Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2012, Article ID 284383, 9 pages
doi:10.1155/2012/284383
Research Article
Total Vertex Irregularity Strength of the Disjoint
Union of Sun Graphs
Slamin,1 Dafik,2 and Wyse Winnona2
1
2
Information System Study Program, University of Jember, Jember 68121, Indonesia
Mathematics Education Study Program, University of Jember, Jember 68121, Indonesia
Correspondence should be addressed to Slamin, [email protected]
Received 11 January 2011; Accepted 31 March 2011
Academic Editor: R. Yuster
Copyright q 2012 Slamin et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment
of positive integer labels {1, 2, . . . , k} to both vertices and edges so that the weights calculated at
vertices are distinct. The total vertex irregularity strength of G, denoted by tvsG is the minimum
value of the largest label k over all such irregular assignment. In this paper, we consider the
total vertex irregularity strengths of disjoint union of s isomorphic
sun graphs, tvssMn , disjoint
union of s consecutive nonisomorphic sun graphs, tvs si1 Mi2 , and disjoint union of any two
nonisomorphic sun graphs tvsMk ∪ Mn .
1. Introduction
Let G be a finite, simple, and undirected graph with vertex set V and edge set E. A vertex
irregular total k-labeling on a graph G is an assignment of integer labels {1, 2, . . . , k} to both
vertices and edges such that the weights calculated at vertices are distinct. The weight of a
vertex v ∈ V in G is defined as the sum of the label of v and the labels of all the edges
incident with v, that is,
wtv λv uv∈E
λuv.
1.1
The notion of the vertex irregular total k-labeling was introduced by Bača et al. 1. The
total vertex irregularity strength of G, denoted by tvsG, is the minimum value of the largest
label k over all such irregular assignments.
2
International Journal of Combinatorics
The total vertex irregular strengths for various classes of graphs have been determined.
For instances, Bača et al. 1 proved that if a tree T with n pendant vertices and no vertices
of degree 2, then n 1/2 ≤ tvsT ≤ n. Additionally, they gave a lower bound and an
upper bound on total vertex irregular strength for any graph G with v vertices and e edges,
minimum degree δ and maximum degree Δ, |V | δ/Δ 1 ≤ tvsG ≤ |V | Δ − 2δ 1.
In the same paper, they gave the total vertex irregular strengths of cycles, stars, and complete
graphs, that is, tvsCn n 2/3, tvsK1,n n 1/2 and tvsKn 2.
Furthermore, the total vertex irregularity strength of complete bipartite graphs Km,n
for some m and n had been found by Wijaya et al. 2, namely, tvsK2,n n 2/3 for
n > 3, tvsKn,n 3 for n ≥ 3, tvsKn,n1 3 for n ≥ 3, tvsKn,n2 3 for n ≥ 4, and
tvsKn,an na 1/n 1 for all n and a > 1. Besides, they gave the lower bound on
tvsKm,n for m < n, that is, tvsKm,n ≥ max{m n/m 1, 2m n − 1/n}. Wijaya and
Slamin 3 found the values of total vertex irregularity strength of wheels Wn , fans Fn , suns
Mn and friendship graphs fn by showing that tvsWn n 3/4, tvsFn n 2/4,
tvsMn n 1/2, tvsfn 2n 2/3.
Ahmad et al. 4 had determined total vertex irregularity strength of Halin graph.
Whereas the total vertex irregularity strength of trees, several types of trees and disjoint
union of t copies of path had been determined by Nurdin et al. 5–7. Ahmad and Bača 8
investigated the total vertex irregularity strength of Jahangir graphs Jn,2 and proved that
tvsJn,2 n 1/2, for n ≥ 4 and conjectured that for n ≥ 3 and m ≥ 3,
tvsJm,n ≥ max
nm 2
nm − 1 2
,
.
3
4
1.2
They also proved that for the circulant graph, tvsCn 1, 2 n 4/5, and conjectured
that for the circulant graph Cn a1 , a2 , . . . , am with degree at least 5, 1 ≤ ai ≤ n/2,
tvsCn a1 , a2 , . . . , am n r/1 r.
