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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 15, Issue 1

# Deficiency of forests

Sana Javed
/ Mujtaba Hussain
/ Ayesha Riasat
/ Salma Kanwal
/ Mariam Imtiaz
Published Online: 2017-12-09 | DOI: https://doi.org/10.1515/math-2017-0122

## Abstract

An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xyE(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. 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 Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].

MSC 2010: 05C78

## 1 Basic definitions, notations and preliminary results

Let G = (V, E) be a finite, simple, undirected graph having |V(G)| = n and |E(G)| = m, where V(G) and E(G) denote the vertex set and edge set, respectively. A general orientation for graph theoretic concepts can be seen in [10]. A labeling (or valuation) of a graph is a map that carries graph elements to numbers (usually to positive integers). A labeling that uses the vertex set only (or the edge set only), is known as vertex labeling (or the edge labeling). If the domain of the labeling includes all vertices and edges, then such a labeling is called total labeling. Cordial, graceful, harmonious and anti-magic are few types of labeling. A bijective labeling is called an edge-magic total if it satisfies the following property, given any edge xyE(G), $λ(x)+λ(y)+λ(xy)=c,$(1)

for some constant c. In other words, an edge-magic total labeling of a graph G is a bijective map λ from V(G) ∪ E(G) onto the integers {1, 2, …, n + m} satisfying (1). The constant c is known as the magic constant and a graph that admits an edge-magic total labeling is called an edge-magic total graph. In [8, 9], Kotzig and Rosa have given the origin of edge-magic total labeling of graphs. Recently, Enomoto et al. [1] brought in the name, super edge-magic labeling in the sense of Kotzig and Rosa, with the additional property that the vertices receive the smallest labels. In [1] they put forward the following conjecture:

#### Conjecture 1.1

([1]). “Every tree is super edge-magic total.”

In this paper we are focused on super edge-magic total labeling. A number of classification problems on edge-magic total graphs have been extensively investigated. For further details see recent survey of graph labelings [6]. Kotzig and Rosa in [9] show that there exists an edge-magic total graph H for any graph G such that H ≅ G ∪ nK1 for some non-negative integer n. This verity provides the base for the concept of edge-magic total deficiency of a graph G [9], denoted by μ(G), which is the minimum non-negative integer n such that GnK1 is edge-magic total i.e., $μ(G)=min{n≥0:G∪nK1isedge−magictotal}.$

In the same paper Kotzig and Rosa provide an upper bound of the edge-magic deficiency of a graph G having order n, $μ(G)≤Fn+2−2−n−12n(n−1),$

where Fn denotes the n-th Fibonacci number.

The super edge-magic deficiency of a graph G, denoted by μs(G) [4], is mathematically expressed as if $M(G)={n≥0:G∪nK1issuperedge−magicgraph},$

Then $μs(G)=minM(G),if M(G)≠∅∞,if M(G)=∅.$

It is easy to see that μ(G)≤ μs(G). In [2], Figueroa-Centeno et al. conjectured, “Every forest with two components has the super edge-magic deficiency at most 1”. Moreover, in the same paper they showed that μs(PmK1,n) is 1 if m = 2 and n is odd or m = 3 and n≢ 0(mod 3) and 0 otherwise. In [7], S. Javed et al. gave the upper bound of deficiencies of disjoint union of graphs consisting of comb, generalized comb and star. In this paper, we frequently use the following two Lemmas.

#### Lemma 1.2

([5]). “A graph G with n-vertices and m-edges is super edge-magic total if and only if there exists a bijection λ:V(G) → {1,2, …, n} such that the set S = {λ(x) + λ(y)|xyE(G)} consists of m consecutive integers. In such a case, λ extends to a super edge-magic total labeling of G.”

The above condition is often easier to use than the original one. The following lemma was found first in [1].

#### Lemma 1.3

([1]). “If a graph G with n vertices and m edges is super edge-magic total then m≤ 2n−3.”

#### Definition 1.4

A comb is a graph derived from the path Pn:u1,u2, …, un, n ≥ 3, by adding n−1 new edges ui+1wi; 1≤ in−1 and this is denoted by Cbn.

