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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar

Salma Kanwal
/ Ayesha Riasat
• Department of Basic Sciences and Humanities, University of Engineering and Technology, KSK Campus Lahore, Pakistan
• Email
• Other articles by this author:
/ Mariam Imtiaz
• Department of Basic Sciences and Humanities, University of Engineering and Technology, KSK Campus Lahore, Pakistan
• Email
• Other articles by this author:
/ Zurdat Iftikhar
/ Sana Javed
/ Rehana Ashraf
Published Online: 2018-11-15 | DOI: https://doi.org/10.1515/math-2018-0111

## Abstract

A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υνE(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as ${\mu }_{s}\left(\mathrm{\wp }\right)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.

Keywords: SEMT labeling; SEMT deficiency; SEMT strength; Fork; H-tree

MSC 2010: 05C78

## 1 Preliminaries

All graphs examined here are finite, simple, planar and undirected. The graph ℘ has vertex-set V(℘) and edge-set E(℘) . Let p =|V (℘) | and q =| E(℘) | . A bijection ϒ :V (℘)∪E(℘)→{1, 2,, p + q} is called an EMT labeling of a graph ℘ if ϒ(υ) + ϒ(υν) + ϒ(ν) = a, where “a” is the constant called the magic constant of ℘. The graph that satisfies such a labeling is said to be an EMT graph. An EMT labeling ϒ is called a SEMT labeling if ϒ(V (℘)) = {1, 2,, p}. A graph that admits this type of labeling is called a SEMT graph. Kotzig and Rosa [1] and Enomoto et al. [2] were the first to introduce the concepts of EMT and SEMT graphs- Wallis [3] called this labeling a strong EMT labeling- respectively and conjectured that every tree is EMT [1], and every tree is SEMT [2]. These conjectures have become very prominent in the area of graph labeling. Many classes of trees have been verified to admit (super) EMT labelings, such as trees with upto 17 vertices by a computer search [4], stars [5], [6], paths, caterpillars [1] and subdivided stars [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], etc. However, in general, these conjectures are still open.

The (super) EMT strength of a graph ℘, denoted by ( sm(℘) ) m(℘), is defined as the minimum of all magic constants a(ϒ), where the minimum is taken over all the (super) EMT labelings of ℘. This minimum is defined only if the graph has at least one such (super) EMT labeling. One can easily perceive that, since the labels of graph ℘(V, E) are from the set {1, 2,, p + q},

$p+q+3≤sm(℘)≤3p.$

Avadayappan et al. first introduced the notions of EMT strength [17] and SEMT strength [18] and found EMT strength for path, cycle etc., and also the exact values of SEMT strength for some graphs. In [19], [20], [21], the SEMT strengths of fire crackers, banana trees, unicyclic graphs, paths, stars, bistars, y -trees and the generalized Petersen graphs have been observed.

Kotzig and Rosa [1] verified that for any graph ℘, ∃ an EMT graph χ s.t. χ ≅ ℘∪nK1 for some non-negative integer n . This fact leads to the concept of EMT deficiency of a graph ℘, μ(℘), which is the minimum non-negative integer n s.t. ℘∪nK1 is EMT. In particular,

$μ(℘)=min{n≥0:℘∪nK1isEMT.}$

In the same paper [1], Kotzig and Rosa gave the upper bound for the EMT deficiency of a graph ℘ with n vertices i.e.,

$μ(℘)≤Fn+2−2−n−n(n−1)2$

where Fn is the nth Fibonacci number. Figueroa-Centeno et al. [22] defined a similar concept for SEMT labeling i.e., the SEMT deficiency of a graph ℘, denoted by ${\mu }_{s}\left(\mathrm{\wp }\right)$, is the minimum non-negative integer n s.t. ℘∪nK1 has a SEMT labeling, or +∞ if there is no such n, more precisely,

If M(℘) = {n ≥ 0 :℘∪nK1 is a SEMT graph}, then

$μs(℘)=minM(℘)ifM(℘)≠ϕ+∞ ifM(℘)=ϕ$

It can be seen easily that for every graph ℘, μ(℘) ≤ μs(℘) . In [22], [23], Figueroa-Centeno et al. provided the exact values of SEMT deficiencies of several classes of graphs. They also proved that all forests have finite deficiencies. Ngurah et al. [24], Baig et al. [25] and Javed et al. [26] gave some upper bounds for the SEMT deficiency of various forests. In [27], Figueroa-Centeno et al. conjectured that every forest with two components has SEMT deficiency at most 1. The examination of deficiencies in this paper will put evidence on this conjecture. However, this conjecture is still open too.

