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ISSN: 2347
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Available Online: http://ijmaa.in/
on
ISSN: 2347-1557
d its Applic
Volume 3, Issue 2 (2015), 161–168.
An
al
ou
cs
International Journal of Mathematics And its Applications
International Journal of Mathematics And its Applications
On Construction of Even Vertex Odd Mean Graphs
Research Article
G.Pooranam1 , R.Vasuki2∗ and S.Suganthi3
1 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.
2 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.
3 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.
Abstract:
A
graph
G
with
p
vertices
and
q
edges
is
said
to
have
an
even vertex odd mean labeling if there exists an injective function f : V (G) → {0, 2, 4, . . . , 2q − 2, 2q} such that
f (u)+f (v)
is a bijection. A graph that admits an
the induced map f ∗ : E(G) → {1, 3, 5, . . . , 2q − 1} defined by f ∗ (uv) =
2
even vertex odd mean labeling is called an even vertex odd mean graph. In this paper we discuss the construction of two
kinds of even vertex odd mean graphs. Here we prove that (Pn ; S1 ) for n ≥ 1, (P2n ; Sm ) for m ≥ 2, n ≥ 1, (Pm ; Cn ) for
(2)
n ≡ 0(mod 4), m ≥ 1, [Pm ; Cn ] for n ≡ 0(mod 4), m ≥ 1 and [Pm ; Cn ] for n ≡ 0(mod 4), m ≥ 1 are even vertex odd
mean graphs.
MSC:
05C78.
Keywords: Labeling, even vertex odd mean labeling, even vertex odd mean graph.
c JS Publication.
1.
Introduction
Throughout this paper, by a graph we mean a finite, undirected and simple graph. Let G(V, E) be a graph with p vertices
and q edges. For notations and terminology we follow [4]. Path on n vertices is denoted by Pn and a cycle on n vertices is
denoted by Cn . The graph P2 × P2 × P2 is called a cube and is denoted by Q3 . Let Cn be a cycle with fixed vertex v and
(Pm ; Cn ) the graph obtained from m copies of Cn and the path Pm : u1 , u2 , . . . , um by joining ui with the vertex v of the
ith copy of Cn by means of an edge for 1 ≤ i ≤ m.
Let Sm be a star with vertices v0 , v1 , v2 , . . . , vm and let (P2n ; Sm ) be the graph obtained from 2n copies of Sm and the path
P2n : u1 , u2 , . . . , u2n by joining uj with the vertex v0 of the j th copy of Sm by means of an edge, for 1 ≤ j ≤ 2n, (Pn ; S1 ) the
graph obtained from n copies of S1 and the path Pn : u1 , u2 , . . . , un by joining uj with the vertex v0 of the j th copy of S1
by means of an edge, for 1 ≤ j ≤ n. Suppose Cn : v1 v2 . . . vn v1 be a cycle of length n. Let [Pm ; Cn ] be the graph obtained
from m copies of Cn with vertices v11 , v12 , . . . , v1n , v21 , . . . , v2n , . . . , vm1 , . . . , vmn and joining vij and v(i+1)j by means of
an edge, for some j and 1 ≤ i ≤ m − 1.
∗
E-mail: [email protected]
161
On Construction of Even Vertex Odd Mean Graphs
(2)
(2)
(2)
Let Cn be a friendship graph. Define [Pm ; Cn ] to be the graph obtained from m copies of Cn and the path Pm : u1 u2 . . . um
(2)