#### Definition 1.5

A two-sided generalized comb, denoted by $\begin{array}{}C{b}_{n,m}^{2}\end{array}$, consists of the vertex set, $V(Cbn,m2)={ui,j;1≤i≤n,1≤j≤m}∪{u0,m+12}$

and the edge set, $E(Cbn,m2)={ui,jui,j+1;1≤i≤n,1≤j≤m−1}∪{ui,m+12ui+1,m+12;0≤i≤n−1},$

i.e., $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ is deduced from n paths Pi,m:ui,1,ui,2, …, ui,m;ui,jui,j+1E( $\begin{array}{}C{b}_{n,m}^{2}\end{array}$); 1 ≤ in;1 ≤ jm−1 and n ≥ 2 of length m−1, where m is odd, by adding one new vertex $\begin{array}{}{u}_{0,\frac{m+1}{2}}\end{array}$ and n new edges $\begin{array}{}{u}_{i,\frac{m+1}{2}}{u}_{i+1,\frac{m+1}{2}};\phantom{\rule{thinmathspace}{0ex}}0\le i\le n-1.\phantom{\rule{thinmathspace}{0ex}}C{b}_{n,m}^{2}\end{array}$ for n = m = 5 is shown in Fig. 1.

Fig. 1

$\begin{array}{}C{b}_{5,5}^{2}\end{array}$

The graph obtained by $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ by deleting the set of vertices $\begin{array}{}\left\{{u}_{i,j};1\le i\le n,\frac{m+3}{2}\le j\le m\right\}\end{array}$ and their adjacent edges is referred to as generalized comb, denoted by $\begin{array}{}C{b}_{n}\left(\underset{n-times}{\underset{⏟}{l,l,\dots ,l}}\right)\end{array}$. The labeling of $\begin{array}{}C{b}_{n}\left(\underset{n-times}{\underset{⏟}{l,l,\dots ,l}}\right)\end{array}$ is discussed in [7].

#### Definition 1.6

A bistar on n vertices, denoted by BS(p,q);p,q ≥ 1, p + q+2 = n, is obtained from two stars K1,p and K1,q by joining their central vertices by an edge.

In this paper we formulate the super edge-magic total labeling of two sided generalized comb. Moreover, we determine an upper bound for super edge-magic total deficiency of forests containing comb, bistar and 2-sided generalized comb.

## 2 Super edge-magic deficiencies of forests of combs and bistar

In this section, we will provide precise value for super edge-magic deficiency of some specific number of copies of the comb Cbn, we will also give an upper bound for super edge-magic deficiency for disjoint union of bistar BS(k,k) and Cbn with some restrictions on the parameters k and n.

#### Theorem 2.1

For n-odd, n ≥ 3, m-even and m ≡ 2(mod 4), the graph GmCbn is super edge magic total.

#### Proof

Consider the graph GmCbn. Then |V(G)| = m(2n−1) and |E(G)| = m(2n−2), where $V(G)={uik;1≤i≤n,1≤k≤m}∪{wjk;1≤j≤n−1,1≤k≤m}$

and $E(G)={uikui+1k;1≤i≤n−1,1≤k≤m}∪{ui+1kwik;1≤i≤n−1,1≤k≤m}.$

Define a labeling f:V(G) → {1,2, …, m(2n−1)} as follows: $f(wn−1k)=3m−4k+64;m2+1≤k≤3m+245m−4k+64;3m+64≤k≤m$

For 1 ≤ k$\begin{array}{}\frac{m}{2}\end{array}$, k-odd $f(uik)=m(i−1)+k+12;i≡0(mod2)m(n−1)−k+1+mi2;i≡1(mod2)f(wjk)=mj+k+12;j≡0(mod2)mn−k+1+m(j−1)2;j≡1(mod2)$

For 2 ≤ k$\begin{array}{}\frac{m}{2}\end{array}$−1, k-even $f(uik)=m(2i−1)+2(k+1)4;i≡0(mod2)m(n−1)−k+1+mi2;i≡1(mod2)f(wjk)=m(2j+1)+2(k+1)4;j≡0(mod2)mn−k+1+m(j−1)2;j≡1(mod2)$

