IMRAN JAVAID, FARIHA KHALID, ALI AHMAD and M. IMRAN
Communicated by the former editorial board
In this paper, we introduce super weak sum labeling and weak sum labeling of a graph Gwith vertex setV and edge setE, defined as follows: A super weak sum (briefly sw-sum) labeling is a bijection L : V → {1,2, ...,|V|} such that for every edge (u, v) in G, there is a vertexwinGwithL(u) +L(v) = L(w).
A graph that can be sw-sum labeled is called an sw-sum graph. It is obvious that an sw-sum graph cannot be connected. There must be at least one isolated vertex, namely the vertex with the largest label. The sw-sum number, ω(H), of a connected graph H is the least numberr of isolated verticesKr such that G= H∪Kr is an sw-sum graph. If the set {1,2, ...,|V|} is replaced by some subsetSofZ+in the definition of sw-sum labeling, then such a labeling will be referred to as weak sum (briefly w-sum) labeling and the minimum number of isolates in such a labeling as w-sum number. We show that a lower bound for the sw-sum number is the minimum degreeδof a vertex in the graph. Graphs achieving this bound will be referred to asδ-optimal sw-summable. We provide labeling schemes for different families of graphs showing that they areδ-optimal sw-summable. We show that not all the graphs areδ-optimal sw-summable and conjecture that all the graphs areδ-optimal w-summable.
AMS 2010 Subject Classification: 05C78, 05C62.
Key words: sum graph, super weak sum labeling, weak sum labeling.
1. INTRODUCTION
All the graphs considered in this paper are simple, finite and undirected.
If a graph G has p vertices and q edges, then G will be referred to as (p, q)- graph and by [p], we mean the set {1,2, . . . , p}. For a vertex v, the set of vertices adjacent with v are referred to as the neighborhood of v, denoted by N(v) and |N(v)|is the degree of v.
A graph Gis called a sum graph if there exists a labeling of the vertices of G by distinct positive integers such that the vertices u and v are adjacent if and only if there exists a vertex whose label is equal to the sum of labels of u and v. The sum number, σ(H), of a graph H is the least number r of iso- lated vertices needed so thatG=H∪Kris a sum graph [1]. All sum graphs are
MATH. REPORTS16(66),3(2014), 413–420
necessarily disconnected. There must be at least one isolated vertex, namely the vertex with the largest label, so that the sum number r of a connected graph is always more than or equal to one.
Sum labelings have important applications in graph storage. Many vari- ations of sum labelings have been studied, for example integer sum labelings [2], mod sum labeling [2], exclusive sum labelings [4], sum* labeling [5] and mod sum* labeling [5].
In this paper, we are introducing a weaker version of sum labeling of a (p, q)-graphG, namely super weak sum (briefly sw-sum) labeling using integers from the set [p] in the following way: A labeling L :V → [p] is called super weak sum labeling if for any (u, v)∈E(G), there exists a vertex w inG such that L(u) +L(v) = L(w). sw-sum graphs are necessarily disconnected so in order to sw-sum label a connected graph H, it becomes necessary to add a set of isolated vertices known as isolates as a disjoint union and the labeling scheme that requires the fewest isolates is termed asoptimal. By this method, any graph can be embedded in an sw-sum graph by adding sufficient isolates.
The smallest number of isolates required for a graph H to support an sw-sum labeling is known as the sw-sum number of a graph, denoted by ω(H). It is evident that ω(H) ≤ q. A lower bound for the sw-sum number of a graph is the minimum degreeδ of a vertex in the graph. We prove this in the following lemma:
Lemma 1.1. A lower bound for the sw-sum number ω(H) of a graph H is the minimum degree δ of a vertex in the graph.
Proof. Let v ∈ V(H) be a vertex with maximum label. Then it has at least δ neighbors v1, v2, ..., vδ. Since sum of labels of v and vi; i = 1,2, ..., δ must be a label of another vertex, so we must have δ isolates to sw-sum label this graph. Henceδ ≤ω(H).
An sw-sum graph is termed asδ-optimal sw-summableif it needsδisolates to sw-sum label a graph.
If the set [p] is replaced by some subset S of Z+ in the definition of sw- sum labeling, then such a labeling is referred to as weak sum (briefly w-sum) labeling. Since w-sum graphs are generalization of sw-sum graphs, so all the terminology mentioned above for sw-sum graphs holds for w-sum graphs as well.
