semt labelings and deﬁciencies of forests with two...

Punjab UniversityJournal of Mathematics (ISSN 1016-2526) Vol. 51(5)(2019) pp. 137-149

SEMT Labelings and Deficiencies of Forests with Two Components (I)

Salma Kanwal1, Zurdat Iftikhar2, Aashfa Azam3

1,2,3 Department of Mathematics,Lahore College for Women University Lahore-Pakistan.

Email:[email protected],[email protected],[email protected]

Received: 07 April, 2018 / Accepted: 06 June, 2018 / Published online: 20 December,2018

Abstract. A set of nodes called vertices V accompanied with the linesthat bridge these nodes called edges E compose an explicit figure termedas a graph G(V,E). |V (G)| = ν and |E(G)| = ε specify its order andsize respectively. A (ν, ε)-graph G determines an edge-magic total (EMT)labeling when Γ : V (G) ∪ E(G) → 1, ν + ε is bijective so as theweights at every edge are the same constant (say) c i.e., for x, y ∈ V (G);Γ(x) + Γ(xy) + Γ(y) = c, independent of the choice of any xy ∈ E(G),such a number is interpreted as a magic constant. If all vertices gain thesmallest of the labels then an EMT labeling is called a super edge-magictotal (SEMT) labeling. If a graph G allows at least one SEMT labelingthen the smallest of the magic constants for all possible distinct SEMT la-belings of G describes super edge-magic total (SEMT) strength, sm(G),of G. For any graph G, SEMT deficiency is the least number of isolatedvertices which when uniting with G yields a SEMT graph. In this paper,we will find SEMT labeling and deficiency of forests consisting of twocomponents, where one of the components for each forest is generalizedcomb Cbτ (`, `, . . . , `︸︷︷︸

τ−times

) and other component is a star, bistar, comb or path

respectively, moreover, we will investigate SEMT strength of aforesaidgeneralized comb.

AMS (MOS) Subject Classification Codes: 05C78Key Words: SEMT labeling, SEMT strength, SEMT Deficiency, generalized comb, star,

bistar, comb, path.

137

138 Salma Kanwal, Zurdat Iftikhar and Aashfa Azam

1. PRELIMINARIES

Labeling is a technique that allots labels to the components of a graph. Total labelinggives us both components (vertices and edges) labelled. A (ν, ε)-graph G determines anedge-magic total (EMT) labeling when Γ : V (G) ∪ E(G) → 1, ν + ε is bijective so asthe weights at every edge are the same constant (say) c, such a number c is interpreted asa magic constant. If all vertices gain the smallest of the labels then an EMT labeling iscalled a super edge-magic total (SEMT) labeling. Kotzig and Rosa [17] and Enomoto et al.[7] first introduced the notions of EMT and SEMT graphs respectively and presented theconjectures: every tree is EMT [17], and every tree is SEMT [7].

If a graph G allows at least one SEMT labeling then the smallest of the magic constantsfor all possible distinct SEMT labelings of G describes super edge-magic total (SEMT)strength, sm(G), of G. Avadayappan et al. first introduced the notion of SEMT strength[4] and found exact values of SEMT strength for some graphs.

In [17], the notion of EMT deficiency was proposed by authors and Figueroa-Centenoet al. [8] continued it to SEMT graphs. For any graph G, the SEMT deficiency, signifiedas µs(G), is the least number n of isolated vertices that we have to take in union with Gso that the resulting graph G ∪ nK1 is SEMT, the case +∞ will arise if no isolated vertexfulfils this criteria. More specifically,

µs(G) =

minM(G) if M(G) 6= ∅+∞ if M(G) = ∅

where M(G) = n ≥ 0 : G ∪ nK1 is a SEMT graph.

Exact values for SEMT deficiencies of several classes of graphs are provided in [9, 8].The authors also proposed a conjecture which tells us about the confined deficiencies ofthe forests. In [10], an assumption was made as a special case of a previous one that says,the deficiency of each two-tree forest is not more than 1. Baig et al. [6] determined SEMTdeficiencies of various forests made up of banana trees, stars etc. In [13, 21], S. Javedet al. and Ngurah et al. gave some upper bounds for SEMT deficiency of forests com-posed of stars, fans, combs, double fans, wheels and generalized combs. The results in[1, 2, 3, 5, 14, 15, 16, 18, 19, 20] migth found useful in the aspect of examined labelinghere. A general reference to graph theoretic terminologies can be found in [22]. For morereview, see the recent survey of graph labelings by Gallian [12].

