Chapter 9
AV Labeling on Disjoint Union of Star
Hypergraphs
9.1
Introduction
Sonntag (2002) proved the existence of an AV labeling on the linear h-uniform star
hypergraphs. This chapter studies the results on an AV labeling for the disjoint union
of isomorphic copies of a linear h-uniform star hypergraph under certain conditions.
Let’s begin with the definition of a linear h-uniform star hypergraph and the construction
of a graph with disjoint union of k isomorphic copies of a linear h-uniform star hypergraph.
h
Definition 9.1.1. A linear h-uniform star hypergraph S n,m
(V, E) is a simple hypergraph
with n vertices and m edges, where E = {e1 , e2 , ..., em }, ei = {v11 , vi2 , vi3 , ..., vih : 1 ≤ i ≤ m}
m
∪
and V = ei , where the vertex v11 is called center of the star hypergraph.
i=1
Definition 9.1.2. Suppose k isomorphic copies of a linear h-uniform star hypergraph
h
S n,m
(V, E). A graph G is obtained by taking disjoint union of k isomorphic copies of a
h
linear h-uniform star hypergraph. Thus, the set of vertices and edges of G kS n,m
(V, E)
∪ p
p
p
p
are defined as E(G) = {ei : 1 ≤ i ≤ m, 1 ≤ p ≤ k}, where ei = {v11 : 1 ≤ p ≤ k} {vil :
k ∪
m
∪
1 ≤ i ≤ m, 1 ≤ p ≤ k, 1 ≤ l ≤ h} and V(G) = ( eip ) such that |V(G)| = mk(h − 1) + k.
p=1 i=1
h
(V, E) with n = 10, m = 3 and
As an example, Figure 9.1 presents a star hypergraph S n,m
h = 4. The star hypergraph in Figure 9.2 contains n = 9, m = 4 and h = 3. Moreover,
Figure 9.3 shows the disjoint union of 2 isomorphic copies of a linear h-uniform star
4
h
with n = 20, m = 3 and h = 4.
hypergraph S 10,3
denoted by 2S n,m
103
9. AV Labeling on Disjoint Union of Star Hypergraphs
v23
v13
v33
v22
v12
v32
v21
v11
v31
v11
4
Figure 9.1: A linear 4-uniform star hypergraph S 10,3
.
v32
v22
v12
v42
v11
v21 v31
v41
v
11
3
Figure 9.2: A linear 3-uniform star hypergraph S 9,4
.
104
9.1 Introduction
1
v23
v113
v133
v212
1
v12
1
v32
v1
v111
21
1
v31
v1
11
2
v23
2
13
v
2
v33
v222
2
v12
v121
2
v21
2
v32
v312
v121
4
Figure 9.3: A linear 4-uniform star hypergraph 2S 20,3
.
105
9. AV Labeling on Disjoint Union of Star Hypergraphs
9.2 Main Results
h
In this section, it is proved that kS n,m
(V, E) admits an AV labeling for even k ≥ 2, m = 3
1
and m = 2 (k + 2) (Javaid and Akhlaq 2012 (b)).
h
Theorem 9.2.1. For k = 2, h = 3 and m = 3 , G kS n,m
(V, E) admits an AV labeling,
where n = 7.
Proof. Define λ : V(G) → {1, 2, ..., 14} as follows:
p
)=p
λ(v11
for 1 ≤ p ≤ 2


6l − 3p − 3



p
λ(v1l ) = 


 6l − 3p − 12


6l − 3p − 2



p
λ(v2l ) = 


 6l + 3p − 11
for
1 ≤ p ≤ 2, l = 2,
for
1 ≤ p ≤ 2, l = 3.
1 ≤ p ≤ 2 l = 2,
for
for 1 ≤ p ≤ 2 l = 3 odd.
and


6l − 3p − 1



p
λ(v3l ) = 


 6l + 3p − 10
for 1 ≤ p ≤ 2, l = 2,
for 1 ≤ p ≤ 2, l = 3.
The edge labeling is obtained by r∗ (e j ) = 15 + j for 1 ≤ j ≤ 6, which gives an AP starting
from 16 with difference 1. Consequently, λ is an AV labeling.
h
Theorem 9.2.2. For k = 4, h = 5 and m = 3 , G kS n,m
(V, E) admits an AV labeling,
where n = 13.
Proof. Define λ : V(G) → {1, 2, ..., 52} as follows:
p
λ(v11
)=p


12l − 3p − 7



p
λ(v1l ) = 


 12l + 3p − 22


12l − 3p − 6



λ(v2lp ) = 


 12l + 3p − 21
for 1 ≤ p ≤ 4
for
1 ≤ p ≤ 4, l = 2, 4
for
1 ≤ p ≤ 4, l = 3, 5.
for
1 ≤ p ≤ 4 l = 2, 4
for 1 ≤ p ≤ 4 l = 3, 5 odd.
and