Note that, if all the labels x ∈ [p] of vertices in a graph G are replaced by kx for some k∈Z+, then this graph receives the labels from k[p] and the sum of labels of every two distinct vertices is a label of another vertex in G.
Hence, we have the following lemma:
Lemma 1.2. Every (p, q)-graph which is sw-summable is w-summable.
Observation 1.3. In a δ-optimal sw-summable graph, if the degree of a vertexvreceiving the largest label isd, then vertices in N(v) receive labels from the set [d].
If a (p, q)-graph is w-summable, then it may not beδ-optimal sw-summable.
In order to show this first we define Cayley graph: Let X be a group and S ⊆X\{1}, an inverse closed subset. TheCayley graph Cay(X, S) is a graph with the vertex set X and two verticesx, y∈X adjacent whenever xy−1 ∈S.
Consider the w-sum labeling ofCay(Z8,{±1,±2})∪K4 in Figure 1. However, this graph does not support sw-sum labeling. By Observation 1.3, suppose that the vertex v0 receives label 8, then the vertices v1, v2, v6 and v7 will re- ceive labels from the set [4] and the vertices v3, v4 and v5 will receive labels from the set {5,6,7}. It can be seen that there exists an edge say (x, y) with one vertex say x having label 7 such thatL(x) +L(y)∈/ [12].
Fig. 1 – Weak sum labeling ofCay(Z8,{±1,±2})∪K4.
Let L and L0 be two optimal w-sum labelings of a graph G. Labeling L is said to be smaller than L0 if the largest label under L is less than the largest label under L0. From this definition, it follows that sw-sum labeling is the smallest w-sum labeling.
In the next section, we provide labeling schemes showing that paths are 1- optimal sw-summable, cycles are 2-optimal sw-summable, wheels are 3-optimal sw-summable, complete graphs are (n−1)-optimal sw-summable and complete multipartite graphsKn1,n2,...,nq aret-optimal sw-summable, wheretis the min- imum degree of a vertex in Kn1,n2,...,nq.
Throughout the paper, the vertices are identified by their labels underL.
2.SUPER WEAK SUM LABELING
Let Pn and Cn be path and cycle onn vertices. We show that paths are 1-optimal summable and cycles are 2-optimal summable by providing sw-sum labeling schemes of Pn∪K1 and Cn∪K2.
Labeling scheme for Pn ∪K1: The vertex set of Pn∪ K1 is given as {v1, v2, . . . , vn} ∪ {s1}. Letn= 2korn= 2k+ 1 depending upon whethernis even or odd. Definev2i =ifori∈[k] and v2i+1 =n−i forifrom 0 to k−1 ork depending upon whethern is even or odd, respectively, and s1 =n+ 1.
Labeling scheme forCn∪K2: LetV(Cn∪K2) ={w1, w2, . . . , wn}∪{t1, t2}.
Let n = 2k or n = 2k+ 1 depending upon whether n is even or odd. Define w2i =n−i+ 1 fori∈[k] andw2i+1 =i+ 1 forifrom 0 tok−1 orkdepending upon whethern is even or odd, respectively, andtj =n+j forj= 1,2.
It is easy to see thatPn∪K1 and Cn∪K2 are sw-sum graphs. Hence, we have the following result:
Theorem 2.1. ω(Pn) = 1 for alln≥2 and ω(Cn) = 2 for all n≥3.
For every integer n ≥ 3, a wheel Wn = (V, E) is a graph with V = {c, v0, v1, . . . , vn−1}, E ={(c, vi),(vi, vi+1)|i= 0,1, ..., n−1} where indices of the vertices are considered modulo n. The vertex c is called the center of the wheel, each edge (c, vi), for i = 0,1,2, ...., n−1, is called a spoke, the verticesv0, v1, . . . , vn−1 are referred to asrim vertices and each edge (vi, vi+1) for i = 0,1, . . . , n−1 is called a rim edge. Now, we show that wheels are 3-optimal sw-summable.
Theorem 2.2. ω(Wn) = 3 for all n≥3.
Proof. By Lemma 1.1, ω(Wn) ≥ 3. Following is the labeling scheme for the wheels with three isolates: Let n = 2k or n = 2k+ 1 depending upon whether nis even or odd. Label the central vertex byc= 1. Set d=n+ 1.
Assign labels to the vertices as v2i = i+ 2 and v2i+1 = d−i, where i∈[k−1]∪ {0}for n even. For n odd we definev2i =i+ 2, wherei∈[k]∪ {0}
and v2i+1=d−i, wherei∈[k−1]∪ {0}.