In this paper, we established the results on SEMT labelings and deficiencies of fork, H -tree and disjoint union of fork with star, bistar and path. Also the SEMT strengths of fork and H -tree are discussed. A useful survey to know about the numerous graph labeling methods is the one by J. A Gallian [28] and for all graph-theoretic terminologies and notions we refer the reader to [29], [30],

## 2 The results

A star on n vertices is isomorphic to complete graph K1,n−1 . A bistar BS(υ,ν) on n vertices is obtained from two stars K1,υ and K1,ν by joining their central vertices through an edge, where υ,ν ≥1, υ+ν = n − 2. A path denoted by Pn is a graph consisting of n vertices and n −1 edges. The subdivided star T(n1, n2, , nρ) is a tree obtained by inserting ni − 1 vertices to each of the ith edge of the star ${K}_{1,\rho }$, where 1≤ iρ, ni ≥ 1 and ρ ≥ 3 . The vertex-set and edge-set are defined as

$V(T(n1,n2,…,nρ))={k}∪{xıℓı:1≤ı≤ρ;1≤ℓı≤nı}$

and

$E(T(n1,n2,…,nρ))={kxı1:1≤ı≤ρ}∪{xıℓıxıℓı+1:1≤ı≤ρ;1≤ℓı≤nı−1}$

respectively. Moreover, $\mathrm{\forall }{n}_{ı}=1,T\underset{\rho -times}{\underset{⏟}{\left(1,1,\dots ,1\right)}}\cong {K}_{1,\rho }.$

Definition 2.1 A fork, denoted by Fr, ℓN \{1}, is a tree deduced from 3 equally sized paths of length ℓ that is P : x1,j, x2,j, x3,j ,1 ≤ jℓ, a single new vertex x2,0 is added to the path x2,j ; 1 ≤ jℓ through an edge, these three paths are joined together by two edges that are xi,1 xi+1,1, 1≤ i ≤ 2. Precisely, the set of vertices and the set of edges of fork are as respectively:

$V(Frℓ)={xı,ȷ:1≤ı≤3,1≤ȷ≤ℓ}∪{x2,0}$

$E(Frℓ)={xı,ȷxı,ȷ+1:1≤ı≤3,1≤ȷ≤ℓ−1}∪{xı,1xı+1,1:1≤ı≤2}∪{x2,0x2,1},$

illustrated in Figure 1.

Fig 1

Fork $F{r}_{6}$

Definition 2.2 H-tree is represented as H , ℓN consisting of four equally sized paths joined together by two new vertices forming alphabet H shape, illustrated in Figure 2. The vertex and edge sets of H -tree are as respectively:

Fig 2

H -tree H4

$V(℘)={xı,ȷ:ı=1,2,1≤ȷ≤2ℓ+1}$

$E(℘)={xı,ȷxı,ȷ+1:1≤ı≤2,1≤ȷ≤2ℓ}∪{x1,ℓ+1x2,ℓ+1}.$

Note 1. Fork Fr can also be written as T(1,, −1,) where N \{1}, as we can see that it is basically a subdivision of star K1,4 . Javed, Hussain, Ali and Shaker [8] have discussed the SEMT labelings on subdivisions of star K1,4 but the advantage of SEMT labeling scheme presented in this paper over the previous ones mentioned in [8] is that it holds for all positive integers >1, not only for odd positive integers. H -tree can be taken as a subdivision of bistar BS(2, 2) and this subdivision is carried out for all positive integers but the point to remember is that all the four legs of H should be equal in order.

The following lemma gives us a necessary and sufficient condition for a graph to be SEMT and in proving the main results, we will frequently use this. Conditions given in this Lemma are easier to work with than the original definition.

#### Lemma 2.3 ([6])

A ( p, q) -graphis SEMT if and only if there exists a bijective function ϒ :V (℘)→{1, 2,, p} such that the set

$S={Υ(υ)+Υ(ν):υν∈E(℘)}$

consists of q consecutive integers. In such a case, ℘ extends to a SEMT labeling of ℘ with the magic constant a = p + q + min(S), where

$S={a−(p+q),a−(p+q)+1,…,a−(p+1)}.$

Note 2. ([18])., Let ϒ be a SEMT labeling of ℘ with the magic constant a(ϒ).

Then, adding all the magic constants obtained at each edge, we get

$qa(Υ)=∑ν∈V(℘)deg℘(ν)Υ(ν)+∑e∈E(℘)Υ(e),q=|E(℘)|$(1)

This condition holds also for EMT labelings. The term deg(ν) in above expression is the degree of vertex νV (℘), which can be defined as the number of vertices that are adjacent to ν, form a set denoted by N(ν), and deg(ν) =| N(ν)| is the degree of ν in ℘.

There may exist a variety of SEMT labeling schemes for a single graph- if any graph admits a SEMT labeling then another distinct SEMT labeling will surely exist for the same graph because of the dual super labeling detailed in [31]- and of course there will be as many different magic constants as the distint labeling schemes. Many researchers have found the lower and upper bounds of magic constants for various graphs. In [7], Ngurah et al. obtained lower and upper bounds of the SEMT magic constants for subdivision of star K1,3 i.e.,

#### Lemma 2.4 ([7])

If T(m, n, k) is a SEMT graph, then magic constant “a” is in the following interval: $\frac{1}{2t}\left(5{t}^{2}+3t+6\right)\le a\le \frac{1}{2t}\left(5{t}^{2}+11t-6\right)$ where t = m+ n + k.