In this paper, we formulated the results on SEMT labeling and deficiency of forests con-sisting of two components, where one component in each forest is a generalized comb andthe other component is a star, bistar, comb or path respectively. Moreover, SEMT strengthof a generalized comb Cbτ (`, `, . . . , `︸︷︷︸

τ−times

) has also been discussed here. The values of pa-

rameters of the star, bistar, comb and path are totally dependant on the parameters involvedin the generalized comb.

SEMT Labelings and Deficiencies of Forests with Two Components (I) 139

x4,1

x4,2

x4,3

x4,4

x1,0

x1,1

x1,2

x1,3

x1,4

x2,1

x2,2

x2,3

x2,4

x3,1

x3,2

x3,3

x3,4

FIGURE 1. Generalized comb Cb4(4, 4, 4, 4)

2. MAIN RESULTS

A star on n vertices is isomorphic to K1,n−1. When we join two stars K1,g , K1,h througha bridge, where g, h ≥ 1 and g+h = n− 2, the resulting tree is termed a bistar BS(g, h).The graph Pn denotes the path of order n and size n− 1, with vertices labelled from x1 toxn along Pn. The comb Cbn [6] is an acyclic graph consisting of Pn together with n − 1new pendant vertices y1, y2, ..., yn−1 adjacent to x2, x3, ..., xn respectively, thus the newedges obtained are xı+1yı : ı ∈ 1, n− 1. A generalized comb [13] is basically adetailing (or subdivision) of a comb’s pendant vertices hanging from the main horizontalpath to form τ hanging paths of order `ı, this is denoted by Cbτ (`1, `2, . . . , `τ ). When`1 = `2 = . . . = `τ = `, then a generalized comb transforms into a balanced generalizedcomb Cbτ (`, `, . . . , `︸︷︷︸

τ−times

), which can precisely be denoted as Cbτ (`, `, . . . , `), as elaborated

in fig. 1.

The following Lemma is an elementary tool for proving graphs to be SEMT. It will beused as a base in each result presented in this work.

Lemma 2.1. [11] A (ν, ε)-graph G is SEMT if and only if ∃ a bijective map Γ : V (G) →1, ν s.t. the set of edge-sums

S = Γ(l) + Γ(m) : lm ∈ E(G)constructs ε consecutive Z+. In that case, G can extend to a SEMT labeling of G withmagic constant c = ν + ε+min(S) and

S = c− (ν + ε), c− (ν + ε) + 1, . . . , c− (ν + 1).

To understand the lemma 2.1, we consider an example, see fig. 2, where it is shown thatif a graph constitutes consecutive edge-sums then its super edge-magicness is assured.


xxx

xxxx

xxxx

xxxxxxx

xxxxxxxxxxxx

xxxxxxxxxxxx

(i) (ii)

FIGURE 2. (i) A Bistar BS(5, 4) with consecutive edge-sums,(ii) A SEMT Bistar BS(5, 4) with magic constant c = 28

It can be seen easily that the following result about SEMT graphs also holds i.e.,

Note. [4] Let c(Γ) be a magic constant of a SEMT labeling Γ of G(V,E), then we end upon this statement:

ε c(Γ) =∑v∈V

degG(v)Γ(v) +∑p∈E

Γ(p), ε = |E(G)| (2. 1)

For a single graph, many SEMT labelings might exist and of-course for a different la-beling, there will be a different magic constant. Hereafter, we are going to find the boundsfor magic constants of SEMT labelings of the generalized comb.