A sun graph Mn is defined as the graph obtained from a cycle Cn by adding a
pendant edge to every vertex in the cycle. In this paper, we determine the total vertex
irregularity strength of disjoint union of the isomorphic sun graphs tvssMn , disjoint
union of consecutive nonisomorphic sun graphs tvs si1 Mi2 and disjoint union of two
nonisomorphic sun graphs tvsMk ∪ Mn , as described in the following section.
2. Main Results
We start this section with a lemma on the lower bound of total vertex irregularity strength of
disjoint union of any sun graphs as follows.
Lemma 2.1. The total vertex irregularity strength of disjoint union of any sun graphs is
tvs si1 Mni ≥ si1 ni 1/2, for s ≥ 1, ni1 ≥ ni , and 1 ≤ i ≤ s.
Proof. The disjoint union of the isomorphic sun graphs si1 Mni has si1 ni vertices ui,j of
s
degree 1 and i1 ni vertices vi,j of degree 3. Note that the smallest weight of vertices of
s
follows that the largest weight of si1 ni vertices of degree 1 is at least
i1 Mni must be 2. It
si1 ni 1 and of si1 ni vertices of degree 3 is at least 2 si1 ni 1. As a consequence,
at least one vertex ui,j or one edge incident with ui,j has label at least si1 ni 1/2.
International Journal of Combinatorics
3
Moreover, at least one vertex vi,j or one edge incident with vi,j has label at least 2 si1 ni 1/4. Then
tvs
s
i1
Mni
s
≥ max
i1
s
ni 1
2 i1 ni 1
,
.
2
4
2.1
Because of
s
i1
s
ni 1
2 i1 ni 1
,
2
4
2.2
then
s
s
i1 ni 1
tvs
Mni ≥
.
2
i1
2.3
We now present a theorem on the total vertex irregularity strength of disjoint union of
the isomorphic sun graphs tvssMn as follows.
Theorem 2.2. The total vertex irregularity strength of the disjoint union of isomorphic sun graphs is
tvssMn sn 1/2, for s ≥ 1 and n ≥ 3.
Proof. Using Lemma 2.1, we have tvssMn ≥ sn 1/2. To show that tvssMn ≤ sn 1/2, we label the vertices and edges of sMn as a total vertex irregular labeling. Suppose the
disjoint union of the isomorphic sun graphs sMn has the set of vertices
V sMn ui,j | 1 ≤ i ≤ s, 1 ≤ j ≤ n ∪ vi,j | 1 ≤ i ≤ s, 1 ≤ j ≤ n
2.4
and the set of edges
EsMn ui,j vi,j | 1 ≤ i ≤ s, 1 ≤ j ≤ n ∪ vi,j vi,j1 | 1 ≤ i ≤ s, 1 ≤ j ≤ n .
The labels of the edges and the vertices of sMn are described in the following formulas:
λ ui,j ⎧
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
1 j i − 1n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
sn 1
⎪
⎩−
2
s−1
; j 1, 2, . . . , n
2
s1
sn 1
s−1
and i , j 1, 2, . . . ,
−n
2
2
2
for i 1, 2, . . . ,
for other i, j,
2.5
4
λ vi,j ⎧
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
1 j i − 1n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ sn 1
⎪
⎩−
2
⎧
⎪
⎪
j i − 1n
⎪
⎪
⎪
⎪
⎪
⎪
⎨
λ ui,j vi,j ⎪
⎪
⎪
⎪
⎪
⎪
sn 1
⎪
⎪
⎩
2
International Journal of Combinatorics
s−1
for i 1, 2, . . . ,
; j 1, 2, . . . , n
2
s1
sn 1
s−1
and i , j 1, 2, . . . ,
−n
2
2
2
for other i, j,
s−1
; j 1, 2, . . . , n
2
s1
sn 1
s−1
and i , j 1, 2, . . . ,
−n
2
2
2
for i 1, 2, . . . ,
for other i, j,
sn 1
,
λ vi,j vi,j1 2
for i 1, 2, . . . , s, j 1, 2, . . . , n.