Javaid [32] gave upper and lower bounds of SEMT magic constants for subdivided stars T(n 1, n2, , nr) with any ni ≥ 1, 1≤ ir, in the form of following lemma:

#### Lemma 2.5 ([32])

If T(n1, n2, , nr) is a super (a,0) -EAT graph, then

$\frac{1}{2l}\left(5{l}^{2}+{r}^{2}-2lr+9l-r\right)\le a\le \frac{1}{2l}\left(5{l}^{2}-{r}^{2}+2lr+5l+r\right),where\phantom{\rule{thinmathspace}{0ex}}l=\sum _{ı=1}^{r}{n}_{ı}.$

Now we find the upper and lower bounds of magic constants for H -tree. Clearly, H -tree H ; ≥1 has 4 + 2 vertices and 4 +1 edges. Among these vertices, two vertices have degree 3, four vertices have degree 1, and the remaining vertices have degree 2, see fig 2. Suppose H has an EMT labeling with magic constant “a”, then qa where q = 4 +1, can not be smaller than the sum obtained by assigning the smallest two labels to the vertices of degree 3, the q − 5 next smallest labels to the vertices of degree 2, and four next smallest labels to the vertices of degree 1; in other words:

$qa≥3∑ı=12ı+2∑ı=3q−3ı+∑ı=q−2q+1ı+∑ı=q+22q+1ı=18+2q(q−5)+4(2q−1)+3q(q+1)2=5q2+q+142$

An upper bound for qa can be achieved by giving the largest labels to the vertices of degree 3, and the q − 5 next largest labels to the vertices of degree 2, and four next largest labels to the vertices of degree 1, in other words:

$qa≤3∑ı=2q2q+1ı+2∑ı=q+52q−1ı+∑ı=q+1q+4ı+∑ı=1qı=6(4q+1)+2(3q+4)(q−5)+4(2q+5)+q(q+1)2=7q2+11q−142$

Thus, we have the following result,

#### Lemma 2.6

If H is an EMT graph, then magic constant “a” is in the following interval:

$12q(5q2+q+14)≤a≤12q(7q2+11q−14)$

By a similar argument, it is easy to verify that the following lemma holds.

#### Lemma 2.7

If H is a SEMT graph, then magic constant “a” is in the following interval:

$12q(5q2+q+14)≤a≤12q(5q2+13q−14)$

In the next results of this section, we will construct the SEMT labeling and strength for Fork and H -tree.

#### Theorem 2.8

For ℓ ≥ 2 , the graph ℘≅ Fr is SEMT with magic constant $a=6\ell +⌈\frac{3\ell }{2}⌉+4$.

Proof. Let ℘≅ Fr, ≥ 2, where

$V(℘)={xı,ȷ:1≤ı≤3,1≤ȷ≤ℓ}∪{x2,0}$

$E(℘)={xı,ȷxı,ȷ+1:1≤ı≤3,1≤ȷ≤ℓ−1}∪{xı,1xı+1,1:1≤ı≤2}∪{x2,0x2,1}$

Let p =|V (℘) | and q =| (E(℘) |, then p = 3 +1 and q = 3

Consider the vertex labeling ϒ : V(℘)→{1, 2,, p} as follows:

$Υ(x2,0)=ℓ+1$

$Υ(xı,ȷ)=1+(ℓ+1)(ı−12)+(ȷ−12);ı≡1(mod2),ı=1,3ȷ≡1(mod2),ȷ≥1ℓ−ȷ−22;ı≡0(mod2),ı=2ȷ≡0(mod2),ȷ≥2⌈3ℓ2⌉+ℓ(ı−12)+ȷ−22+2;ı≡1(mod2),ı=1,3ȷ≡0(mod2),ȷ≥2⌈3ℓ2⌉+ℓ−ȷ−12+1;ı≡0(mod2),ı=2ȷ≡1(mod2),ȷ≥1$

The edge-sums generated by the above labeling "ϒ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where $\hslash =⌊\frac{3\ell }{2}⌋+2.$ Thus by Lemma 2.3, "ϒ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + +1, where +1 = min(S).

From this theorem, we obtain the magic constant $a\left(\mathrm{Υ}\right)=6\ell +⌈\frac{3\ell }{2}⌉+4;\ell \ge 2$ for Fork tree and by given lower bound of magic constants in Lemma 2.5, we have $a\left(\mathrm{Υ}\right)\ge \frac{5{q}^{2}+q+12}{2q}$, where q = 3, thus we can conclude:

#### Theorem 2.9

The SEMT strength for Fork Fr ; ≥ 2 (subdivision of star K1,4 ) is in the following interval:

$15ℓ2+ℓ+42ℓ≤sm(Frℓ)≤6ℓ+⌈3ℓ2⌉+4,ℓ≥2.$

#### Theorem 2.10

For ℓ ≥1, the graph ℘≅ H is SEMT with magic constant a = 2(5 + 3).