We can see that, Cbτ (`1, `2, . . . , `τ ), `1 = `2 = . . . = `τ ; τ ≥ 2 has τ` + 1 verticesand τ` edges. From these vertices, τ − 1 vertices have degree 3, ε + 1 − 2τ verticeshave degree 2, and the remaining τ + 1 vertices have degree 1, see fig 1. Consider thatCbτ (`1, `2, . . . , `τ ) has an EMT labeling with magic constant “c”, then εc where ε = τ`,can not be less than the sum we achieve when we allocate the degree-3 vertices with lowestτ − 1 labels, the ε+ 1− 2τ next lowest labels to degree-2 vertices, and τ + 1 next lowestlabels to degree-1 vertices; in other words:

ε c ≥ 3τ−1∑ı=1

ı+ 2ε−τ∑ı=τ

ı+ε+1∑

ı=ε−τ+1

ı+2ε+1∑ı=ε+2

ı

We can get the upper bound for εc by assigning highest τ−1 labels to vertices of degree3, ε − 2τ + 1 next highest labels to vertices of degree 2, and τ + 1 next highest labels tovertices of degree 1, in other words:

ε c ≤ 32ε+1∑

ı=2ε−τ+3

ı+ 22ε−τ+2∑ı=ε+τ+2

ı+ε+τ+1∑ı=ε+1

ı+ε∑ı=1

ı


Consequently, we end up with the following result;

Lemma 2.2. If Cbτ (`1, `2, . . . , `τ ); τ ≥ 2 is EMT graph, then magic constant “ c” is inthe following interval:

1

2ε(5ε2 + 2τ2 − 2ετ + 7ε− 2τ + 2) ≤ c ≤ 1

2ε(7ε2 − 2τ2 + 2ετ + 5ε+ 2τ − 2); ε = τ`

By similar process, we can get following result for SEMT graphs:

Lemma 2.3. If Cbτ (`1, `2, . . . , `τ ); τ ≥ 2 is SEMT graph, then magic constant “ c” is inthe following interval:

1

2ε(5ε2 + 2τ2 − 2ετ + 7ε− 2τ + 2) ≤ c ≤ 1

2ε(5ε2 − 2τ2 + 2ετ + 7ε+ 2τ − 2); ε = τ`

3. SEMT STRENGTH OF GENERALIZED COMB

From SEMT labeling for a generalized comb Cbτ (`1, `2, . . . , `τ ), `1 = `2 = . . . = `τ ;τ ≥ 2, [13], we have magic constant c = 2τ`+ d τ`2 e+ 3 and by the given lower bound ofmagic constants in Lemma 2.3, we have:

Theorem 3.1. The SEMT strength for generalized combG ∼= Cbτ (`, `, . . . , `), τ ≥ 2 is, for ε = τ`:

5ε2 + 2τ2 − 2ετ + 7ε− 2τ + 2

2ε≤ sm(G) ≤ 2ε+

⌈ε2

⌉+ 3

4. SEMT LABELING AND DEFICIENCY OF FORESTS FORMED BY GENERALIZEDCOMB AND STAR, GENERALIZED COMB AND BISTAR

In this section, it is shown that the two forests, made up of two components i.e., general-ized comb and star, generalized comb and bistar, are SEMT. The first result of this sectioncan be concluded as follows:

Theorem 4.1. For ` ≥ 2, τ ≥ 2(a): Cbτ (`, `, . . . , `) ∪K1,$ is SEMT.(b): µs(Cbτ (`, `, . . . , `) ∪K1,$−1) ≤ 1; (τ, `) 6= (2, 2),where $ ≥ 1 and is given by $ = b τ`−12 c.

Proof. (a): Consider the graph G ∼= Cbτ (`, `, . . . , `) ∪K1,$.Here V (K1,$) = yp; 1 ≤ p ≤ $ + 1 and E(K1,$) = y1yp; 2 ≤ p ≤ $ + 1.Let ν = |V (G)| and ε = |E(G)|, so ν = τ`+$ + 2 and ε = τ`+$.Valuation Γ : V (Cbτ (`, `, . . . , `))→ 1, `τ + 1 is described as follows:

Γ(xı,) =

ı+12 + `(−1)

2 ; ı, ≡ 1(mod 2)`2 −

ı2 + 1 ; ı, ≡ 0(mod 2)

Now consider the labeling Ω : V (G)→ 1, ν.For 1 ≤ p ≤ $ + 1

Ω(yp) =

d τ`2 e+ 1 ; p = 1τ`+ p ; p 6= 1


FIGURE 3. SEMT labeling of Cb5(5, 5, 5, 5, 5) ∪K1,12

Let A = d τ`2 e+ 1 and B = τ`+$ + 1, then

Γ(xı,) =

A+ `(−1)

2 + ı2 ; ı ≡ 0(mod 2), ≡ 1(mod 2)

A+ `2 −

ı−12 ; ı ≡ 1(mod 2), ≡ 0(mod 2)

Γ(x1,0) = B + 1 = τ`+$ + 2

Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ.The edge-sums of G induced by the above labeling Ω form consecutive integers starting

from ~ + 1 and ending on ~ + ε, where ~ = d τ`2 e+ 2. Hence from Lemma 2.1, we end upon a SEMT graph with c = d τ`2 e+ 2τ`+ 2$ + 5.

