Let $\mathbb{N}_0$ denote the set of all non-negative integers and
$\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer
(IASI) of a given graph $G$ is an injective function $f:V(G)\to
\mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to
\mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also
injective. An IASI $f$ of a graph $G$ is said to be a weak IASI of $G$ if
$|f^+(uv)|=\max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a
weak IASI may be called a weak IASI graph. The sparing number of a graph $G$ is
the minimum number of edges with singleton set-labels required for a graph $G$
to admit a weak IASI. In this paper, we introduce the notion of $k$-sieve
graphs of a given graph and study their sparing numbers.Comment: 9 pages, 3 figures, Publishe

Germina, Augustine

Sudev, Naduvath

English

arXiv

International audienceLet N0 denote the set of all non-negative integers and P(N0) be its power set. An integer additive set-indexer (IASI) of a given graph G is an injective function f : V (G) ! P(N0) such that the induced function f + : E(G) ! P(N0) de…ned by f + (uv) = f (u) + f (v) is also injective. An IASI f of a graph G is said to be a weak IASI of G if f + (uv) = max(jf (u)j; jf (v)j) for all u; v 2 V (G). A graph which admits a weak IASI may be called a weak IASI graph. The sparing number of a graph G is the minimum number of edges with singleton set-labels required for a graph G to admit a weak IASI. In this paper, we introduce the notion of k-sieve graphs of a given graph and study their sparing numbers

Naduvath, Sudev

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A Note On The Sparing Number Of The Sieve Graphs
Of Certain Graphs
Sudev Naduvath, Augustine Germina
To cite this version:
Sudev Naduvath, Augustine Germina. A Note On The Sparing Number Of The Sieve Graphs Of
Certain Graphs. Applied Mathematics E-Notes, 2015. ￿hal-02263323￿
Applied Mathematics E-Notes, 15(2015), 29-37 c ISSN 1607-2510
Available free at mirror sites of http://www.math.nthu.edu.tw/amen/
A Note On The Sparing Number Of The Sieve
Graphs Of Certain Graphs
Naduvath Sudevy, Augustine Germinaz
Received 22 September 2014
Abstract
Let N0 denote the set of all non-negative integers and P(N0) be its power set.
An integer additive set-indexer (IASI) of a given graph G is an injective function
f : V (G) ! P(N0) such that the induced function f+ : E(G) ! P(N0) dened
by f+(uv) = f(u) + f(v) is also injective. An IASI f of a graph G is said to be
a weak IASI of G if
f+(uv) = max(jf(u)j; jf(v)j) for all u; v 2 V (G). A graph
which admits a weak IASI may be called a weak IASI graph. The sparing number
of a graph G is the minimum number of edges with singleton set-labels required
for a graph G to admit a weak IASI. In this paper, we introduce the notion of
k-sieve graphs of a given graph and study their sparing numbers.
1 Introduction
For all terms and denitions, not dened specically in this paper, we refer to [6, 13]
and for di¤erent graph classes, we refer to [2]. Unless mentioned otherwise, all graphs
considered here are simple, nite and have no isolated vertices.
The notion of a set-valued graph has been introduced in [1] as a graph, the labels
of whose vertices and edges are the subsets of a given set. Since then, several studies
have been done on set-valuations of graphs. The sumset of two non-empty sets A;B,
denoted by A+B, is dened as A+B = fa+ b : a 2 A; b 2 Bg. Using the terminology
of sumsets of sets, the notion of an integer additive set-indexer of a given graph is
introduced in [4] as follows.
Let N0 denote the set of all non-negative integers and P(N0) be its power set.
An integer additive set-indexer (IASI, in short) of a graph G is an injective function
f : V (G) ! P(N0) such that the induced function f+ : E(G) ! P(N0) dened by
f+(uv) = f(u) + f(v) is also injective.
The cardinality of the set-label of an element (vertex or edge) of a graph G is called
the set-indexing number of that element.