Proof. Let ℘≅ H, ≥1, where

$V(℘)={xı,ȷ:ı=1,2,1≤ȷ≤2ℓ+1}$

$E(℘)={xı,ȷxı,ȷ+1:1≤ı≤2,1≤ȷ≤2ℓ}∪{x1,ℓ+1x2,ℓ+1}$

Let v =|V (℘) | and e =| E(℘) |, then p = 4 + 2 and q = 4 +1.

Consider the vertex labeling ϒ :V (℘)→{1, 2,, p} as follows:

$Υ(xı,ȷ)=1+ȷ−12;ı≡1(mod2),ı= 1ȷ≡1(mod2),ȷ≥12ℓ−ȷ−22+1;ı≡0(mod2),ı=2ȷ≡0(mod2),ȷ≥22(ℓ+1)+ȷ−22;ı≡1(mod2),ı=1ȷ≡0(mod2),ȷ≥24ℓ−ȷ−12+2;ı≡0(mod2),ı=2ȷ≡1(mod2),ȷ≥1$

The edge-sums generated by the above labeling "ϒ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where = 2( +1). Thus by Lemma 2.3, "ϒ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + +1, where +1 = min(S).

This theorem gives us the magic constant a(ϒ) = 2(5 + 3), ≥1 for H -tree and by given lower bound of magic constants in Lemma 2.7, we have $\frac{5{q}^{2}+q+14}{2q}$, where q = 4 +1. Thus we can conclude:

#### Theorem 2.11

The SEMT strength for H -tree H , ℓ ≥1 (subdivision of bistar BS(2, 2) ) is in the following interval:

$40ℓ2+22ℓ+104ℓ+1≤sm(Hℓ)≤10ℓ+6,ℓ≥1.$

In the next section, we will study the SEMT labelings and deficiencies of forests consisting of fork, star, bistar and path.

## 2.1 Semt labeling and deficiency of forests formed by fork, star, bistar and path

#### Theorem 2.12

For ℓ ≥ 2 ,

1. (a): $F{r}_{\ell }\cup {K}_{1,\varpi }$ is SEMT.

2. (b): ${\mu }_{s}\left(F{r}_{\ell }\cup {K}_{1,\varpi -1}\right)\le 1.$

where ϖ = −1.

Proof. (a): Consider the graph ℘≅ FrK1,ϖ.

Let p =|V (℘) | and q =| (E(℘) |, then

$p=3ℓ+ϖ+2$

$q=3ℓ+ϖ$

We define a labeling ϒ :V (Fr )→{1, 2, ,3 +1}, as

$Υ(xı,ȷ)=⌊ℓ2⌋+ℓ(ı−12)−ȷ−22;ı≡1(mod2),ı=1,3ȷ≡0(mod2),ȷ≥2⌊ℓ2⌋+ȷ−12+1;ı≡0(mod2),ı=2ȷ≡1(mod2),ȷ≥1$

Now consider the labeling Ψ:V (℘)→{1,2,, p}.

For 1≤ kϖ +1,

$Ψ(yk)=⌊3ℓ2⌋+1;k=13ℓ+k;k≠1$

Let $A=⌊\frac{3\ell }{2}⌋+1$

$Υ(xı,ȷ)=A+⌈ℓ2⌉+ℓ(ı−12)−ȷ−12;ı≡1(mod2),ı=1,3ȷ≡1(mod2),ȷ≥1A+⌈ℓ2⌉+ȷ−22+1;ı≡0(mod2),ı=2ȷ≡0(mod2),ȷ≥2$

$Υ(x2,0)=Ψ(x2,0)=3ℓ+ϖ+2$

$Ψ(xı,ȷ)=Υ(xı,ȷ),1≤ı≤3,1≤ȷ≤ℓ$

The edge-sums generated by the above labeling "Ψ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where $\hslash =⌊\frac{3\ell }{2}⌋+2.$ Thus by Lemma 2.3, "Ψ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + +1, where +1 = min(S).

(b): Let [origin = c]180 ≅ FrK1,ϖ−1K1 . Here

$V(℧)=V(Frℓ)∪V(K1,ϖ−1)∪{z}$

$V(K1,ϖ−1)={yk:1≤k≤ϖ}andE(K1,ϖ−1)={y1yk:2≤k≤ϖ}.$

Let p′ =| (V ([origin = c]180 )| and q′ =| E([origin = c]180 )|, then

$p′=3ℓ+ϖ+2$

and

$q′=3ℓ+ϖ−1$

Before formulating the labeling Ψ′ :V ([origin = c]180 )→{1, 2,, p′}, keep in view the labeling ϒ defined in (a). We define the labeling Ψ′ as follows:

$Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤3,1≤ȷ≤ℓ$

with A = Ψ(y1) = Ψ′(y1)

$Ψ′(yk)=Ψ(yk);1≤k≤ϖ$

$Ψ′(z)=3ℓ+ϖ+1$

$Ψ′(x2,0)=Ψ(x2,0)=Υ(x2,0)=3ℓ+ϖ+2$

The edge-sums generated by the above labeling "Ψ′" are the set of consecutive positive integers $S=\left\{\hslash +1,\hslash +2,\dots ,\hslash +q\right\},where\phantom{\rule{thinmathspace}{0ex}}\hslash =⌈\frac{3\ell }{2}⌉+2.$ Thus by Lemma 2.3, "Ψ′" can be extended to a SEMT labeling of [origin = c]180 and we obtain the magic constant a = p′ + q′ + +1, where +1 = min(S).