(b): Let G ∼= Cbτ (`, `, . . . , `) ∪K1,$−1 ∪K1; (τ, `) 6= (2, 2)so,

V (G) = V (Cbτ (`, `, . . . , `)) ∪ V (K1,$−1) ∪ zV (K1,$−1) = yp; 1 ≤ p ≤ $E(K1,$−1) = y1yp; 2 ≤ p ≤ $

Let ν = |V (G)| = τ` + $ + 2 and ε = |E(G)| = τ` + $ − 1. Keeping in mind thevaluation Γ defined in (a), we describe the labeling Ω : V (G)→ 1, ν as

Ω(x1,0) = τ`+$ + 2

Ω(z) = τ`+$ + 1

Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ.The edge-sums of G induced by the above labeling Ω form consecutive integers starting

from ~ + 1 and ending on ~ + ε, where ~ = d τ`2 e+ 2. Hence from Lemma 2.1, we end upon a SEMT graph with c = ν + ε+ ~ + 1.

In the formulation of next results, we will use the labeling Γ provided in previous theo-rem 4.1.


Theorem 4.2. For `, τ ≥ 2;ω,$ ≥ 1,(a): Cbτ (`, `, . . . , `) ∪BS(ω,$) is SEMT, (`, τ) 6= (2, 2).(b): µs(Cbτ (`, `, . . . , `) ∪BS(ω,$ − 1)) ≤ 1; (`, τ) /∈ (3, 2), (2, 3) and $ ≥ 2.Where $ is given by $ = b τ`−32 c.

Proof. (a): Consider the graph G ∼= Cbτ (`, `, . . . , `) ∪ BS(ω,$); `, τ ≥ 2, ω,$ ≥ 1.V (BS(ω,$)) = zut : u = 1, 2; 0 ≤ t ≤ ρ, where

ρ =

ω ;u = 1$ ;u = 2

and E(BS(ω,$)) = z10z1`; 1 ≤ ` ≤ ω ∪ z10z20 ∪ z20z2m; 1 ≤ m ≤ $.Let ν = |V (G)| and ε = |E(G)|, so we get ν = τ`+ω+$+ 3 and ε = τ`+ω+$+ 1.Keeping in mind the valuation Γ defined in Theorem 4.1 with A = d τ`2 e + ω + 1 andB = τ`+ ω +$ + 2, we describe the labeling Ω : V (G)→ 1, ν as

Ω(zut) =

d τ`2 e+ t ;u = 1, t = r, 1 ≤ r ≤ ωd τ`2 e+ ω + 1 ;u = 2, t = 0τ`+ ω + 2 ;u = 1, t = 0τ`+ ω + 2 + t ;u = 2, t = r, 1 ≤ r ≤ $

Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ

Ω(x1,0) = B + 1 = τ`+ ω +$ + 3.

The edge-sums of G induced by the above labeling Ω form consecutive integers startingfrom ~ + 1 and ending on ~ + ε, where ~ = d τ`2 e + ω + 2. Hence from Lemma 2.1, weend up on a SEMT graph.

(b): Let G ∼= Cbτ (`, `, . . . , `) ∪ BS(ω,$ − 1) ∪ K1; `, τ ≥ 2, $ ≥ 2. HereV (G) = V (Cbτ (`, `, . . . , `)) ∪ V (BS(ω,$ − 1)) ∪ z and where V (BS(ω,$ − 1)) =zut : u = 1, 2; 0 ≤ t ≤ ρ, where

ρ =

ω ;u = 1$ − 1 ;u = 2

and E(BS(ω,$− 1)) = z10z1t; 1 ≤ t ≤ ω∪ z10z20∪ z20z2t; 1 ≤ t ≤ $ − 1. Letν = |V (G)| and ε = |E(G)|, so we get ν = τ`+ω+$+ 3 and ε = τ`+ω+$. Keepingin mind the valuation Γ defined in Theorem 4.1 with A and B the same as in part (a), wedescribe the labeling Ω : V (G)→ 1, ν as

Ω(x1,0) = B + 2, Ω(z) = B + 1

Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ.

