on a weaker version of sum labeling of graphs

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ON A WEAKER VERSION OF SUM LABELING OF GRAPHS
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 G with vertex set V and edge set E, 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 vertex w in G with L(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 number r of isolated vertices Kr such that
G = H ∪ Kr is an sw-sum graph. If the set {1, 2, ..., |V |} is replaced by some
subset S of Z+ 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 degree δ of a vertex in the graph. Graphs
achieving this bound will be referred to as δ-optimal sw-summable. We provide
labeling schemes for different families of graphs showing that they are δ-optimal
sw-summable. We show that not all the graphs are δ-optimal sw-summable and
conjecture that all the graphs are δ-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 G is 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 isolated vertices needed so that G = H ∪Kr is a sum graph [1]. All sum graphs are
MATH. REPORTS 16(66), 3 (2014), 413–420
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Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran
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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 variations 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)-graph G, 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 in G 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 as optimal. 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 degree δ 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. Hence δ ≤ ω(H). An sw-sum graph is termed as δ-optimal sw-summable if it needs δ isolates
to sw-sum label a graph.
If the set [p] is replaced by some subset S of Z+ in the definition of swsum 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:
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On a weaker version of sum labeling of graphs
415
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
vertex v receiving the largest label is d, then vertices in N (v) receive labels from
the set [d].
If a (p, q)-graph is w-summable, then it may not be δ-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. The Cayley graph Cay(X, S) is a graph
with the vertex set X and two vertices x, y ∈ X adjacent whenever xy −1 ∈ S.
Consider the w-sum labeling of Cay(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 receive 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 that L(x) + L(y) ∈
/ [12].
Fig. 1 – Weak sum labeling of Cay(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 1optimal 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 graphs Kn1 ,n2 ,...,nq are t-optimal sw-summable, where t is the minimum degree of a vertex in Kn1 ,n2 ,...,nq .
Throughout the paper, the vertices are identified by their labels under L.
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Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran
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2. SUPER WEAK SUM LABELING
Let Pn and Cn be path and cycle on n 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 }. Let n = 2k or n = 2k + 1 depending upon whether n is
even or odd. Define v2i = i for i ∈ [k] and v2i+1 = n − i for i from 0 to k − 1
or k depending upon whether n is even or odd, respectively, and s1 = n + 1.
Labeling scheme for Cn ∪K2 : Let V (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 for i ∈ [k] and w2i+1 = i + 1 for i from 0 to k − 1 or k depending
upon whether n is even or odd, respectively, and tj = n + j for j = 1, 2.
It is easy to see that Pn ∪ K1 and Cn ∪ K2 are sw-sum graphs. Hence, we
have the following result:
Theorem 2.1. ω(Pn ) = 1 for all n ≥ 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
vertices v0 , v1 , . . . , vn−1 are referred to as rim 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 n is even or odd. Label the central vertex by c = 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 define v2i = i + 2, where i ∈ [k] ∪ {0}
and v2i+1 = d − i, where i ∈ [k − 1] ∪ {0}.
After labeling all the vertices of the graph Wn , d is 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 and d + 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 wheel Wn is 3-optimal sw-summable. Let Kn be a complete graph with vertex set {v1 , v2 , . . . , vn }. Now, we
show that ω(Kn ) = n − 1 for all n ≥ 2.
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On a weaker version of sum labeling of graphs
417
Theorem 2.3. For all n ≥ 2, ω(Kn ) = n − 1.
Proof. Following is the sw-sum labeling scheme for complete graph Kn
with n−1 isolates: Let n = 2k or n = 2k +1 depending upon whether n is 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 vertex vj then the edges incident at vj are (vi , vj ) with i 6= j
and i ∈ [n] \ {j}, j ∈ [n] \ {i}. Note that if i + j < n, then there is a vertex vk
with k = i+j such that, vk = vi +vj and if i+j > n, then vi +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 graph Kn is a label
of another vertex. By Lemma 1.1, we conclude that ω(Kn ) ≥ n − 1. This gives
that ω(Kn ) = n − 1 for all n. A complete multipartite graph is a graph whose vertex set can be partitioned into q subsets V1 , V2 , . . . , Vq such 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 the q-partitions in such a way that n1 ≤
n2 ≤ . . . ≤ nq where ni ’s are the number of vertices in Vi -class. We name the
vertices from the classes V1 , V2 , . . . , Vq as v1 , v2 , . . . , vn1 , vn1 +1 , . . . , vn1 +n2 , . . . , vs
where s = n1 +n2 +. . .+nq . Now, since Vq is the class having maximum number
of vertices nq , so they attain the minimum degree. Let t denotes the minimum
degree of a vertex in Kn1 ,n2 ,...,nq then t = s − nq . In the following theorem, we
shall prove that ω(Kn1 ,n2 ,...,nq ) = s − nq .
Theorem 2.4. Kn1 ,n2 ,...,nq is δ-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 i 6= 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 + i and they results
in minimum isolates. This shows that Kn1 ,n2 ,...,nq can be sw-sum labeled using
t isolates, so ω(Kn1 ,n2 ,...,nq ) ≤ t. Hence, by Lemma 1.1, ω(Kn1 ,n2 ,...,nq ) = t for
all ni , i = 1, 2, . . . , q. 418
Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran
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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 OF Cay(Zn , {±1, ±2})
We note that paths, cycles, wheels, complete graphs, complete multipartite graphs and stars are all δ-optimal sw-summable graphs. Earlier it was
shown that Cay(Z8 , {±1, ±2}) ∪ K4 is not sw-summable but w-summable. In
this section, we show that Cay(Zn , {±1, ±2})∪K4 is w-summable for all n ≥ 5.
Theorem 3.1. For all n ≥ 5, Cay(Zn , {±1, ±2}) is 4-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 4optimal w-summable, we define the labeling as follows:
i + 1(0 ≤ i ≤ k − 1), r = 0,
v3i =
i + 1(0 ≤ i ≤ k),
r = 1, 2,
v3i+1 = n − i(0 ≤ i ≤ k − 1), r = 0, 1, 2,