#### Theorem 2.13

For ℓ ≥ 2 ,

1. (a): $F{r}_{\ell }\cup BS\left(\zeta ,\xi \right)$ is SEMT.

2. (b): ${\mu }_{s}\left(F{r}_{\ell }\cup BS\left(\zeta ,\xi -1\right)\right)\le 1.$

where ξ = − 2, ζ ≥ 0.

Proof. (a): Consider the graph ℘≅ FrBS(ζ ,ξ).

Let p =|V (℘) | and q =| E(℘) |, then

$p=3ℓ+ζ+ξ+3$

$q=3ℓ+ζ+ξ+1$

Before formulating the labeling Ψ:V (℘) →{1,2,, p}, keep in view the labeling ϒ defined in theorem 2.12. We define the labeling Ψ as follows:

$Υ(xı,ȷ)=Ψ(xı,ȷ);1≤ı≤3,1≤ȷ≤ℓ$

with $A=⌊\frac{3\ell }{2}⌋+\zeta +1$

$Ψ(z♭t)=⌊3ℓ2⌋+t;♭=1,1≤t≤ζ⌊3ℓ2⌋+ζ+1;♭=2,t=03ℓ+ζ+2;♭=1,t=03ℓ+ζ+t+2;♭=2,1≤t≤ξ$

$Υ(x2,0)=Ψ(x2,0)=3ℓ+ζ+ξ+3$

The edge-sums generated by the above labeling "Ψ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where $\hslash =⌊\frac{3\ell }{2}⌋+\zeta +2.$ Thus by Lemma 2.3, "Ψ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + +1, where +1 = min(S).

(b): Let [origin = c]180 ≅ FrBS(ζ ,ξ −1) ∪ K1 . Here

$V(℧)=V(Frℓ)∪V(BS(ζ,ξ−1))∪{z}$

Let p′ =| V ([origin = c]180 )| and q′ =| E([origin = c]180 )|, then

$p′=3ℓ+ζ+ξ+3$

and

$q′=3ℓ+ζ+ξ$

Before formulating the labeling Ψ′ :V ([origin = c]180 )→{1, 2,, p′}, keep in view the labeling ϒ defined in theorem 2.12. We define the labeling Ψ′ as follows:

$Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤3,1≤ȷ≤ℓ$

with $A=\mathrm{\Psi }\left({z}_{20}\right)={\mathrm{\Psi }}^{\prime }\left({z}_{20}\right)=⌊\frac{3\ell }{2}⌋+\zeta +1$

$Ψ′(z1t)=Ψ(z1t);0≤t≤ζ$

$Ψ′(z2t)=Ψ(z2t);0≤t≤ξ−1$

$Ψ′(z)=3ℓ+ζ+ξ+2$

$Ψ′(x2,0)=Ψ(x2,0)=Υ(x2,0)=3ℓ+ζ+ξ+3$

The edge-sums generated by the above labeling "Ψ′" are the set of consecutive positive integers S = { +1, + 2,, + q′}, where $\hslash =⌊\frac{3\ell }{2}⌋+\zeta +2.$ Thus by Lemma 2.3, "Ψ′" can be extended to a SEMT labeling of [origin = c]180 and we obtain the magic constant a = p′ + q′ + +1, where +1 = min(S).

In the next two theorems, we will present two distinct SEMT labelings- which are non-dual of each other- for disjoint union of path Pm and fork.

#### Theorem 2.14

For ℓ ≥ 2

• (a)(i): FrPr is SEMT.

• (a)(ii): FrPr−1 is SEMT.

• (b)(i): μs( FrPr−2) ≤ 1.

• (b)(ii): μs( FrPr−3) ≤ 1.

where r = 2 −1.

Proof. (a): Consider the graph $\mathrm{\wp }\cong F{r}_{\ell }\cup {P}_{\varrho }$, where

$V(Pϱ)={xt:1≤t≤ϱ}$

and

$E(Pϱ)={xtxt+1:1≤t≤ϱ−1}$

Let p =|V (℘) | and q =| E(℘) |, so we get

$p=3ℓ+ϱ+1$

$q=3ℓ+ϱ−1$

where

$ϱ=r;for a(i)r−1;for a(ii)$

Before formulating the labeling Ψ:V (℘)→{1,2,, p}, keep in view the labeling ϒ defined in Theorem 2.12.