Mathematics Subject Classications: 05C78.
yDepartment of Mathematics, Vidya Academy of Science and Technology, Thalakkottukara,
Thrissur-680501, Kerala, India
zPG & Research Department of Mathematics, Mary Matha Arts & Science College, Mananthavady,
Wayanad-670645, Kerala, India
29
30 The Sparing Number of the Sieve Graphs of Certain Graphs
LEMMA 1.1 ([5]). Let A and B be two non-empty nite sets of non-negative
integers. Then max (jAj ; jBj)  jA+Bj  jAj jBj. Therefore, for any integer additive
set-indexer f of a graph G, we have
max (jf(u)j ; jf(v)j)  f+(uv) = jf(u) + f(v)j  jf(u)j jf(v)j ;
where uv 2 E(G).
DEFINITION 1.2 ([5]). An IASI f of a graph G is said to be a weak IASI iff+(uv) = jf(u) + f(v)j = max(jf(u)j; jf(v)j)
for all u; v 2 V (G). A graph which admits a weak IASI is called a weak IASI graph. A
weak IASI f is said to be weakly k-uniform IASI if jf+(uv)j = k, for all u; v 2 V (G)
and for some positive integer k.
If A and B are two non-empty sets of non-negative integers, then jA+Bj = jAj
if and only if jBj = 1 and jA+Bj = jBj if and only if jAj = 1. Hence, we have the
following Theorem 1.3.
THEOREM 1.3 ([5]). A graph G admits a weak IASI if and only if at least one end
vertex of every edge of G has a singleton set-label.
DEFINITION 1.4 ([8]). A mono-indexed element (a vertex or an edge) of an IASI
graph G is an element of G whose set-indexing number is 1. The sparing number of a
graph G is dened to be the minimum number of mono-indexed edges required for G
to admit a weak IASI and is denoted by '(G).
THEOREM 1.5 ([8]). An odd cycle Cn contains an odd number of mono-indexed
edges and an even cycle contains an even number of mono-indexed edges.
THEOREM 1.6 ([8]). The sparing number of an odd cycle Cn is 1 and that of an
even cycle is 0.
THEOREM 1.7 ([8]). The sparing number of a bipartite graph is 0.
THEOREM 1.8 ([8]). The sparing number of a complete graphKn is 12 (n 1)(n 2).
Now, recall the denition of graph powers.
DEFINITION 1.9 ([3]). The r-th power of a simple graph G is the graph Gr whose
vertex set is V , two distinct vertices being adjacent in Gr if and only if their distance
in G is at most r. The graph G2 is referred to as the square of G, the graph G3 as the
cube of G.
The following is an important theorem on graph powers.
N. K. Sudev and K. A. Germina 31
THEOREM 1.10 ([12]). If d is the diameter of a graph G, then Gd is a complete
graph.
The admissibility of weak IASIs by certain graph classes and graph powers and the
determination of their corresponding sparing numbers have been done in [10, 11, ?].
The admissibility of weak IASIs by the graph operations and certain graphs associated
with the given IASI graphs have been discussed in [7] and [9]. As a continuation to
these studies, in this paper, we discuss the sparing number of a particular type of
graphs obtained by adding some edges to the given graphs according to certain rules.
2 Sparing Number of the k-Sieve of a Graph
Motivated by the terminology of graph powers, we introduce the notion of a k-sieve of
a given graph as follows.
DEFINITION 2.1. A k-sieve graph or simply a k-sieve of a given graph G, denoted
by G(k), is the graph obtained by joining the non-adjacent vertices of G which are at
a distance k. A cycle obtained by joining two vertices of G, which are at a distance k
in G, is called a k-ringlet of the graph G.
Note that every k-ringlet of a graph is a cycle of length k + 1. The number of
k-ringlets in a graph G is the number of distinct k-paths in G. The number of edges
in G(k) that are not in G is the number of k-ringlets in G.
REMARK 2.2. Note that G(2) = G2, the square of the graph G. But, for k > 2,
G(k) and Gk are non-isomorphic graphs. The sparing number of the square of certain
graphs have already been studied and communicated. Hence, in this paper, we need
to consider k  3.
PROPOSITION 2.3. Let l be the length of a maximal path in G. If k > l, then the
sparing number of the k-sieve of G is equal to the sparing number of G itself.