by joining ui with the center vertex of the ith copy of Cn
for 1 ≤ i ≤ m. The graceful labelings of graphs was first
introduced by Rosa in 1961 [1] and R.B. Gnanajothi introduced odd graceful graphs [3]. The concept of mean labeling was
first introduced by S. Somasundaram and R. Ponraj [7]. Further some more results on mean graphs are discussed in [6, 8, 9].
A graph G is said to be a mean graph if there exists an injective function f from V (G) to {0, 1, 2, . . . , q} such that the
l
m
(v)
induced map f ∗ from E(G) to {1, 2, 3, . . . , q} defined by f ∗ (uv) = f (u)+f
is a bijection.
2
In [5], K. Manickam and M. Marudai introduced odd mean labeling of a graph. A graph G is said to be odd mean
if there exists an injective function f from V (G) to {0, 1, 2, 3, . . . , 2q − 1} such that the induced map f ∗ from E(G) to
l
m
(v)
{1, 3, 5, . . . , 2q − 1} defined by f ∗ (uv) = f (u)+f
is a bijection. The concept of even mean labeling was introduced and
2
studied by B. Gayathri and R. Gopi [2]. A function f is called an even mean labeling of a graph G with p vertices and q
edges if f is an injection from the vertices of G to the set {2, 4, . . . , 2q} such that when each edge uv is assigned the label
f (u)+f (v)
,
2
then the resulting edge labels are distinct. A graph which admits an even mean labeling is said to be even mean
graph.
Motivated by these, R. Vasuki et al. introduced the concept of even vertex odd mean labeling [10] and discussed the even
vertex odd mean behaviour of some standard graphs. A graph G with p vertices and q edges is said to have an even
vertex odd mean labeling if there exists an injective function f : V (G) → {0, 2, 4, . . . , 2q − 2, 2q} such that the induced map
f ∗ E(G) → {1, 3, 5, . . . , 2q − 1} defined by f ∗ (uv) =
f (u)+f (v)
2
is a bijection. A graph that admits an even vertex odd mean
labeling is called an even vertex odd mean graph. An even vertex odd mean labeling of K2,5 is shown in Figure 1.
0
20
b
b
b
2
6
b
b
10
b
14
b
18
Figure 1.
In this paper, we prove that, the graphs (Pn ; S1 ) for n ≥ 1, (P2n ; Sm ) for m ≥ 2, n ≥ 1, (Pm ; Cn ) for n ≡ 0(mod 4), m ≥ 1,
(2)
[Pm ; Cn ] for n ≡ 0(mod 4), m ≥ 1, and [Pm ; Cn ] for n ≡ 0(mod 4), m ≥ 1 are even vertex odd mean graphs.
2.
Even Vertex Odd Mean Graphs (Pm ; G)
Let G be a graph with fixed vertex v and let (Pm ; G) be the graph obtained from m copies of G and the path Pm :
u1 , u2 , . . . , um by joining ui with the vertex v of the ith copy of G by means of an edge for 1 ≤ i ≤ m.
Theorem 2.1. (P2n ; Sm ) is an even vertex odd mean graph, m ≥ 2, n ≥ 1.
Proof.
Let u1 , u2 , . . . , u2n be the vertices of P2n . Let v0j , v1j , v2j , . . . , vmj be the vertices of the j th copy of Sm , where v0j
is the center vertex, 1 ≤ j ≤ 2n.
162
G.Pooranam, R.Vasuki and S.Suganthi
Define f : V (P2n ; Sm ) → {0, 2, 4, . . . , 2q − 2, 2q = 4n(m + 2) − 2} as follows:
f (uj ) =
f (v0j ) =
f (vij ) =