$Ψ(xı,ȷ)=Υ(xı,ȷ);1≤ı≤3,1≤ȷ≤ℓ$

with $A=⌊\frac{3\ell }{2}⌋+⌊\frac{\varrho +1}{2}⌋$ We define the labeling Ψ as follows:

$Ψ(xt)=⌊3ℓ2⌋+k;t=2k−1,1≤k≤⌊ϱ+12⌋3ℓ+⌊3ℓ2⌋+k−⌊ℓ2⌋;t=2k,1≤k≤⌊ϱ2⌋,fora(i)3ℓ+⌊3ℓ2⌋+k−⌊ℓ2⌋−1;t=2k,1≤k≤ϱ2,fora(ii)$

$Ψ(x2,0)=Υ(x2,0)=3ℓ+⌊3ℓ2⌋+⌊ϱ2⌋−⌊ℓ2⌋+1;fora(i)3ℓ+⌊3ℓ2⌋+ϱ2−⌊ℓ2⌋;fora(ii)$

The edge-sums generated by the above labeling "Ψ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where $\hslash =⌊\frac{3\ell }{2}⌋+⌊\frac{\varrho +1}{2}⌋+1.$ Thus by Lemma 2.3, "Ψ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + min(S), where min(S) = +1.

(b): Let [origin = c]180 ≅ FrPρK1, where

$V(Pϱ)={xt:1≤t≤ϱ},$

$V(K1)={z}$

and

$E(Pϱ)={xtxt+1:1≤t≤ϱ−1}$

Let p′ =| V ([origin = c]180 )| and q′ =| E([origin = c]180 )|, so we get

$p′=3ℓ+ϱ+2$

$q′=3ℓ+ϱ−1$

where

$ϱ=r−2;for b(i)r−3;for b(ii)$

Before formulating the labeling Ψ′ :V ([origin = c]180 )→{1, 2,, p′}, keep in view the labeling ϒ defined in theorem 2.12.

$\mathrm{Υ}\left({x}_{ı,ȷ}\right)=\mathrm{\Psi }\left({x}_{ı,ȷ}\right)={\mathrm{\Psi }}^{\prime }\left({x}_{ı,ȷ}\right);\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le ı\le 3,1\le ȷ\le \ell ,for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}b\left(i\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}b\left(ii\right)both$ with $A=⌊\frac{3\ell }{2}⌋+⌊\frac{\varrho +1}{2}⌋$

$Ψ′(xt)=Ψ(xt),t≡1(mod2)$

$Ψ′(xt)=3ℓ+⌊3ℓ2⌋+k−⌊ℓ2⌋−2;t=2k−1,1≤k≤ϱ−12,forb(i)3ℓ+⌊3ℓ2⌋+k−⌊ℓ2⌋−3;t=2k−1,1≤k≤⌈ϱ−12⌉,forb(ii)$

$LetB=3\ell +⌊\frac{3\ell }{2}⌋+\frac{\varrho -1}{2}-⌊\frac{\ell }{2}⌋-1\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}C=3\ell +⌊\frac{3\ell }{2}⌋+⌈\frac{\varrho -1}{2}⌉-⌊\frac{\ell }{2}⌋-2$then

$Ψ′(z)=B+1;for b(i)C+1;for b(ii)$

$Ψ′(x2,0)=B+2;for b(i)C+2;for b(ii)$

The edge-sums generated by the above labeling "Ψ′" are the set of consecutive positive integers S = { +1, + 2,, + q′}, where $\hslash =⌊\frac{3\ell }{2}⌋+⌊\frac{\varrho +1}{2}⌋+1.$ Thus by Lemma 2.3, "Ψ′" can be extended to a SEMT labeling of [origin = c]180 and we obtain the magic constant a = p′ + q′ + min(S), where min(S) = +1.

#### Theorem 2.15

For ℓ ≥ 2

• (a)(i): $F{r}_{\ell }\cup {P}_{r}$ is SEMT.

• (a)(ii): $F{r}_{\ell }\cup {P}_{r-1}$ is SEMT, ≠ 2

• (b)(i): ${\mu }_{s}\left(F{r}_{\ell }\cup {P}_{r-2}\right)\le 1,$

• (b)(ii): ${\mu }_{s}\left(F{r}_{\ell }\cup {P}_{r-3}\right)\le 1;\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\ell \ne 2,3$

where r = 2 − 2.

Proof. (a): Consider the graph $\mathrm{\wp }\cong F{r}_{\ell }\cup {P}_{\varrho }$, where

$V(Pϱ)={xt:1≤t≤ϱ}$

and

$E(Pϱ)={xtxt+1:1≤t≤ϱ−1}$

Let p =|V (℘) | and q =| E(℘) |, so we get

$p=3ℓ+ϱ+1$

$q=3ℓ+ϱ−1$

where

$ϱ=r;fora(i)r−1;fora(ii)$

Before formulating the labeling Ψ:V (℘)→{1,2,, p}, keep in view the labeling ϒ defined in theorem 2.12.