For $\begin{array}{}\frac{m}{2}\end{array}$+1 ≤ km, k-even $f(uik)=m(2n+2i−5)+2(k+1)4;i≡0(mod2)m(3n+i−1)2−k+1;i≡1(mod2)f(wjk)=m(2n+2j−3)+2(k+1)4;j≡0(mod2),j≠n−1m(3n+j)2−k+1;j≡1(mod2)$

For $\begin{array}{}\frac{m}{2}\end{array}$+2 ≤ km−1, k-odd $f(uik)=m(n+i−2)+k+12;i≡0(mod2)m(3n+i−1)2−k+1;i≡1(mod2)f(wjk)=m(n+j−1)+k+12;j≡0(mod2),j≠n−1m(3n+j)2−k+1;j≡1(mod2)$

The labeling f gives the following set of consecutive integers $\begin{array}{c}\left\{\frac{4mn-m+6}{4},\frac{4mn-m+10}{4},\dots ,\frac{12mn-9m+2}{4}\right\}\end{array}$ that appears as the weights of the edges in the graph. □

#### Theorem 2.2

For n, m-even, n ≥ 4 and m ≡ 2(mod 4), μs(mCbn) ≤ $\begin{array}{}\frac{m}{2}\end{array}$.

#### Proof

Consider the graph GmCbn ∪ ( $\begin{array}{}\frac{m}{2}\end{array}$)K1. Then |V(G)| = 2mn$\begin{array}{}\frac{m}{2}\end{array}$ and |E(G)| = m(2n−2), where V(G) = V(mCbn) ∪ {zl; 1 ≤ l$\begin{array}{}\frac{m}{2}\end{array}$} and E(G) = E(mCbn).

Define a labeling g:V(G) → {1,2, …, 2mn$\begin{array}{}\frac{m}{2}\end{array}$} as follows: $g(unk)=3m−4k+64;m2+1≤k≤3m+245m−4k+64;3m+64≤k≤mg(zl)=3mn2−m+l;1≤l≤m2$

For 1 ≤ k$\begin{array}{}\frac{m}{2}\end{array}$, $g(uik)=f(uik)∀i;1≤i≤ng(wjk)=f(wjk)∀j;1≤j≤n−1,$

where the labeling f is defined in Theorem 2.1. For $\begin{array}{}\frac{m}{2}\end{array}$+1 ≤ km, k-even $g(uik)=m(2n+2i−5)+2(k+1)4;i≡0(mod2),i≠nm(3n+i)2−k+1;i≡1(mod2)g(wjk)=m(2n+2j−3)+2(k+1)4;j≡0(mod2)m(3n+j+1)2−k+1;j≡1(mod2)$

For $\begin{array}{}\frac{m}{2}\end{array}$+2 ≤ km−1, k-odd $g(uik)=m(n+i−2)+k+12;i≡0(mod2),i≠nm(3n+i)2−k+1;i≡1(mod2)g(wjk)=m(n+j−1)+k+12;j≡0(mod2)m(3n+j+1)2−k+1;j≡1(mod2)$

The labeling g constitutes the following set of edge weights $\begin{array}{}\left\{\frac{4mn-m+6}{4},\frac{4mn-m+10}{4},\dots ,\frac{12mn-9m+2}{4}\right\}.\end{array}$

In the next Theorem we will compute an upper bound for the edge-magic deficiency of a forest consisting of bistar BS(k,k) and comb Cbn.

#### Theorem 2.3

For k ≥ 2, consider the graph GBS(k,k) ∪ Cbn. Then

1. The graph G is super edge-magic total for nk+2 and k-odd.

2. μs(G) ≤ 1 for nk+3 and k-even.

#### Proof

Consider the graph GBS(k,k) ∪ Cbn. We have $V(G)={ui:1≤i≤n}∪{wj:1≤j≤n−1}∪{z1,z2}∪{zi,t:1≤i≤2,1≤t≤k}$

and E(G) = {z1z1,t,z2z2,t:1 ≤ tk} ∪ {uiui+1:1 ≤ in−1} ∪ {ui+1wi:1 ≤ in−1} ∪ {z1z2}, which give |V(G)| = 2(n + k)+1 and E(G) = 2(n + k)−1.