2.6
The weights of the vertices ui,j and vi,j of sMn are
wt ui,j 1 j i − 1n,
for i 1, 2, . . . , s, j 1, 2, . . . , n,
sn 1
,
wt vi,j 1 j i − 1n 2
2
2.7
for i 1, 2, . . . , s, j 1, 2, . . . , n.
It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex
irregular total. Therefore tvssMn sn 1/2 for s ≥ 1 and n ≥ 3.
Figure 1 illustrates the total vertex irregular labeling of the disjoint union 5 copies sun
graphs M5 .
If we substitute s 1 into the theorem above, we obtain a result that has been proved
by Wijaya and Slamin 3 as follows.
Corollary 2.3. The total vertex irregularity strength of sun graph tvsMn n 1/2, for s 1
and n ≥ 3.
The following theorem shows the total vertex irregularity strength of disjoint union of
nonisomorphic sun graphs with consecutive number of pendants.
Theorem 2.4. The total vertex irregularity strength of disjoint union of consecutive nonisomorphic
sun graphs is tvs si1 Mi2 ss 5/4, for s ≥ 1.
Proof. Using Lemma 2.1, we have tvs si1 Mi2 ≥ ss 5 2/4. To show that
s
tvs i1 Mi2 ≤ ss 5 2/4, we label the vertices and edges of si1 Mi2 as a total
International Journal of Combinatorics
5
∗1
2
1
28
1
13
13
3 1
1 6
5
13
13
1
1
31
13
30
4
3
5
1
∗1
2
1 29
32 1
4
1
∗1
12
7
11
6
13
33
1
13
13
1 11
8 1
10
37 1
13
13 1
13
9
40
13
∗9
15
2
9
1
17
14
1
22
13
13
13
13
13
8
∗4
43
4
1
2
41
35
13
12
13
1
1
10
1
1 39
42 3
13
13
36
13
3 16
7
1 34
38
1
13
13
48
9
13
23 10
18 5 13 26
8 21
13
5 44
47 8
13
13
13
20
7
13
13
10 49
52 13
13
45
13
13
13
19
6
13
13
12
51
6
7
46
13
25
12
11
50
13
24
11
Figure 1: Vertex irregular total 13 labelings of 5M5 .
vertex irregular labeling. Suppose the disjoint union of the nonisomorphic sun graphs with
consecutive number of pendants si1 Mi2 has the set of vertices
V ∪si1 Mi2 {u1,1 , u1,2 , u1,3 , u2,1 , u2,2 , u2,3 , u2,4 , . . . , us,1 , us,2 , . . . , us,s2 ,
v1,1 , v1,2 , v1,3 , v2,1 , v2,2 , v2,3 , v2,4 , . . . , vs,1 , vs,2 . . . , vs,s2 }
2.8
6
International Journal of Combinatorics
and the set of edges
E
s
Mi2
{u1,1 v1,1 , . . . , u1,3 v1,3 , u2,1 v2,1 , . . . , u2,4 v2,4 , . . . , us,1vs,1 , us,2 vs,2 , . . . , us,s2 vs,s2 }
i1
∪ {v1,1 v1,2 , . . . , v1,3 v1,1 , v2,1 v2,2 , . . . , v2,4 v2,1 , . . . , vs,1 vs,2 , . . . , vs,s2 vs,1 }.
2.9
The labels of the edges and the vertices of
λ ui,j ⎧
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
s
i1
Mi2 are described in the following formulas:
2s
for i 1, 2, . . . ,
− 1 , j 1, 2, . . . , i 2,
3
2s
;
i
3
ss 5 2
2s/3 − 12s/3 4
j 1, 2, . . . ,
−
4
2
⎪
⎪
⎪
⎪
i2 3i − 4
⎪
⎪
⎪
1j
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ss 5 2
⎪
⎪
for other i, j,
⎩−
4
⎧
2s
⎪
⎪
−
1
, j 1, 2, . . . , i 2,
1
for
i
1,
2,
.