$Ψ(xı,ȷ)=Υ(xı,ȷ);1≤ı≤3,1≤ȷ≤ℓ$

with $A=⌊\frac{3\ell }{2}⌋+⌈\frac{\varrho -1}{2}⌉$. We define the labeling Ψ as follows:

$Ψ(xt)={⌊3l2⌋+k ;t=2k,1≤k≤⌈ϱ−12⌉3l+⌊3l2⌋+k−⌊l2⌋−1 ;t=2k−1,1≤k≤ϱ2,for a(i) 3l+⌊3l2⌋+k−⌊l2⌋−2 ;t=2k−1,1≤k≤⌈ϱ2⌉,for a(ii)$

$Ψx2,0=Υx2,0=3ℓ+3ℓ2+ϱ2−ℓ2;forai3ℓ+3ℓ2+ϱ2−ℓ2−1;foraii$

The edge-sums generated by the above labeling "Ψ" are the set of consecutive positive integers S = { +1, + 2,, + q}, where $\hslash =⌊\frac{3\ell }{2}⌋+⌈\frac{\varrho -1}{2}⌉+1.$ Thus by Lemma 2.3, "Ψ" can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + min(S), where min(S) = +1.

(b): Let [origin = c]180 ≅ FrPρK1, where

$V(Pϱ)={xt:1≤t≤ϱ},$

$V(K1)={z}$

and

$E(Pϱ)={xtxt+1:1≤t≤ϱ−1}$

Let p′ =| V ([origin = c]180 )| and q′ =| E([origin = c]180 )|, so we get

$p′=3ℓ+ϱ+2$

$q′=3ℓ+ϱ−1$

where

$ϱ=r−2;for b(i)r−3;for b(ii)$

Before formulating the labeling Ψ′ :V ([origin = c]180 )→{1, 2,, p′}, keep in view the labeling ϒ defined in theorem 2.12.

$\mathrm{Υ}\left({x}_{ı,ȷ}\right)=\mathrm{\Psi }\left({x}_{ı,ȷ}\right)={\mathrm{\Psi }}^{\prime }\left({x}_{ı,ȷ}\right);\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le ı\le 3,1\le ȷ\le \ell ,for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}b\left(i\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}b\left(ii\right)both$with$A=⌊\frac{3\ell }{2}⌋+⌈\frac{\varrho -1}{2}⌉$

$Ψ′(xt)=Ψ(xt),t≡0(mod2)$

$Ψ′xt=3ℓ+3ℓ2+k−ℓ2−2;t=2k−1,1≤k≤ϱ+12,forbi3ℓ+3ℓ2+k−ℓ2−3;t=2k−1,1≤k≤ϱ+12,forbii$

Let $B=3\ell +⌊\frac{3\ell }{2}⌋+⌊\frac{\varrho +1}{2}⌋-⌊\frac{\ell }{2}⌋-2\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}C=3\ell +⌊\frac{3\ell }{2}⌋+\frac{\varrho +1}{2}-⌊\frac{\ell }{2}⌋-3$, then

$Ψ′(z)=B+1;forb(i)C+1;forb(ii)$

$Ψ′(x2,0)=B+2;forb(i)C+2;forb(ii)$

The edge-sums generated by the above labeling "Ψ′" are the set of consecutive positive integers S = { +1, + 2,, + q′}, where $\hslash =⌊\frac{3\ell }{2}⌋+⌈\frac{\varrho -1}{2}⌉+1.$ Thus by Lemma 2.3, "Ψ′" can be extended to a SEMT labeling of [origin = c]180 and we obtain the magic constant a = p′ + q′ + min(S), where min(S) = +1.

## Concluding remarks

In this paper, we defined new terminologies for a particular class of subdivided stars and subdivided bistars named as fork Fr and H -tree H respectively. Furthermore, we established the results on SEMT labelings and deficiencies of fork, H -tree and disjoint union of fork with star, bistar and path.

Javaid [32] gave upper and lower bounds of SEMT magic constants for subdivided stars T(n1, n2, , nr) with any ni ≥ 1, 1≤ ir . This paper extended the key concept for evaluating the bounds for H -tree. Consequently, we ended up on the SEMT strengths of fork and H -tree. We conclude the paper with the subsequent open problems:

Open problem 1. Make SEMT forests of existing trees with newly defined trees in this manuscript.

Open problem 2. Find the SEMT labeling for disjoint union of any number of isomorphic or non-isomorphic copies of Fork tree and H -tree and determine the bounds for their deficiencies.

Open problem 3. Find the exact values of the SEMT strength for Fr and H.

Open problem 4. Find the SEMT strength of forests with more than one component, mentioned in this paper.

The authors are deeply indebted to the referees for their valuable thoughts and remarks to improve the original version of this manuscript. The research contents of this paper are partially supported by HEC (5420/Federal/NRPU/R & D/HEC/2016).

## References

• [1]

Kotzig A., Rosa A., Magic valuations of finite graphs, Canad. Math. Bull. 1970, 13, 451-461.