1. Define a labeling f:V(G) → {1,2, …, |V(G)|} in the following way: $f(z1)=n+k+1f(z2)=k+2f(z1t)=t+1;1≤t≤kf(z2t)=n+k+t+1;1≤t≤kf(ui)=k+i+1;2≤i≤k+1,i≡0(mod2)k+i;k+3≤i≤n,i≡0(mod2)n+2k+i+1;1≤i≤n,i≡1(mod2)f(wj)=k+j+2;2≤j≤k−1,j≡0(mod2)k+j+1;k+3≤j≤n−1,j≡0(mod2)n+2k+j+2;1≤j≤n−1,j≡1(mod2)1;j=k+1$

The set of edge weights formed under the labeling f consists of the following consecutive integers {n + k+3, n + k+4, …, 3(n + k)+1}.

2. Consider the graph HGK1, where V(K1) = {u}.

Define a labeling g:V(H) → {1,2, …, 2(n + k+1)} as follows: $g(u)=2g(z1)=n+k+2g(z2)=k+3g(z1t)=t+2;1≤t≤kg(z2t)=n+k+t+2;1≤t≤kg(ui)=k+i+2;2≤i≤k+2,i≡0(mod2)k+i+1;k+4≤i≤n,i≡0(mod2)n+2k+i+2;1≤i≤n,i≡1(mod2)g(wj)=k+j+3;2≤j≤n−1,j≡0(mod2),j≠k+2n+2k+j+3;1≤j≤n−1,j≡1(mod2)1;j=k+2$

The labeling g gives the following set of consecutive integers {n + k+5, n + k+6, …, 3(n + k+1)} as the edge weights. □

## 3 Super edge-magic total labeling of two-sided comb

#### Theorem 3.1

For n ≥ 2, m ≥ 3 and m-odd, the graph G$\begin{array}{}C{b}_{n,m}^{2}\end{array}$ is super edge-magic total.

#### Proof

Consider the graph G$\begin{array}{}C{b}_{n,m}^{2}\end{array}$. Then |V(G)| = mn+1 and |E(G)| = mn, where $V(G)={ui,j;1≤i≤n,1≤j≤m}∪{u0,m+12}$

and $E(G)={ui,jui,j+1;1≤i≤n,1≤j≤m−1}∪{ui,m+12ui+1,m+12;0≤i≤n−1}.$

To show that G is super edge-magic total, we will define a labeling f:V(G) → {1,2, …, mn+1} as follows:

For $\begin{array}{}\frac{m-1}{2}\end{array}$-odd, $f(u0,m+12)=m+54$

For j ≡ 1(mod 2), $f(u1,j)=j+12if 1≤j≤m−12;j+32if m+32≤j≤m.f(ui,j)=m(i−1)+j+32if i≡1(mod2) and j≡1(mod2)for 3≤i≤n and 1≤j≤m;mi−j+42if i≡0(mod2) and j≡0(mod2)for 2≤i≤n and 2≤j≤m−1;⌈mn2⌉+m(i−1)+j+22if i≡1(mod2) and j≡0(mod2)for 1≤i≤n and 2≤j≤m−1;⌈mn2⌉+im−j+32if i≡0(mod2) and j≡1(mod2)for 2≤i≤n and 1≤j≤m.$

The set of edge weights given by the labeling f consists of the following mn consecutive integers $\begin{array}{}\left\{⌈\frac{mn}{2}⌉+3,⌈\frac{mn}{2}⌉+4,\dots ,⌈\frac{3mn}{2}⌉+2\right\}.\end{array}$