PROOF. Given that l is the length of a maximal path in G. Hence, for any pair
of vertices x; y in G, d(x; y)  l. That is, there exists no vertex in G which is at a
distance k from another vertex of G. Therefore, if k > l, then G(k) = G. Hence,
'(G(k)) = '(G).
Invoking the above result, we need to consider the integral values between 3 and l,
including both, for k. If k = l, then the longest path of G becomes a cycle of length
l + 1 in G(k).
We now proceed to determine the sparing number of the sieve graphs of certain
other standard graphs. Let us begin with the path graphs.
THEOREM 2.4. Let Pn be a path of length n. Then, for odd integers k with
2 < k  n, the sparing number of P (k)n is 0 and for even integer k with 2 < k  n, the
32 The Sparing Number of the Sieve Graphs of Certain Graphs
sparing number of P (k)n is
'(P (k)n ) =
8><>:
2(lk + r)  3 if (lk + r)k + s = n where s  r   2 and r  2;
2(lk + r)  2 if (lk + r)k + s = n where s = r   1 and r  1;
2(lk + r)  1 if (lk + r)k + s = n where s = r and r  0:
where l; k and s are non-negative integers.
PROOF. Let Pn be a path on n + 1 vertices. Let V = fv1; v2; v3; : : : ; vn; vn+1g
be the vertex set of Pn. The proof is developed considering various possible cases as
below.
Case 1 Let n be an odd integer. Label the vertices of Pn alternately by distinct singleton
sets and distinct non-singleton sets. Then, each vertex vi is adjacent to vi 1; vi+1
and to the vertex vi+k in P
(k)
n . Then, we have the following subcases.
Case 1-1 If k = n, then by Proposition 2.3, P (k)n = Cn+1. Since n is odd, P
(k)
n is even
cycle. Then, by Theorem 1.8, P (k)n has no mono-indexed edges.
Case 1-2 Let k < n. Then, under the set-labeling we dened above, no two adjacent
vertices simultaneously have singleton set-labels or non-singleton set-labels.
Therefore, no edge in P (k)n has no mono-indexed edges. Therefore, P
(k)
n has
no mono-indexed edges if k  n is an odd integer.
Case 2 Let n be an even integer.
Case 2-1 If k = n, then by Proposition 2.3, P (k)n = Cn+1. Since n is even, P
(k)
n is
odd cycle. Then, by Theorem 1.8, P (k)n has at least one mono-indexed edge.
That is, '(P (k)n ) = 1.
Case 2-2 Let k < n. Then, there exists two integers r and s such that rk + s  n,
where 0  s < k and r = 0; 1; 2; : : :. We can label the vertices in such a way
that no two adjacent vertices have non-singleton set-labels in the following
way.
Label the vertices v1; v3; v5; : : : ; vk 1 of P
(k)
n by distinct non-singleton sets and label
v2; v4; v6; : : : ; vk by distinct singleton sets. Since vk+1 is adjacent to v1, vk+1 can be
labeled only by a singleton set which is not used for labeling any one of the preceding
vertices. Then, the edge vkvk+1 is a mono-indexed edge. If n  2k, then the only
mono-indexed edge in P (k)n is vkvk+1.
If n > 2k, label the vertices v1; v2; : : : ; vk+1 as mentioned above and proceed labeling
vk+2; vk+4; : : : ; v2k by distinct non-singleton sets, that are not used for labeling before,
and the vertices vk+2; vk+4; : : : ; v2k by distinct singleton sets that are not used before
for labeling. Then, vk+1v2k+1 is a mono-indexed edge. Since vk+2 is adjacent to v2k+2;
the vertex v2k+2 must be mono-indexed. Therefore, v2k+1v2k+2 is also a mono-indexed
edge. Proceeding in this way, we arrive at the following cases.
N. K. Sudev and K. A. Germina 33
Case 1 Let s  r   2. Then, r  2. Now, we can nd a path P 0 : vkvk+1v2k+1v2k+2
v3k+2v3k+3 : : : : : : v(r 1)k+(r 2)v(r 1)k+(r 1), all of whose elements have singleton
set-labels, containing all the mono-indexed edges of P (k)n . The length of the path
P 0 is 2r   3.