.
.
,
⎪
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
2s
⎪
⎪
;
i
⎪
⎪
3
⎪
⎪
⎪
⎪
ss 5 2
2s/3 − 12s/3 4
⎨
j
1,
2,
.
.
.
,
−
λ vi,j 4
2
⎪
⎪
⎪
2
⎪
3i
−
4
i
⎪
⎪
⎪
1j ⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ss 5 2
⎪
⎪
for other i, j,
⎩−
4
⎧ 2
i 3i − 4
2s
⎪
⎪
⎪
j
for i 1, 2, . . . ,
− 1 , j 1, 2, . . . , i 2,
⎪
⎪
2
3
⎪
⎪
⎪
⎪
2s
⎪
⎪
⎪
i
;
⎨
3
λ ui,j vi,j ⎪
ss 5 2
2s/3 − 12s/3 4
⎪
⎪
−
j 1, 2, . . . ,
⎪
⎪
⎪
4
2
⎪
⎪
⎪
⎪
⎪
ss 5 2
⎪
⎩
for other i, j,
4
ss 5 2
, for i 1, 2, . . . , s, j 1, 2, . . . , i 2.
λ vi,j vi,j1 4
2.10
International Journal of Combinatorics
7
∗1
∗1
2
1
6
5
1
4
25
1
22
1
10
24 1
4
1
28
1 23
7
2
∗1
1
9
10
3 13
1
10
27
6
7
1
i=2
∗4
5
14
15
10
10
8
29
1
10
8
3
1
i=1
1
5
26
1
10
10
10
3
10
10
10
34
4
10
35
5
10
10 1
10
9 19
10
2
32
10
1
10
10
31
18
8
11
i=3
8
38
10
10
10
12
6 36 10
39 9
10
10
2
9
1 30
33 3
1
16 6
10
7
37
10
i=4
17
7
Figure 2: Vertex irregular total 10 labelings of M3 ∪ M4 ∪ M5 ∪ M6 .
The weights of the vertices ui,j and vi,j of
s
i1 Mi2
are
i2 3i 2j − 2
wt ui,j , for i 1, 2, . . . , s, j 1, 2, . . . , i 2,
2
i2 3i 2j − 2
ss 5 2
wt vi,j 2
for i 1, 2, . . . , s, j 1, 2, . . . , i 2.
2
4
2.11
It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex
irregular total. Therefore tvs si1 Mi2 ss 5 2/4 for s ≥ 1 dan n ≥ 3.
Figure 2 illustrates the vertex irregular total 10 labelings of the disjoint union 4
consecutive nonisomorphic sun graphs M3 ∪ M4 ∪ M5 ∪ M6 .
Finally, we conclude this section with a result on the total vertex irregularity strength
of disjoint union of two nonisomorphic sun graphs as follows.
Theorem 2.5. The total vertex irregularity strength of disjoint union of two nonisomorphic sun
graphs is tvsMk ∪ Mn k n 1/2, for n > k ≥ 3.
8
International Journal of Combinatorics
∗1
∗1
1
3
2
1
5
1
5 11
6
6
2 20
6
4
22
3
i=1
19
1
23 5
1
16
6
6
6
6
4
18
1
15
1
6
1
17
7
6
5
2
6
14
1
1
6
4
1
6
8 2
6
3
21
6
6
10
4
6
i=2
9
3
Figure 3: Vertex irregular total 6 labelings of M4 ∪ M6 .
Proof. Using Lemma 2.1, we have tvs si1 Mi2 ≥ kn1/2. To show that tvsMk ∪Mn ≤
kn1/2, we label the vertices and edges of Mk ∪Mn as a vertex irregular total k-labeling.