For $\begin{array}{}\frac{m-1}{2}\end{array}$ -even, $f(u0,m+12)=m+34$

For j ≡ 0(mod 2), $f(u1,j)=j2if 2≤j≤m−12;j+22if m+32≤j≤m−1.f(ui,j)=mi−j+32if i≡0(mod2) and j≡1(mod2)for 2≤i≤n and 1≤j≤m;m(i−1)+j+22if i≡1(mod2) and j≡0(mod2)for 3≤i≤n and 2≤j≤m−1;⌊mn2⌋+m(i−1)+j+32if i≡1(mod2) and j≡1(mod2)for 1≤i≤n and 1≤j≤m;⌊mn2⌋+mi−j+42if i≡0(mod2) and j≡0(mod2)for 2≤i≤n and 2≤j≤m−1.$

The set of edge weights formed under the labeling f is $\begin{array}{}\left\{⌊\frac{mn}{2}⌋+3,⌊\frac{mn}{2}⌋+4,\dots ,⌊\frac{3mn}{2}⌋+2\right\}.\end{array}$

## 4 Super edge-magic deficiency of copies of two-sided comb

#### Theorem 4.1

For n-even, n ≥ 2, m-odd and m ≥ 3, the graph G ≅ 2 $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ is super edge-magic total.

#### Proof

Define the graph G ≅ 2 $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ in the following way: $V(G)={ui,jk;1≤i≤n,1≤j≤m,1≤k≤2}∪{u0,m+12k;1≤k≤2}$

and $E(G)={ui,jkui,j+1k;1≤i≤n,1≤j≤m−1,1≤k≤2}∪{ui,m+12kui+1,m+12k;0≤i≤n−1,1≤k≤2}.$

Then |V(G)| = 2(mn+1) and |E(G)| = 2mn.

For $\begin{array}{}\frac{m-1}{2}\end{array}$ -even, $f(u1,11)=2(mn+1)$

For 1 ≤ k ≤ 2, $f(u0,m+12k)=(mn2+1)(k−1)+m+34$

For j ≡ 0(mod 2), $f(u1,jk)=(mn2+1)(k−1)+j2if 2≤j≤m−12;(mn2+1)(k−1)+j+22if m+32≤j≤m−1.f(ui,jk)=(mn2+1)(k−1)+mi−j+32if i≡0(mod2) and j≡1(mod2)for 2≤i≤n and 1≤j≤m;(mn2+1)(k−1)+m(i−1)+j+22if i≡1(mod2) and j≡0(mod2)for 3≤i≤n−1 and 2≤j≤m−1;mn2(k+1)+m(i−1)+j+32if i≡1(mod2) and j≡1(mod2)for 1≤i≤n−1,1≤j≤m and (i,j,k)≠(1,1,1);mn2(k+1)+mi−j+42if i≡0(mod2) and j≡0(mod2)for 2≤i≤n and 2≤j≤m−1.$

The set of edge weights of G, formed by the labeling defined by f, is {mn + 4,mn + 5, …, 3(mn + 1)}.

For $\begin{array}{}\frac{m-1}{2}\end{array}$ -odd, $f(un,12)=1$

For 1 ≤ k ≤ 2, $f(u0,m+12k)=mn2(k+1)+k+m+14$

For j ≡ 1(mod 2), $f(u1,jk)=mn(k+1)+j−12+kif 1≤j≤m−12;mn(k+1)+j+12+kif m+32≤j≤m.f(ui,jk)=mn2(k−1)+m(i−1)+j+22if i≡1(mod2) and j≡0(mod2)for 1≤i≤n−1 and 2≤j≤m−1;mn2(k−1)+mi−j+32if i≡0(mod2) and j≡1(mod2)for 2≤i≤n,1≤j≤mand (i,j,k)≠(n,1,2);mn(k+1)+im−j2+k+1if i≡0(mod2) and j≡0(mod2)for 2≤i≤n and 2≤j≤m−1;mn(k+1)+m(i−1)+j+12+kif i≡1(mod2) and j≡1(mod2)for 3≤i≤n−1 and 1≤j≤m.$

The labeling f gives the following set of edge weights {mn+3,mn+4, …, 3mn+2}. □

#### Theorem 4.2

For n, m-odd, n ≥ 3 and m > 3, the graph G ≅ 2 $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ has the super edge-magic deficiency at most 1.