The edge-sums of G induced by the above labeling Ω form consecutive integers startingfrom ~ + 1 and ending on ~ + ε, where ~ = d τ`2 e + ω + 2 = ~. Hence from Lemma 2.1,we end up on a SEMT graph.


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

2324

25

26

27

28

2930

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56 57

58

59

60

61 62

6364

65

6667

6869

70

71

72

7374

75

7677

7879 80

81

82

83

84

85

86

87

88

89909192

93

94

95

96

97

9899

100

101

102

104 105103

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

145

144

146

147

148

FIGURE 4. SEMT labeling of Cb7(6, 6, 6, 6, 6, 6, 6) ∪BS(11, 19)

5. SEMT FORESTS FORMED BY GENERALIZED COMB AND COMB, GENERALIZEDCOMB AND PATH

The motivation of this section is to continue the work of exploring forests with twocomponents that are SEMT. In the previous section, we have determined two forests thatwere SEMT and also provided the situations for their SEMT deficiencies. The next tworesults of this section give us SEMT labeling for the disjoint union of the generalized combsCbτ (`, `, . . . , `) with comb Cbω and path P$ respectively.

Theorem 5.1. For τ, ` ≥ 2;ω ≥ 1(a): Cbτ (`, `, . . . , `) ∪ Cbω is SEMT.(b): µs(Cbτ (`, `, . . . , `) ∪ Cbω−1) ≤ 1; ω ≥ 2, (`, τ) 6= (2, 2), where

ω =

⌊τ`− 1

2

⌋.

Proof. (a): Consider the graph G ∼= Cbτ (`, `, . . . , `) ∪ Cbω , whereV (Cbω) = xp; 0 ≤ p ≤ ω ∪ yq; 1 ≤ q ≤ ω,E(Cbω) = xpxp+1; 0 ≤ p ≤ ω − 1 ∪ xpyp; 1 ≤ p ≤ ω.Let ν = |V (G)| and ε = |E(G)|, so we get ν = τ`+ 2ω+ 2 and ε = τ`+ 2ω. Keeping inmind the valuation Γ defined in Theorem 4.1 withA = d τ`2 e+ω+1 andB = τ`+2ω+1,


FIGURE 5. SEMT labeling of Cb6(4, 4, 4, 4, 4, 4) ∪ Cb11

we describe the labeling Ω : V (G)→ 1, ν as

For 0 ≤ p ≤ ω, 1 ≤ q ≤ ω,

Ω(xp) =

d τ`2 e+ p+ 1 ; p is evenτ`+ ω + 1 + p ; p is odd

and

Ω(yq) =

τ`+ ω + 1 + q ; q is evend τ`2 e+ q + 1 ; q is odd

Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τand

Ω(x1,0) = B + 1.


(b): Let G ∼= Cbτ (`, `, . . . , `) ∪ Cbω−1 ∪ K1, where V (K1) = z, V (Cbω−1) =xp; 0 ≤ p ≤ ω − 1 ∪ yq; 1 ≤ q ≤ ω − 1, E(Cbω−1) = xpxp+1; 0 ≤ p ≤ ω − 2 ∪xpyp; 1 ≤ p ≤ ω − 1.

Let ν = |V (G)| and ε = |E(G)|, so we get ν = τ`+2ω+1 and ε = τ`+2ω−2. Keep-ing in mind the valuation Γ defined in Theorem 4.1 withA = d τ`2 e+ω andB = τ`+2ω−1,we describe the labeling Ω : V (G)→ 1, ν as

For 0 ≤ p ≤ ω − 1, 1 ≤ q ≤ ω − 1

Ω(xp) =

d τ`2 e+ p+ 1 = Ω(xp) ; p is evenτ`+ ω + p = Ω(xp)− 1 ; p is odd


also

Ω(yq) =

τ`+ ω + q = Ω(yq)− 1 ; q is evend τ`2 e+ q + 1 = Ω(yq) ; q is odd

Ω(xı,) = Γ(xı,), 1 ≤ ı ≤ `, 0 ≤ ≤ τ

Ω(x1,0) = B + 2, Ω(z) = B + 1.