 (2m + 4)(j − 1) + 2,

 (2m + 4)j − 4,


 (2m + 4)(j − 1),
1 ≤ j ≤ 2n and j is odd
1 ≤ j ≤ 2n and j is even
1 ≤ j ≤ 2n and j is odd

 (2m + 4)j − 2,
1 ≤ j ≤ 2n and j is even


 (2m + 4)(j − 1) + 4i + 2,
1 ≤ i ≤ m, 1 ≤ j ≤ 2n and j is odd

 (2m + 4)(j − 2) + 4i,
1 ≤ i ≤ m, 1 ≤ j ≤ 2n and j is even.
For the vertex labeling f, the induced edge labeling f ∗ is obtained as follows:
f ∗ (uj uj+1 ) = (2m + 4)j − 1, 1 ≤ j ≤ 2n − 1


 (2m + 4)(j − 1) + 1,
1 ≤ j ≤ 2n and j is odd
f ∗ (uj v0j ) =

 (2m + 4)j − 3,
1 ≤ j ≤ 2n and j is even



(2m + 4)(j − 1) + 2i + 1,
1 ≤ i ≤ m, 1 ≤ j ≤ 2n






and j is odd
f ∗ (v0j vij ) =


(2m + 4)(j − 1) + 2i − 1,
1 ≤ i ≤ m, 1 ≤ j ≤ 2n






and j is even.
It can be verified that f is an even vertex odd mean labeling and hence (P2n ; Sm ) is an even vertex odd mean graph for
n ≥ 1 and m ≥ 2. For example, an even vertex odd mean labeling of (P6 ; S4 ) is shown in Figure 2.
b
0
b
6
Figure 2.
b
26
20
2
b
b
b
b
10 14
b
b
b
18
4
8
44
b
22
b
b
b
12 16
b
b
24
b
b
b
b
30 34 38 42
b
46
b
68
50
b
b
28 32 36
b
b
40
b
b
b
48
b
b
b
54 58 62 66
b
70
b
b
b
52 56 60 64
(P6 ; S4 )
Theorem 2.2. The graph (Pn ; S1 ), n ≥ 1 is an even vertex odd mean graph.
Proof.
Let u1 , u2 , . . . , un be the vertices of Pn . Let v0j , and v1j be the vertices.
Define f : V (Pn ; S1 ) → {0, 2, 4, . . . , 2q − 2, 2q = 6n − 2} as follows:


 6j − 6,
f (uj ) =

 6j − 2,
1 ≤ j ≤ n and j is odd
1 ≤ j ≤ n and j is even
f (v0j ) = 6j − 4, 1 ≤ j ≤ n


 6j − 2,
1 ≤ j ≤ n and j is odd
f (v1j ) =

 6j − 6,
1 ≤ j ≤ n and j is even.
163
On Construction of Even Vertex Odd Mean Graphs
The induced edge labels are obtained as follows:
f ∗ (uj uj+1 ) = 6j − 1, 1 ≤ j ≤ n − 1


 6j − 5,
1≤j≤n
f ∗ (uj v0j ) =

 6j − 3,
1≤j≤n


 6j − 3,
1≤j≤n
f ∗ (v0j v1j ) =

 6j − 5,
1≤j≤n
and j is odd
and j is even
and j is odd
and j is even.
Thus, f is an even vertex odd mean labeling. Hence (Pn ; S1 ) is an even vertex odd mean graph for any n ≥ 1. For example,
an even vertex odd mean labeling of (P7 ; S1 ) is shown in Figure 3.
0
b
2
Figure 3.
10
b
8
b
b
b
b
4
6
12
b
20
14
b
b
26
b
b
b
16
34
24
22
b
b
32
b
b
b
38
b
b
28
18
36
b
b
30
40
(P7 ; S1 )
Theorem 2.3. (Pm ; Cn ) is an even vertex odd mean graph, for n ≡ 0(mod 4) and m ≥ 1.
Proof.
Let vi1 , vi2 , . . . , vin be the vertices in the ith copy of Cn , 1 ≤ i ≤ m and u1 , u2 , . . . , um be the vertices of Pm . In
(Pm ; Cn ), ui is joined with vi1 by an edge, for each i, 1 ≤ i ≤ m.
Define f : V (Pm ; Cn ) → {0, 2, 4, . . . , 2q − 2, 2q = (2n + 4)m − 2} as follows.
f (ui ) =


 2(n + 2)(i − 1),
1 ≤ i ≤ m and i is odd

 2(n + 2)i − 2,
1 ≤ i ≤ m and i is even.
For 1 ≤ i ≤ m and i is odd,



2(n + 2)(i − 1) + 2j,



f (vij ) =
2(n + 2)(i − 1) + 2j + 4,




 2(n + 2)(i − 1) + 2j,
1≤j≤
n
2
n
2
+ 1 ≤ j ≤ n and j is odd
n
2
+ 2 ≤ j ≤ n and j is even.
For 1 ≤ i ≤ m and i is even,



2(n + 2)i − 2(j + 1),



f (vij ) =
2(n + 2)i − 2(j + 3),





2(n + 2)i − 2(j + 1),
164
1≤j≤
n
2
n
2
+ 1 ≤ j ≤ n and j is odd
n
2
+ 2 ≤ j ≤ n and j is even.
G.Pooranam, R.Vasuki and S.Suganthi
The induced edge labels are obtained as follows:
f ∗ (ui ui+1 ) = 2i(n + 2) − 1,
1 ≤ i ≤ m − 1.
For 1 ≤ i ≤ m and i is odd,