Case 2 If s = r   1, then r  1. Now, there exists a path
P 00 : vkvk+1v2k+1v2k+2v3k+2v3k+3 : : : : : : v(r 1)k+(r 1)vrk+(r 1);
all of whose edges are mono-indexed, containing all the mono-indexed edges of
P
(k)
n . The length of the path P 00 is 2r   2.
Case 3 If s = r, then the path
P 000 : vkvk+1v2k+1v2k+2v3k+2v3k+3 : : : vrk+(r 1)v(r+1)k;
all of whose edges are mono-indexed, containing all the mono-indexed edges in
P
(k)
n . Therefore, the length of P 000 is 2r   1.
If r = s = k, then rk+ s = (k+1)k and if r = lk and s = k, then rk+ s = (lk+1)k
and hence we can proceed the labeling procedure in the same manner as explained
above. Then, the result follows.
Figure 1 illustrates a weak IASI for the 4-sieve of the path of length 16. The
mono-indexed edges are represented in dotted lines.
Figure 1: 4-sieve of a path with a weak IASI dened on it.
The above theorem arouses an interest in determining the sparing number of the
k-sieve of a cycle. Here, note that a maximal path between any pair of vertices in a
cycle Cn is bn2 c. Therefore, a k-sieve exists for Cn if and only if n  2k+1. Moreover,
Cn(k) is a 4-regular graph. Then, we have the following results.
34 The Sparing Number of the Sieve Graphs of Certain Graphs
THEOREM 2.5. Let Cn be a Cycle that admits a weak IASI. Then, for an odd
integer k, 1 < k  l,
'(C(k)n ) =
(
0 if Cn is an even cycle,
k + 1 if Cn is an odd cycle;
where l is the length of a largest path in G.
PROOF. Let k be an odd integer. Then, every k-subcycle of C(k)n , obtained by
joining the vertices of Cn which are at a distance k in Cn, is an even cycle of length
k + 1. Let C 0 be such an even cycle of length k + 1 in C(k)n , which has exactly one
edge, say e0, which is not in Cn. If Cn is an even cycle, then by Theorem 1.6, it need
not contain mono-indexed edges. Therefore, as a result of Theorem 1.5, e0 can not be
mono-indexed. Therefore, C 0 does not contain any mono-indexed edges. If Cn is an odd
cycle, then by Theorem 1.6, it must have one mono-indexed edge. If C 0 contains this
mono-indexed edge of Cn, then as a result of Theorem 1.5, e0 must be mono-indexed.
There exist such k cycles containing this mono-indexed edge of Cn. Therefore, the
sparing number of C(k)n is k + 1. The proof is complete.
Figure 2 illustrates Theorem 2. The rst subgure is the 3-sieve of an even cycle
with a weak IASI dened on it and the second subgure is 3-sieve of an odd cycle with
a weak IASI on it. Mono-indexed edges in the second graph are represented by dotted
lines.
Figure 2: 3-sieve of C12 and a 3-sieve of C12 with weak IASIs dened on them.
Next let us consider the case when k is an even integer.
THEOREM 2.6 Let Cn be a cycle of length n. For an even integer k; 2 < k  n,
the sparing number of C(k)n is
'(P (k)n ) =
8><>:
3 if n = 2k;
2
h
(lk + r)  2
j
(l 1)k+(r 1)
2
ki
if n = lk + r;
2l if n = l (k + 1) ;
N. K. Sudev and K. A. Germina 35
where l; k and r are non-negative integers such that l  2 and r < l.
PROOF. First let n = 2k. Then C(k)n is a cubic graph. Let us begin the labeling
process by labeling the rst vertex v1 by a non-singleton set and then label the following
vertices alternatively by distinct singleton sets and distinct non-singleton sets. Then,
the vertex vk is a mono-indexed vertex. Being adjacent to the vertex v1, vk+1 must also
be mono-indexed. That is, the edge vkvk+1 is mono-indexed. Now, label the vertex vk+2
by a non-singleton set and then label the following vertices alternatively by distinct
singleton sets and distinct non-singleton sets. Here, the vertex v2k 1 is mono-indexed.