Theorem 5.2. For τ ≥ 2, ` ≥ 2(a)(i): Cbτ (`, `, . . . , `) ∪ P$ is SEMT.(a)(ii): Cbτ (`, `, . . . , `) ∪ P$−1 is SEMT.(b)(i): µs(Cbτ (`, `, . . . , `) ∪ P$−2) ≤ 1; (`, τ) 6= (2, 2).(b)(ii): µs(Cbτ (`, `, . . . , `) ∪ P$−3) ≤ 1; (`, τ) 6= (2, 2). where

$ =

τ` ; τ, ` ≡ 1(mod 2)τ`− 1 ; otherwise

Proof. (a): Consider the graphG ∼= Cbτ (`, `, . . . , `)∪Pt, where V (Pt) = xp; 1 ≤ p ≤ tand E(Pt) = xpxp+1; 1 ≤ p ≤ t− 1. Let ν = |V (G)| and ε = |E(G)|, so we getν = τ`+ t+ 1 and ε = τ`+ t− 1, where

t =

$ ; for(a(i))$ − 1 ; for(a(ii))

Keeping in mind the valuation Γ defined in Theorem 4.1 with

A =

2d τ`2 e ; for(a(i))d τ`2 e+ b τ`−12 c ; for(a(ii))

and for a(i)

B =

2τ` ; ` is odd, τ is odd2τ`− 1 ; otherwise

and for a(ii)

B =

2τ`− 1 ; ` is odd, τ is odd2τ`− 2 ; otherwise

We describe the labeling Ω : V (G)→ 1, ν as

Ω(xp) =

d τ`2 e+ r ; p = 2r − 1; 1 ≤ r ≤ b t+1

2 cτ`+ d τ`2 e+ t ; p = 2t, 1 ≤ t ≤ b t2c, for(a(i))τ`+ d τ`2 e+ t− 1 ; p = 2t, 1 ≤ t ≤ b t2c, for(a(ii))

furthermore,Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ

Ω(x1,0) = B + 1.

The edge-sums of G induced by above labeling Ω form consecutive integersFor (a)(i) ~ + 1, ~ + ε


FIGURE 6. SEMT labeling of Cb7(5, 5, 5, 5, 5, 5, 5) ∪ P35

For (a)(ii) ~, ~ + ε− 1where

~ =

τ`+ 2 ; τ is odd, ` is oddτ`+ 1 ; otherwise

Hence from Lemma 2.1, we end up on a SEMT graph.(b): Let G ∼= Cbτ (`, `, . . . , `) ∪ Pt ∪K1, where V (Pt) = xp; 1 ≤ p ≤ t, V (K1) =

z and E(Pt) = xpxp+1; 1 ≤ p ≤ t− 1. Let ν = |V (G)| and ε = |E(G)|, so we getν = τ`+ t+ 2 and ε = τ`+ t− 1, where

t =

$ − 2 ; for(b(i))$ − 3 ; for(b(ii))

Keeping in mind the valuation Γ defined in Theorem 4.1 with

A =

d τ`2 e+ b τ`−12 c ; for(b(i))d τ`2 e+ b τ`−32 c ; for(b(ii))

and for b(i)

B =

2τ`− 2 ; τ is odd, ` is odd2τ`− 3 ; otherwise

and for b(ii)

B =

2τ`− 3 ; τ is odd, ` is odd2τ`− 4 ; otherwise

We describe the labeling Ω : V (G)→ 1, ν as


Ω(xp) =

d τ`2 e+ r ; p = 2r − 1; 1 ≤ r ≤ b t+1

2 cτ`− 1 + d τ`2 e+ t ; p = 2t, 1 ≤ t ≤ b t2c, for(b(i))τ`− 2 + d τ`2 e+ t ; p = 2t, 1 ≤ t ≤ b t2c, for(b(ii))

Furthermore,Ω(xı,) = Γ(xı,); 1 ≤ ı ≤ `, 0 ≤ ≤ τ

Ω(x1,0) = B + 2, Ω(z) = B + 1.