After labeling all the vertices of the graph Wn, dis the maximum label on the graph. Note that v1 = d, so v1 +c = d+ 1. v0 +v1 = d+ 2 and v1+v2=d+ 3 are larger than the maximum label in the graph. Hence, there must be three isolates with labels d+ 1, d+ 2 andd+ 3.
It is easy to see that the sum of the labels of every spoke and rim edge is a label of another vertex. Hence, the wheelWnis 3-optimal sw-summable.
Let Kn be a complete graph with vertex set {v1, v2, . . . , vn}. Now, we show thatω(Kn) =n−1 for alln≥2.
Theorem 2.3. For all n≥2, ω(Kn) =n−1.
Proof. Following is the sw-sum labeling scheme for complete graph Kn withn−1 isolates: Letn= 2korn= 2k+ 1 depending upon whethernis even or odd. Starting from any vertex, label the vertices as: vi =i,i = 1,2, ..., n.
Since vi is adjacent with n−1 vertices, so there must be n−1 isolates.
Consider any vertexvj then the edges incident atvj are (vi, vj) withi6=j and i∈[n]\ {j},j∈[n]\ {i}. Note that if i+j < n, then there is a vertex vk withk=i+jsuch that,vk=vi+vjand ifi+j > n, thenvi+vj ∈[2n−1]\[n], which means (vi, vj) is labeled by an isolate. Hence (vi, vj) is an edge.
We see that the sum of the labels of every edge on the graphKnis a label of another vertex. By Lemma 1.1, we conclude thatω(Kn)≥n−1. This gives that ω(Kn) =n−1 for alln.
A complete multipartite graph is a graph whose vertex set can be parti- tioned intoq subsetsV1, V2, . . . , Vqsuch that every (u, v) is an edge if and only if u and v belong to different partite sets. If |Vi| = ni, 1 ≤ i ≤ q, then we denote complete multipartite graph as Kn1,n2,...,nq.
For labeling purpose, we arrange theq-partitions in such a way thatn1≤ n2 ≤. . .≤nq whereni’s are the number of vertices in Vi-class. We name the vertices from the classesV1, V2, . . . , Vqasv1, v2, . . . , vn1, vn1+1, . . . , vn1+n2, . . . , vs wheres=n1+n2+. . .+nq. Now, sinceVqis the class having maximum number of verticesnq, so they attain the minimum degree. Lettdenotes the minimum degree of a vertex in Kn1,n2,...,nq thent=s−nq. In the following theorem, we shall prove that ω(Kn1,n2,...,nq) =s−nq.
Theorem 2.4. Kn1,n2,...,nq isδ-optimal super weak summable.
Proof. Let V = {V1, V2, ..., Vq} be the vertex set of Kn1,n2,...,nq and s= n1+n2+...+nq be the total number of vertices in the graph. Now, assign labels to the vertices as: vi = i, i ∈ [s]. The maximum label is s which is the label of the vertex vs. Now, the sum of vs+vi =s+i, i∈ [t]. Since vs
has maximum label on the graph, so the labels vs+vi are greater than the maximum label s, so they must be the isolates.
Labels of the vertices of Kn1,n2,...,nq form the sequence {1,2, . . . , s, s+ 1, . . . , s+t}, so vs−j+vi=s−j+i∈ {1,2, . . . , s, s+ 1, . . . , s+t},i∈[s−1], j ∈[s] and i6=s−j. This shows that the sum of labels of every vertex is the label of another vertex on the graph. Since s−j+i < s+iand they results in minimum isolates. This shows thatKn1,n2,...,nq can be sw-sum labeled using t isolates, soω(Kn1,n2,...,nq)≤t. Hence, by Lemma 1.1,ω(Kn1,n2,...,nq) =t for all ni,i= 1,2, . . . , q.
Corollary 2.5. Let Km,n be a bipartite graph then ω(Km,n) = m if m≤n. In particular, ω(Sn) =ω(K1,n) = 1, where Sn is a star.
3. WEAK SUM LABELING OFCay(Zn,{±1,±2})
We note that paths, cycles, wheels, complete graphs, complete multipar- tite graphs and stars are all δ-optimal sw-summable graphs. Earlier it was shown thatCay(Z8,{±1,±2})∪K4 is not sw-summable but w-summable. In this section, we show thatCay(Zn,{±1,±2})∪K4is w-summable for alln≥5.
Theorem3.1. For alln≥5,Cay(Zn,{±1,±2})is4-optimal w-summable.