Since, v2k is adjacent to the vertex v1, v2k must be mono-indexed. Therefore, the edge
v2k 1v2k is mono-indexed. Also, the edge vkv2k is also mono-indexed. Therefore, the
number of mono-indexed edges in this case is 3.
Note that if n > 2k the k-sieve of every cycle is a 4-regular graph, for any integer
k. Then we have the following cases.
Case 1 Assume that n = lk + r; r < l, l and r being positive integers and l  2. Then,
the total number of edges in C(k)n isE(C(k)n ) = 12X d(v) = 2(lk + r):
Now, label the vertex v1 by a non-singleton set and then label the remaining
vertices by distinct singleton sets and distinct non-singleton sets such that no two
adjacent vertices having non-singleton set-labels. Then, the last k + 1 vertices
must be 1-uniform, as each of them are adjacent to one vertex having a non-
singleton set-label. Out of the remaining (l 1)k+(r 1) vertices, b (l 1)k+(r 1)2 c
vertices have non-singleton set-label. The number of edges that are not mono-
indexed is 4 b (l 1)k+(r 1)2 c. The total number of mono-indexed edges is 2[(lk +
r)  2 b (l 1)k+(r 1)2 c].
Case 2 Assume that n = l(k + 1), l being a positive integer. Let C be a partition of
V (G), where each set in C contains exactly k + 1 vertices. Therefore, each set
in C consists of exactly k2 vertices have non-singleton set-labels and 1 +
k
2 mono-
indexed vertices in C(k)n . Therefore, the number of vertices having non-singleton
set-labels is l k2 . Therefore, the number of edges that are not mono-indexed is 2lk.
The number edges in C(k)n isE(C(k)n ) = 12X d(v) = 2l(k + 1):
Therefore, the number of mono-indexed edges in C(k)n is 2l(k + 1)  2lk = 2l.
Figure 3 illustrates a weak IASI for the 4-sieve of a cycle on 20 vertices.
36 The Sparing Number of the Sieve Graphs of Certain Graphs
Figure 3: 4-sieve of C20 which is a weak IASI graph.
3 Conclusion
It can be observed that a complete graph Kn can have a k-sieve graph as every vertex
of Kn is at a distance 1 from all other vertices of G. Similarly, a complete bipartite
graph Km;n (or a complete r-partite graph, Kn1;n2;:::;nr , for r > 2) also does have
a k-sieve graph, for k  3, as any two vertices in Km;n (or in Kn1;n2;:::;nr )are at a
distance at most 2.
Let k be an odd integer. If G be a tree, then G(k) is a graph all of whose cycles
are of length k + 1, an even integer. Then, G(k) is a bipartite graph. Therefore, by
Theorem 1.7, the number of mono-indexed edges in G(k) is also 0.
If G be a bipartite graph (containing cycles), then G has no odd cycles. Then, since
every k-ringlet of G is an even cycle, every cycle in G(k) is of even length. Therefore,
G(k) is also a bipartite graph, for odd k. Therefore, by Theorem 1.7, the number of
mono-indexed edges in G(k) is also 0.
But, for even integers k, to determine the sparing number of G(k), for a graph G
which is bipartite (cyclic or acyclic), we need to use Theorem 2.4, Theorem 2.5 and
Theorem 2.6, for distinct paths and cycles in G.
Evaluating the sparing number of the k-sieves of bipartite graphs, Eulerian graphs,
armed crown graphs etc. are some of the open problems in this area. Determining the
sparing number of the k-sieves of graph operations and graph products are also worth
for further exploration.
More properties and characteristics of weak IASIs, both uniform and non-uniform,
are yet to be investigated. The problems of establishing the necessary and su¢ cient
conditions for various graphs and graph classes to have certain IASIs still remain to be
settled. All these facts highlight a great scope for further studies in this area.
Acknowledgement
The authors gratefully acknowledge the critical comments and suggestions of the
N. K. Sudev and K. A. Germina 37
anonymous referee, which helped to improve the quality of the paper.
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A Note On The Sparing Number Of The Sieve Graphs Of Certain Graphs

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