ARITHMETIC ODDDECOMPOSITION [AOD] OF
SOMECLASS OF GRAPHS
V.G. SMILIN SHALI
Research Scholar, Reg. No. 12609,
Research Department of Mathematics,
Nesamony Memorial Christian College,
Marthandam, Kanyakumari District, Tamil Nadu.
Affiliated to ManonmaniamSundaranar University,
Abishekapatti, Tirunelveli-627012, Tamil Nadu,India.
S.ASHA
Assistant Professor,
Research Department of Mathematics,
Nesamony Memorial Christian College,
Marthandam, Kanyakumari District, Tamil Nadu.
Affiliated to ManonmaniamSundaranar University,
Abishekapatti, Tirunelveli-627012, Tamil Nadu,India.
AbstractBy a graph , we mean a finite,
undirected graph without loops or multiple edges.
If
1
,
2
,…,
are connected edge-
disjointsubgraphs of G with () = (
1
)∪
(
2
) ∪ … ∪ (
),then{
1
,
2
,…,
} is
said to be a decomposition of . The concept of
arithmetic odd decomposition [AOD] was
introduced by E. Ebin Raja Merly and N.
Gnanadhas. A decomposition {
1
,
2
,…,
}of
is said to be Arithmetic Decomposition [AD]if
|(
) = + ( - 1)d for every i =
1,2,..., anda,d Z
+
. An arithmetic odd
decomposition [AOD] is an arithmetic
decomposition with = 1 and = 2. We denote
the AODby{
1
,
3
,…,
2−1
}. In this paper, we
investigate AOD for some special class of graphs,
namely
,
1,
˄
2
and
˄
3
.
Keywords:Decomposition of Graph, Arithmetic
Decomposition, Arithmetic Odd Decomposition
[AOD].
Introduction
Let = ( , )be a simple connected
graph with p vertices and q edges.
If
1
,
2
,…,
are connected edge-disjoint
subgraphs of G withE(G) = E(
1
) ∪ E(
2
)∪
. .. ∪ E(
), then {
1
,
2
,…,
}is said to be a
decomposition of G. Different types of
decomposition of G have been studied in the
literature by imposing suitable conditions on the
subgraphs G
i
. The concept of Arithmetic Odd
Decomposition [AOD] was introduced by E. Ebin
Raja Merly and N. Gnanadhas. In this paper, we
investigate AOD for some special class of graphs.
Terms not defined here are used in the sense of
Harary [3].
Preliminaries
Definition 2.1. Let = ( , )be a simple graph
of order p and size q. If
1
,
2
,…,
are edge-
disjoint subgraphs of such that E(G) = E(
1
)∪
E(
2
) ∪ . .. ∪ E(
), then {
1
,
2
,…,
}is said
to be a decomposition of G.
Definition 2.2. A decomposition
1
,
2
,…,
of a
connected graph G is said to be a linear
decomposition or arithmetic decomposition if each
is connected and |(
)| = + ( − 1) , for
1≤≤ anda,d ∈ Z. Clearly =
2
[2 +
( − 1)].
Definition 2.3. When a = 1 and d = 1, the size of G
is =
(+1)
2
. When a = 1 and d = 2, the size of G is
q = n
2
. Hence the number of edges of G is a perfect
square. Since the number of edges of G is a perfect
square, q is the sum of first n odd numbers
1, 3, 5, … , (2 − 1). Thus we call the arithmetic
decomposition with a = 1 and d = 2 as Arithmetic
Odd Decomposition (AOD). Since the number of
edges of each subgraph of G is odd, we denote the
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 8, 2019 (Special Issue)
© Research India Publications. http://www.ripublication.com
Page 63 of 67