12l − 3p − 5



p
λ(v3l ) = 


 12l + 3p − 20
for 1 ≤ p ≤ 4, l = 2, 4
for 1 ≤ p ≤ 4, l = 3, 5.
106
9.2 Main Results
The edge labeling is obtained by r∗ (e j ) = 110 + j for 1 ≤ j ≤ 12, which gives an AP
starting from 111 with difference 1. Consequently, λ is an AV labeling.
h
Theorem 9.2.3. For k = 6, h = 7 and m = 3 , G kS n,m
(V, E) admits an AV labeling,
where n = 19.
Proof. Define λ : V(G) → {1, 2, ..., 114} as follows:
p
λ(v11
)=p


18l − 3p − 11



p
λ(v1l ) = 


 18l + 3p − 32


18l − 3p − 10



p
λ(v2l ) = 


 18l + 3p − 31
for 1 ≤ p ≤ 6
for
1 ≤ p ≤ 6, l = 2, 4, 6
for
1 ≤ p ≤ 6, l = 3, 5, 7.
for 1 ≤ p ≤ 6, l = 2, 4, 6
for 1 ≤ p ≤ 6, l = 3, 5, 7 odd.
and


18l − 3p − 9



p
λ(v3l ) = 


 18l + 3p − 30
for 1 ≤ p ≤ 6, l = 2, 4, 6
1 ≤ p ≤ 6, l = 3, 5, 7.
for
The edge labeling is obtained by r∗ (e j ) = 357 + j for 1 ≤ j ≤ 18, which gives an AP
starting from 158 with difference 1. Consequently, λ is an AV labeling.
h
Theorem 9.2.4. For even k ≥ 2, h = k + 1 and m = 3, G kS n,m
(V, E) admits an AV
labeling, where n = 3(h − 1) + 1.
Proof. Define λ : V(G) → {1, 2, ..., |V(G)|} as follows:
p
)=p
λ(v11
f or 1 ≤ p ≤ k


3lk − 2k − 3p + 1



p
λ(v1l ) = 


 3lk − 5k + 3p − 2


3lk − 2k − 3p + 2



p
λ(v2l ) = 


 3lk − 5k + 3p − 1
for
for
for
1 ≤ p ≤ k, l = even
1 ≤ p ≤ k, l ≥ 3 odd.
1 ≤ p ≤ k, l = even,
for 1 ≤ p ≤ k, l ≥ 3 odd.
and


3lk − 2k − 3p + 3



p
λ(v3l ) = 


 3lk − 5k + 3p
for
1 ≤ p ≤ k, l = even,
for 1 ≤ p ≤ k, l ≥ 3 odd.
∑
∑
The edge labeling is obtained by r∗ (e j ) =
(3lk − 2k − 2) +
(3lk − 5k + 1) + j
l=even
l≥3 odd
for 1 ≤ j ≤ mk, which gives an AP starting from hk2 + 1 with difference 1. Consequently,
107
9. AV Labeling on Disjoint Union of Star Hypergraphs
λ is an AV labeling.
Figures 9.4, 9.5 and 9.6 explain the labeling schemes presented in theorems 9.2.1, 9.2.2
and 9.2.3. respectively. These figures also partially justify the labeling scheme presented
in 9.2.4.
10
9
11
7
8
6
1
13
12
14
4
3
5
2
3
Figure 9.4: An AV labeling of 2S 14,3
(disjoint union of 2 isomorphic copies of the linear
3
3-uniform star hypergraph S 7,3 ).
108
9.2 Main Results
42
45
41
43
44
46
39
36
40
38
18
17
14
15
35
19
37
21
20
16
11
48
51
50
49
52
30
33
34
32
24
23
8
31
29
25
27
26
10
9
13
2
1
47
12
22
5
28
7
6
4
3
5
Figure 9.5: An AV labeling of 4S 52,3
(disjoint union of 4 isomorphic copies of the linear
5
5-uniform star hypergraph S 13,3 ).
109
9. AV Labeling on Disjoint Union of Star Hypergraphs
101
98
97
94
61
58
25
22
99
95
96
63
62
59
26
100
91
64
55
28
19
60
27
23
102
92
93
66
65
56
29
57
30
20
21
24
2
1
107
104
105
89
103
88
67
90
69
68
52
31
16
53
32
54
33
17
106
85
70
108
86
87
72
71
49
34
13
50
35
51
36
14
15
18
4
3
113
110
109
82
73
46
37
10
111
83
84
48
74
47
38
54
39
11
112
79
76
43
40
7
80
1
77
81
78
44
41
45
42
8
9
12
6
5
7
Figure 9.6: An AV labeling of 6S 114,3
(disjoint union of 6 isomorphic copies of the linear
7
7-uniform star hypergraph S 19,3 ).
110
9.2 Main Results
h
Theorem 9.2.5. For even k = 2, h = 3 and m = 2, G kS n,m
(V, E) admits an AV labeling,
where n = 5.
Proof. Let us consider throughout the labeling 1 ≤ p ≤ 2 and 1 ≤ i ≤ 2. Define
λ : V(G) → {1, 2, ..., 10} as follows:
p
λ(v11
) = p.