 2(n + 2)(i − 1) + 2j + 1,
f ∗ (vij vi(j+1) ) =
 2(n + 2)(i − 1) + 2j + 3,

1≤j≤
n
2
n
2
−1
≤j ≤n−1
f ∗ (vin vi1 ) = 2(n + 2)(i − 1) + n + 1
f ∗ (ui vi1 ) = 2(n + 2)(i − 1) + 1.
For 1 ≤ i ≤ m and i is even,
f ∗ (vij vi(j+1) ) =


 2(n + 2)i − (2j + 3),
1≤j≤

 2(n + 2)i − (2j + 5),
n
2
n
2
−1
≤j ≤n−1
f ∗ (vin vi1 ) = 2(n + 2)i − n − 3
f ∗ (ui vi1 ) = 2(n + 2)i − 3.
Thus, f is an even vertex odd mean labeling and hence (Pm ; Cn ) is an even vertex odd mean graph for n ≡ 0(mod 4) and
m ≥ 1. For example, an even vertex odd mean labeling of (P4 ; C8 ) is shown in Figure 4.
0
b
3.
2
18
12
Figure 4.
b
b
4
b
6
14
b
b
8
b
b
22
b
26
b
36
20
b
b
34
32
24
b
78
b
b
b
16
40
38
b
b
b
30
b
56
b
b
42
b
b
44
46
58
52
b
54
b
b
b
b
48
b
62
b
76
60
66
b
b
74
72
64
b
b
b
70
(P4 ; C8 )
Even Vertex Odd Mean Graphs [Pm ; G]
Let G be a graph with fixed vertex v and let [Pm ; G] be the graph obtained from m copies of G by joining vij and v(i+1)j
by means of an edge for some j and 1 ≤ i ≤ m − 1.
Theorem 3.1. [Pm ; Cn ] is an even vertex odd mean graph, for n ≡ 0(mod 4) and m ≥ 1.
Proof.
Let u1 , u2 , . . . , um be the vertices of Pm . Let vi1 , vi2 , . . . , vin be the vertices of the ith copy of Cn , 1 ≤ i ≤ m and
joining vij (= ui ) and v(i+1)j (= ui+1 ) by means of an edge for some j.
Define f : V ([Pm ; Cn ]) → {0, 2, 4, . . . , 2q − 2, 2q = (2n + 2)m − 2} as follows:
165
On Construction of Even Vertex Odd Mean Graphs
For 1 ≤ i ≤ m and i is odd,



2(n + 1)(i − 1) + 2j − 2,



f (vij ) =
2(n + 1)(i − 1) + 2j + 2,





2(n + 1)(i − 1) + 2j − 2,
n
2
1≤j≤
n
2
+ 1 ≤ j ≤ n and j is odd
n
2
+ 2 ≤ j ≤ n and j is even.
For 1 ≤ i ≤ m and i is even,



2(n + 1)i − 2j,



f (vij ) =
2(n + 1)i − 2(j + 2),





2(n + 1)i − 2j,
n
2
1≤j≤
n
2
+ 1 ≤ j ≤ n and j is odd
n
2
+ 2 ≤ j ≤ n and j is even.
The induced edge labels are obtained as follows:
f ∗ (vi1 v(i+1)1 ) = 2(n + 1)i − 1,
1≤i≤m−1
For 1 ≤ i ≤ m and i is odd,
f ∗ (vij v(i+1)j ) =


 2(n + 1)(i − 1) + 2j − 1,
1≤j≤

 2(n + 1)(i − 1) + 2j + 1,
n
2
n
2
≤j ≤n−1
f ∗ (vin vi1 ) = 2(n + 1)i − (n + 3).
For 1 ≤ i ≤ m and i is even,
f ∗ (vij v(i+1)j ) =