 k + 2 + i(0 ≤ i ≤ k − 2), r = 0,
k + 2 + i(0 ≤ i ≤ k − 1), r = 1,
v3i+2 =

k + 3 + i(0 ≤ i ≤ k − 1), r = 2,
for r = 0, v3k−1 = k + 1 and for r = 2, v3k+1 = k + 2.
Note that v1 = n is the largest label of a vertex in the graph and v0 +v1 =
n + 1, v1 + v3 = n + 2 for r = 0, 1, 2,

 4k + 2, r = 0,
4k + 3, r = 1,
v1 + v2 =

4k + 5, r = 2,

 4k + 1, r = 0,
4k + 2, r = 1,
v1 + vn−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
n + 2(1 ≤ i ≤ k − 1), r = 0,
v3i + v3i−2 =
n + 2(1 ≤ i ≤ k),
r = 1, 2,
k + 2i + 2(0 ≤ i ≤ k − 1), r = 0, 1,
v3i + v3i−1 =
k + 2i + 3(0 ≤ i ≤ k − 1), r = 3,
v3i + v3i+1 = n + 1(0 ≤ i ≤ k − 1), r = 0, 1, 2,
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On a weaker version of sum labeling of graphs
419

 k + 3 + 2i(0 ≤ i ≤ k − 2), r = 0,
k + 3 + 2i(0 ≤ i ≤ k − 1), r = 1,
v3i + v3i+2 =

k + 4 + 2i(0 ≤ i ≤ k − 1), r = 2,
n + k + 1(0 ≤ i ≤ k − 1), r = 0, 1,
v3i−1 + v3i+1 =
n + k + 2(0 ≤ i ≤ k − 1), r = 3,

 n + k + 2(0 ≤ i ≤ k − 2), r = 0,
n + k + 2(0 ≤ i ≤ k − 1), r = 1,
v3i+1 + v3i+2 =

n + k + 3(0 ≤ i ≤ k − 1), r = 2,
2k + 2, r = 0, 1,
v0 + vn−2 =
k + 2, r = 2,
3k + 2, r = 0, 1,
vn−2 + vn−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 of Cay(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.
REFERENCES
[1] F. Harary, Sum graphs and difference graphs. Congr. Numer. 72 (1990), 101–108.
[2] F. Harary, Sum graphs over all the integers. Discrete Math. 124 (1994), 99–105.
[3] K.M. Koh, M. Miller, W.F. Smyth and Y. Wang, On optimal summable graphs. Submitted
for publication.
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Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran
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[4] M. Miller, J.F. Ryan, Slamin, K. Sugeng and M. Tuga, Exclusive sum graph labelings.
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[5] M. Sutton, Summable graph labelings and their applications. Ph.D. Thesis, October 2000.
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
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