#### Proof

Consider the graph HGK1, where V(K1) = {z}. To show that H is super edge-magic total, define the labeling g:V(H) → {1,2, …, 2mn+3} as follows:

For $\begin{array}{}\frac{m-1}{2}\end{array}$ -even, $g(z)=3mn+52g(u1,11)=mn+1g(un,m2)=1$

For 1 ≤ k ≤ 2, $g(u0,m+12k)=2mn(k+1)+6k+m+14$

For j ≡ 0(mod 2), $g(u1,jk)=mn(k+1)+3k+j−12if 2≤j≤m−12;mn(k+1)+3k+j+12if m+32≤j≤m−1.g(ui,jk)=(mn+1)(k−1)+m(i−1)+j+12if i≡1(mod2) and j≡1(mod2)for 1≤i≤n and 1≤j≤m,(i,j,k)≠(1,1,1) and (i,j,k)≠(n,m,2);(mn+1)(k−1)+mi−j+22if i≡0(mod2) and j≡0(mod2)for 2≤i≤n−1,2≤j≤m−1;mn(k+1)+3k+im−j+22if i≡0(mod2) and j≡1(mod2)for 2≤i≤n−1 and 1≤j≤m;mn(k+1)+m(i−1)+3k+j+12if i≡1(mod2) and j≡0(mod2)for 3≤i≤n and 2≤j≤m−1.$

The set of edge weights produced by the labeling g consists of the following set of 2mn consecutive integers {mn+4,mn+5, …, 3(mn+1)}.

For $\begin{array}{}\frac{m-1}{2}\end{array}$ -odd, $g(z)=3mn+52g(u1,21)=2mn+3$

For 1 ≤ k ≤ 2, $g(u0,m+12k)=(mn+3)(k−1)2+m+54$

For j ≡ 1(mod 2), $g(u1,jk)=mn(k−1)+3k+j−22if 1≤j≤m−12;mn(k−1)+3k+j2if m+32≤j≤m.g(ui,jk)=(mn+3)(k−1)−j+mi+42if i≡0(mod2) and j≡0(mod2)for 2≤i≤n−1 and 2≤j≤m−1;mn(k−1)+m(i−1)+3k+j2if i≡1(mod2) and j≡1(mod2)for 3≤i≤n,1≤j≤m;mn(k+1)+m(i−1)+k+j+32if i≡1(mod2) and j≡0(mod2)for 1≤i≤n and 2≤j≤m−1and (i,j,k)≠(1,2,1);mn(k+1)+mi−j+k+42if i≡0(mod2) and j≡1(mod2)for 2≤i≤n−1 and 1≤j≤m.$

The set of edge weights under the labeling g is {mn+6,mn+7, …, 3mn+5}. □

## Concluding remarks

In [3], Figueroa-Centeno et al. discovers that if a graph is super edge-magic, then an odd number of copies of the graph is also super edge-magic. In this paper, we extend this concept for an even number of copies of comb, so the result in [3] significantly generalizes our results. It is also shown that the two-sided generalized comb, denoted by $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ is super edge-magic total. Moreover we have found upper bounds for the super edge-magic deficiency of forests mCbn, CbnBS(k,k) and 2 $\begin{array}{}C{b}_{n,m}^{2}\end{array}$ for different values of the parameters k, m and n. In this context we formulate some open problems:

1. Let n-odd, n ≥ 3 and m ≥3. Determine the exact value of the super edge-magic deficiency of 2 $\begin{array}{}C{b}_{n,m}^{2}\end{array}$.

2. For k1,k2 ≥ 2,k1k2 and n ≥ 3, find an upper bound of the super edge-magic deficiency of BS(k1,k2) ∪ Cbn.

3. For n ≥ 3 and m ≡ 0(mod 4). Calculate the upper bound of the super edge-magic deficiency of mCbn.

## Acknowledgement

We are thankful to the referees for their useful discussions and remarks on our manuscript.

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Accepted: 2017-10-03

Published Online: 2017-12-09

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1431–1439, ISSN (Online) 2391-5455,

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