 2(n + 1)i − 2j − 1,
1≤j≤

 2(n + 1)i − 2j − 3,
n
2
n
2
−1
≤j ≤n−1
f ∗ (vin vi1 ) = 2(n + 1)i − (n + 1).
Thus, f is an even vertex odd mean labeling and hence, [Pm ; Cn ] is an even vertex odd mean graph for n ≡ 0(mod 4) and
m ≥ 1. For example, an even vertex odd mean labeling of [P5 ; C8 ] is shown in Figure 5.
0
34
b
b
2
14
16
b
b
b
Figure 5.
10
36
b
b
b
6
b
4 18
20 32
b
b
b
24
28
b
30 52
b
50 38
42
b
b
b
b
12
22
48
40 54
b
b
68
56
b
46
b
b
b
b
b
72
70
b
b
b
b
b
b
64
60
b
66
b
b
82 78
b
58
b
84
[P5 ; C8 ]
(2)
Theorem 3.2. [Pm ; Cn ] is an even vertex odd mean graph, for n ≡ 0(mod 4) and m ≥ 1.
166
b
86 74
88
b
b
76
G.Pooranam, R.Vasuki and S.Suganthi
Proof.
Let u1 , u2 , . . . , um be the vertices of Pm and each vertex ui , 1 ≤ i ≤ m is attached with the common vertex in the
(2)
(2)
ith copy of Cn . Let vi0j and vi00j for 1 ≤ j ≤ n be the vertices in the ith copy of Cn , in which vi01 = vi001 , 1 ≤ i ≤ m.
(2)
Define f : V ([Pm ; Cn ]) → {0, 2, 4, . . . , 2q − 2, 2q = (4n + 2)m − 2} as follows:
For 1 ≤ i ≤ m,



(4n + 2)i − 2(n + j),





 (4n + 2)(i − 1) + 2j − 6,
f (vi0j ) =
1≤j≤2
3≤j≤


(4n + 2)(i − 1) + 2j − 2,





 (4n + 2)(i − 1) + 2j − 6,



(4n + 2)i − 2(n − j + 2),



00
f (vij ) =
(4n + 2)i − 2n + 2j,





(4n + 2)i − 2(n − j + 2),
n
2
+2
n
2
+ 3 ≤ j ≤ n and j is odd
n
2
+ 3 ≤ j ≤ n and j is even
2≤j≤
n
2
n
2
+ 1 ≤ j ≤ n and j is odd
n
2
+ 2 ≤ j ≤ n and j is even
The induced edge labels are obtained as follows:
For 1 ≤ i ≤ m,


 (4n + 2)(i − 1) + 2j − 5,
0
)
=
f ∗ (vi0j v(i+1)
j

 (4n + 2)(i − 1) + 2j − 3,
3≤j≤
n
2
n
2
+1
+2≤j ≤n
f ∗ (vi01 vi02 ) = (4n + 2)i − (2n + 3)
f ∗ (vi02 vi03 ) = (4n + 2)i − 3(n + 1)


 (4n + 2)i − 2n + 2j − 5,
∗ 00 00
f (vij v(i+1)j ) =

 (4n + 2)i − 2n + 2j − 1,
1≤j≤
n
2
n
2
−1
≤j≤n
f ∗ (vi00n vi001 ) = (4n + 2)i − (n + 3).
(2)
Thus, f is an even vertex odd mean labeling. Hence, [Pm ; Cn ] is an even vertex odd mean graph for n ≡ 0(mod 4) and
(2)
m ≥ 1. For example, an even vertex odd mean labeling of [P4 ; C8 ] is shown in Figure 6.
62
28
b
b
b
22 26
b
20
b
b
18
32 54
b
b
b
50
14
10
12
b
b
2
b
4
Figure 6.
b
b
6
b
66 88
b
64
b
0
46
b
38
b
34 80
b
74 70
b
72
b
bb
b
b
112 116
b
b
b
b
78 82
b
b
124 128
134
122
120 132
118
b
b
b
b
b
b
b
bb
44 48
40 36
100
86 98
84
b
b
bb
b
b
b
b
90 94
b
bb
b
b
b
b
56 60
52
16 30
130
96
b
b
b
68 114
b
b
102
108 104
b
b
b
106
(2)
[P4 ; C8 ]
167
On Construction of Even Vertex Odd Mean Graphs
References
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149-153.
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273-284.
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22(4)(2009), 663-675.
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168