Proof. Let v0, v1, . . . , vn−1 be the vertices of Cay(Zn,{±1,±2}), where n = 3k+r, k ∈ Z+ and r = 0,1,2. To show that Cay(Zn,{±1,±2}) is 4- optimal w-summable, we define the labeling as follows:
v3i=
i+ 1(0≤i≤k−1), r= 0, i+ 1(0≤i≤k), r= 1,2, v3i+1 =
n−i(0≤i≤k−1), r = 0,1,2, v3i+2=
k+ 2 +i(0≤i≤k−2), r = 0, k+ 2 +i(0≤i≤k−1), r = 1, k+ 3 +i(0≤i≤k−1), r = 2, forr = 0,v3k−1 =k+ 1 and for r= 2,v3k+1=k+ 2.
Note thatv1 =nis the largest label of a vertex in the graph andv0+v1= n+ 1,v1+v3 =n+ 2 for r= 0,1,2,
v1+v2=
4k+ 2, r= 0, 4k+ 3, r= 1, 4k+ 5, r= 2, v1+vn−1 =
4k+ 1, r = 0, 4k+ 2, r = 1, 4k+ 4, r = 2.
Now, it remains to show that vi+vj ∈ {v1, v2, . . . , vn−1} ∪ {v0+v1, v1+ v2, v1+v3, v1+vn−1} whenever (vi, vj) is an edge.
Note that
v3i+v3i−2 =
n+ 2(1≤i≤k−1), r= 0, n+ 2(1≤i≤k), r= 1,2, v3i+v3i−1 =
k+ 2i+ 2(0≤i≤k−1), r= 0,1, k+ 2i+ 3(0≤i≤k−1), r= 3, v3i+v3i+1 =
n+ 1(0≤i≤k−1), r= 0,1,2,
v3i+v3i+2 =
k+ 3 + 2i(0≤i≤k−2), r= 0, k+ 3 + 2i(0≤i≤k−1), r= 1, k+ 4 + 2i(0≤i≤k−1), r= 2, v3i−1+v3i+1=
n+k+ 1(0≤i≤k−1), r = 0,1, n+k+ 2(0≤i≤k−1), r = 3, v3i+1+v3i+2 =
n+k+ 2(0≤i≤k−2), r= 0, n+k+ 2(0≤i≤k−1), r= 1, n+k+ 3(0≤i≤k−1), r= 2, v0+vn−2 =
2k+ 2, r= 0,1, k+ 2, r= 2, vn−2+vn−1 =
3k+ 2, r= 0,1, 2k+ 3, r= 2,
for r = 0, v3k−3 +v3k−1 = 2k+ 1 and for r = 2, vn−3 +vn−2 = 3(k+ 1), vn−3 +vn−1 = 3k+ 4. We see that the sum of the labels of every edge is a label of another vertex. We conclude that Cay(Zn,{±1,±2}) with n ≥ 5 can be weak sum labeled using only four isolates. Hence, by Lemma 1.1, Cay(Zn,{±1,±2}) is 4-optimal w-summable.
Concluding Remarks: In this paper, we have introduced a weaker version of sum labeling of a (p, q) graph using labels from the set [p]. We have seen that wheels, complete graphs, complete bipartite graphs can be super weak sum labeled using δ isolates. Also, note that |a−b| ∈ [p] for any a, b ∈ [p].
Hence, if we define super difference labeling by replacing L(u) +L(v) with
|L(u)−L(v)|in the definition of sw-sum labeling, then all the classes of graphs mentioned above are super difference graphs. We note that not all the graphs are sw-summable and give w-sum labeling ofCay(Zn,{±1,±2}). We have the following conjecture and open question for further work on this paper.
Conjecture 3.2. All graphs are δ-optimal w-summable.
Open Problem 3.3. Does there exist a graph which is an sw-sum graph but not δ-optimal sw-summable?
Acknowledgments. The authors are grateful to the anonymous referee whose careful reading and valuable suggestions resulted in producing an improved paper.
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Received 31 May 2012 Bahauddin Zakariya University Multan, Center for Advanced Studies in Pure and Applied Mathematics,
Pakistan ijavaidbzu@gmail.com National University of Computer
and Emerging Sciences, FAST, Lahore,
Pakistan L125505@nu.edu.pk
Jazan University, College of Computer and Information System,
Jazan, KSA, ahmadsms@gmail.com
National University of Sciences and Technology,
Center for Advanced Mathematics and Physics, Islamabad, Pakistan,
imrandhab@gmail.com