• [2]

Enomoto H., Lladó A.S., Nakamigawa T., Ringel G., Super edge-magic graphs, SUT J. Math. 1998, 34(2), 105-109. Google Scholar

• [3]

Wallis W.D., Magic Graphs, 2001, Birkhäuser, Boston-Basel-Berlin. Google Scholar

• [4]

Lee S.M., Shah Q.X., All trees with at most 17 vertices are super edge magic, In: 16th MCCCC Conference (Carbondale, University Southern Illinois), 2002, November. Google Scholar

• [5]

Wallis W.D., Baskoro E.T., Miller M., Slamin, Edge-magic total labelings, Australas. J. Combin. 2000, 22, 177-190. Google Scholar

• [6]

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

• [7]

Baskoro E.T., Ngurah A.A.G., Simanjuntak R., On (super) edge-magic total labeling of subdivision of K1,3, SUT J. Math. 2007, 43, 127-136. Google Scholar

• [8]

Javed M., Hassain M., Ali K., Shaker H., On super edge-magic total labeling on subdivision of trees, Util. Math. 2012, 89, 169-177. Google Scholar

• [9]

Ali K., Hussain M., Shaker H., Javed M., Super edge-magic total labeling of subdivided stars, Ars Combin. 2015, 120, 161-167. Google Scholar

• [10]

Lu Y.J., A proof of three-path trees Pmnt being edge-magic, College Mathematica 2001, 17(2), 41-44. Google Scholar

• [11]

Lu Y.J., A proof of three-path trees Pm nt being edge-magic (II), College Mathematica 2004, 20(3), 51-53. Google Scholar

• [12]

Javaid M., Bhatti A.A., On super a d -edge-antimagic total labeling of subdivided stars, Ars Combin. 2012, 105, 503-512. Google Scholar

• [13]

Ali A., Javaid M., Rehman M.A., SEMT labeling on disjoint union of subdivided stars, Punjab Uni. J. Math. 2016, 48(1), 111-122. Google Scholar

• [14]

Raheem A., Baig A.Q., Edge antimagic total labeling of isomorphic copies of subdivided stars, Int. J. Math. Soft Com. 2016, 6(1), 121-131.

• [15]

Javaid M., Bhatti A.A., Aslam M.K., Super (a,d)-edge antimagic total labeling of a subclass of trees, AKCE Int. J. Graph. Combin. 2017, 14, 158-164.

• [16]

Raheem A., Baig A.Q., Javaid M., On (a,d)-EAT labeling of subdivision of K1,r, J. Info. Opt. Sci. 2018, 39(3), 643-650. Google Scholar

• [17]

Avadayappan S., Vasuki R., Jeyanthi P., Magic Strength of a Graph, Indian J. pure appl. Math. 2000, 31(7), 873-883. Google Scholar

• [18]

Avadayappan S., Jeyanthi P., Vasuki R., Super magic strength of a graph, Indian J. pure appl. Math. 2001, 32(11), 1621-1630.

• [19]

Swamminatan V., Jeyanthi P., Super edge-magic strength of fire crackers, banana trees and unicyclic graphs, Discrete Math. 2006, 306, 1624-1636.

• [20]

Akka D.G., Warad N.S., Super magic strength of a graph, Indian J. Pure Appl. Math. 2010, 41(4), 557-568.

• [21]

Hungund N.S., Akka D.G., Super edge-magic strength of some new families of graphs, Bull. Marathwada Math. Soc. 2011, 12(1), 47-54. Google Scholar

• [22]

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

• [23]

Figueroa-Centeno R.M., Ichishima R., Muntaner-Batle F.A., On the super edge-magic deficiency of graphs, Electron. Notes Discrete Math. 2002, 11, 299-314.

• [24]

Ngurah A.A., Baskoro E.T., Simanjuntak R., On the super edge-magic deficiencies of graphs, Australas. J. Combin. 2008, 40, 3–14. Google Scholar

• [25]

Baig A.Q., Ahmad A., Baskoro E.T., Simanjuntak R., On the super edge-magic deficiency of forests, Util. Math. 2011, 86, 147-159. Google Scholar

• [26]

Javed S., Riasat A., Kanwal S., On super edge magicness and deficiencies of forests, Utilitas Math. 2015, 98, 149-169. Google Scholar

• [27]

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

• [28]

Gallian J.A., A dynamic survey of graph labeling, Electron. J. Combin. 20th edition, 2017, December, DS6.

• [29]

Baa M., Miller M., Super edge-antimagic graphs (a wealth of problems and some solutions), 2008, Brown Walker Press, Boca Raton, Florida, USA. Google Scholar

• [30]

West D.B., Introduction to Graph Theory (2nd ed.), 2001, Prentice-Hall. Google Scholar

• [31]

Baskoro E.T., Sundarsana I.W., Cholily Y.M., How to construct new super edge-magic graphs from some old ones, J. Indones. Math. Soc. (MIHMI), 2005, 11, 155-162. Google Scholar

• [32]

Javaid M., labeling Graphs and Hypergraphs (PhD thesis), 2013, FAST-NUCES, Lahore Campus, Pakistan. Google Scholar

Accepted: 2018-09-23

Published Online: 2018-11-15

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1313–1325, ISSN (Online) 2391-5455,

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