Suppose the disjoint union of the nonisomorphic sun graphs with different pendant Mk ∪Mn
has the set of vertices
V sMn {u1,1 , u1,2 , . . . , u1,n , u2,1 , u2,2 , . . . , u2,n , v1,1 , v1,2 , . . . , v1,n , v2,1 , v2,2 , . . . , v2,n }
2.12
and the set of edges
EsMn {u1,1 v1,1 , u1,2 v1,2 , . . . , u1,n v1,n , u2,1 v2,1 , u2,2 v2,2 , . . . , u2,n v2,n }
∪ {v1,1 v1,2 , v1,2 v1,3 , . . . , v1,n v1,1 , v2,1 v2,2 , v2,2 v2,3 , . . . , v2,n v2,1 }.
2.13
The labels of the edges and the vertices of Mk ∪ Mn are described in the following formulas:
λ ui,j λ vi,j ⎧
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
for i 1; j 1, 2, . . . , k,
kn1
−1
i 2; j 1, 2, . . . ,
2
⎪
⎪
1 j i − 1k
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎩ k n 1
for other i, j,
2
⎧
1
for i 1; j 1, 2, . . . , k,
⎪
⎪
⎪
⎪
⎪
⎪
kn1
⎪
⎪
⎪
−
1
i
2;
j
1,
2,
.
.
.
,
⎨
2
⎪
⎪
1 j i − 1k
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ k n 1
2
for other i, j,
International Journal of Combinatorics
λ ui,j vi,j ⎧
⎪
j i − 1k
⎪
⎪
⎪
⎪
⎨
9
for i 1; j 1, 2, . . . , k,
kn1
−1
i 2; j 1, 2, . . . ,
2
⎪
⎪
⎪
⎪
kn1
⎪
⎩
for other i, j,
2
kn1
λ vi,j vi,j1 , for i 1; j 1, 2, . . . , k, i 2; j 1, 2, . . . , n.
2
2.14
The weights of the vertices ui,j and vi,j of Mk ∪ Mn are
wt ui,j 1 j i − 1k, for i 1, 2, j 1, 2, . . . , n,
kn1
wt vi,j 1 j i − 1k 2
, for i 1, 2, j 1, 2, . . . , n.
2
2.15
It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex
irregular total. Therefore tvsMk ∪ Mn k n 1/2 for n > k ≥ 3.
Figure 3 illustrates the vertex irregular total 6 labelings of the disjoint union of
2 nonisomorphic sun graphs M4 ∪ M6 .
3. Conclusion
We conclude this paper with the following conjecture for the direction of further research in
this area.
Conjecture 1. The total vertex irregularity strength of disjoint union of any sun graphs is
tvs si1 Mni si1 ni 1/2, for s ≥ 1, ni1 ≥ ni and 1 ≤ i ≤ s.
References
1 M. Bača, S. Jendrol’, M. Miller, and J. Ryan, “On irregular total labellings,” Discrete Mathematics, vol.
307, no. 11-12, pp. 1378–1388, 2007.
2 K. Wijaya, Slamin, Surahmat, and S. Jendrol’, “Total vertex irregular labeling of complete bipartite
graphs,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 55, pp. 129–136, 2005.
3 K. Wijaya and Slamin, “Total vertex irregular labelings of wheels, fans, suns and friendship graphs,”
Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 65, pp. 103–112, 2008.
4 A. Ahmad, Nurdin, and E. T. Baskoro, “On total irregularitystrength of generalized Halin graph,” to
appear in Ars Combinatoria.
5 Nurdin, E. T. Baskoro, A. N. M. Salman, and N. N. Gaos, “On total vertex-irregular labellings for
several types of trees,” Utilitas Mathematica, vol. 83, pp. 277–290, 2010.
6 Nurdin, E. T. Baskoro, A. N. M. Salman, and N. N. Gaos, “On the total vertex irregularity strength of
trees,” Discrete Mathematics, vol. 310, no. 21, pp. 3043–3048, 2010.
7 Nurdin, A. N. M. Salman, N. N. Gaos, and E. T. Baskoro, “On the total vertex-irregular strength of a
disjoint union of t copies of a path,” Journal of Combinatorial Mathematics and Combinatorial Computing,
vol. 71, pp. 227–233, 2009.
8 A. Ahmad and M. Bača, “On vertex irregular total labelings,” to appear in Ars Combinatoria.
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