4l − 2p − 2 + i



p
λ(vil ) = 


 4l + 2p − 8 + i
for l = 2,
for l = 3.
The edge labeling is obtained by r∗ (e j ) = 12 + j for 1 ≤ j ≤ 4, which gives an AP starting
from 13 with difference 1. Consequently, λ is an AV labeling.
h
Theorem 9.2.6. For even k = 4, h = 5 and m = 3, G kS n,m
(V, E) admits an AV labeling,
where n = 13.
Proof. Let us consider throughout the labeling 1 ≤ p ≤ 4 and 1 ≤ i ≤ 3. Define
λ : V(G) → {1, 2, ..., 52} as follows:
p
) = p.
λ(v11


12l − 3p − 8 + i



p
λ(vil ) = 


 12l + 3p − 23 + i
for l = 2, 4
for l = 3, 5.
The edge labeling is obtained by r∗ (e j ) = 110 + j for 1 ≤ j ≤ 12, which gives an AP
starting from 111 with difference 1. Consequently, λ is an AV labeling.
1
h
Theorem 9.2.7. For even k ≥ 2, h = k + 1 and m = 2 (k + 2), G kS n,m (V, E) admits an
AV labeling, where n = m(h − 1) + 1.
Proof. Let us consider throughout the labeling 1 ≤ p ≤ k and 1 ≤ i ≤ m. Define
λ : V(G) → {1, 2, ..., |V(G)|} as follows:
p
λ(v11
) = p.


mk(l − 2) − m(p − 1) + m(k − 1) + k + i
for 1 ≤ p ≤ k, l = 2, 4, ..., h − 1



p
λ(vil ) = 


 mk(l − 3) + m(p − 1) + mk + k + i
for 1 ≤ p ≤ k, l = 3, 5, ..., h
The edge labeling is obtained by r∗ (e j ) =
∑
[mk(l − 2) + m(k − 1) + k + 1] +
l=even
∑
l≥3 odd
[mk(l −
3) + k(m + 1) + 1] + j for 1 ≤ j ≤ mk, which is an AP starting from hk2 + 1 with difference
1. Consequently, λ is an AV labeling.
111
9. AV Labeling on Disjoint Union of Star Hypergraphs
9.3 New Results
h
In this section, it is proved that kS n,m
(V, E) is an AV graph for even k ≥ 4, m = 2 and
m = k.
h
Theorem 9.3.1 For k ≥ 4, h = k + 1 and m = 2, G kS n,m
(V, E) admits an AV labeling,
where n = k(2h − 1).
Proof. Define λ : V(G) → {1, 2, ..., |V(G)|} as follows:
p
λ(v11
)=p
f or 1 ≤ p ≤ k


2lk − 2p − k + 1



λ(v1lp ) = 


 2lk + 2p − 3k − 1
f or
1 ≤ p ≤ k, l = even
f or
1 ≤ p ≤ k, l ≥ 3 odd.
f or
1 ≤ p ≤ k, l = even,
f or
1 ≤ p ≤ k, l ≥ 3 odd.
and


2lk − 2p − k + 2



p
λ(v2l ) = 


 2lk + 2p − 3k
The edge labeling is obtained by r∗ (e j ) = hk2 + j for 1 ≤ j ≤ 2k, which is an AP starting
from hk2 + 1 with difference 1. Consequently, λ is an AV labeling.
h
Theorem 9.3.2. For even k ≥ 4, m = k and h = k + 1, G kS n,m (V, E) admits an AV
vertex labeling.
Proof. Define λ : V → {1, 2, ..., |V(G)|} as follows:
p
λ(v11
)=p
and
f or 1 ≤ p ≤ k
1 ≤ p ≤ k, 1 ≤ i ≤ m


mk(l − 2) − m(p − 1) + m(k − 1) + k + i



λ(vilp ) = 


 mk(l − 3) + m(p − 1) + mk + k + i
∗
The edge labeling is obtained by r (e j ) =
∑
h−1
2
which is an AP starting from
r=1
∑
h−1
2
f or
f or
even l ≥ 2,
odd l ≥ 3.
[2mk(2r − 1) + k + 1] + j for 1 ≤ j ≤ mk,
r=1
[2mk(2r − 1) + k + 2] with difference 1. Consequently, λ
is an AV labeling.
112
9.4 Summary
9.4
Summary
h
In this chapter, the hypergraph kS n,m
(V, E) obtained by the disjoint union of k isomorphic
h
copies of a linear h-uniform star hypergraph S n,m
(V, E) is defined and explained with the
help of suitable examples. Consequently, the following results related to an AV labeling
h
on kS n,m
(V, E) are proved:
h
For even k and h = k + 1, kS n,m
(V, E) admits an AV labeling if
•
•
•
•
m = 2 and k ≥ 4.
m = 3.
m = 12 (k + 2).
m = k and k ≥ 4.
All the results which are proved in the second section of this chapter have been published
(Javaid and Akhlaq 2012 (b)).
113