,8]. The super edge-magic deficiency of some classes of chain graphs, wheels, fans, double fans, and disjoint union of complete bipartite graphs can be found in [13,14].…”

Section: Introductionmentioning

confidence: 99%

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