The edge-sums of G induced by above labeling Ω form consecutive integersFor (b)(i) ~ + 1, ~ + εFor (b)(ii) ~, ~ + ε− 1where

~ =

τ`+ 1 ; τ is odd, ` is oddτ` ; otherwise

Hence from Lemma 2.1, we end up on a SEMT graph.

REFERENCES

[1] S. Ahmad, On the Evaluation of a Subdivision of the Ladder Graph, Punjab Univ. j. math. 47, No. 1 (2015)15-19.

[2] U. Ali, M. Bilal, S. Zafar and Z. Zahid, Some Families of Convex Polytopes Labeled by 3-Total Edge ProductCordial Labeling, Punjab Univ. j. math. 49, No. 3 (2017) 119-132.

[3] A. Ali, M. Javaid and M. A. Rehman, SEMT Labeling on Disjoint Union of Subdivided Stars, Punjab Univ.j. math. 48, No. 1 (2016) 111-122.

[4] S. Avadayappan, P. Jeyanthi and R. Vasuki, Super magic strength of a graph, Indian J. Pure Appl. Math. 32,No. 11 (2001) 1621-1630.

[5] J. B. Cosgrave and K. Dilcher, Gauss Factorials, Jacobi Primes, and Generalized Fermat Numbers, PunjabUniv. j. math. 50, No. 4 (2018) 1-21.

[6] A. Q. Baig, A. Ahmad, E. T. Baskoro and R. Simanjuntak, On the super edge-magic deficiency of forests,Util. Math. 86, (2011) 147-159.

[7] H. Enomoto, A. S. Llado, T. Nakamigawa and G. Ringel, Super edge-magic graphs, SUT J. Math., 34, No.2 (1998) 105-109.

[8] R. M. Figueroa-Centeno, R. Ichishima and F. A. Muntaner-Batle, On the super edge-magic deficiency ofgraphs, Ars Combin. 78, (2006) 33-45.

[9] R. M. Figueroa-Centeno, R. Ichishima and F. A. Muntaner-Batle, On the super edge-magic deficiency ofgraphs, Electron. Notes Discrete Math. 11, (2002) 299-314.

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

[11] R. M. Figueroa-Centeno, R. Ichishima and F. A. Muntaner-Batle, The place of super edge-magic labelingamong other classes of labeling, Discrete Math. 231, (2001) 153-168.

[12] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 19th edition, Dec. (2016) DS6.[13] S. Javed, A. Riasat and S. Kanwal, On super edge magicness and deficiencies of forests, Utilitas Math. 98,

(2015) 149-169.[14] S. Kanwal, A. Azam and Z. Iftikhar, SEMT labelings and deficiencies of forests with two components (II),

to appear in Punjab Univ. j. math.[15] S. Kanwal and I. Kanwal, SEMT Valuations of Disjoint Union of Combs, Stars and Banana Trees, Punjab

Univ. j. math. 50, No. 3 (2018) 131-144.[16] A. Khalid. A, G. Sana, M. Khidmat and A. Q. Baig, Super Edge-Magic Deficiency of Disjoint Union of

Shrub Tree, Star and Path Graphs, Punjab Univ. j. math. 47, No. 2 (2015) 1-10.[17] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13, (1970) 451-461.


[18] M. K. Mahmood and S. Ali, A Novel Labeling Algorithm on Several Classes of Graphs, Punjab Univ. j.math. 49, No. 2 (2017) 23-35.

[19] M. H. Mateen and M. K. Mahmood, Power Digraphs Associated with the Congruence xn = y(modm),Punjab Univ. j. math. 51, No. 5 (2019) 93-102.

[20] F. Malik, N. Saeed, S. Zafar and Z. Zahid, Extremal Prism like Graphs with Respect to the F-Index, PunjabUniv. j. math. 50, No. 4 (2018) 31-37.

[21] A. A. Ngurah, E. T. Baskoro and R. Simanjuntak, On the super edge-magic deficiencies of graphs, Australas.J. Combin. 40, (2008) 3-14.

[22] D. B. West, Introduction to Graph Theory, second edition, Prentice-Hall, (2001).

semt labelings and deﬁciencies of forests with two...

Documents