Proyecciones Journal of Mathematics
Vol. 35, No 3, pp. 251-262, September 2016.
Universidad Católica del Norte
Antofagasta - Chile
DOI: 10.4067/S0716-09172016000300003
Total edge irregularity strength of disjoint union
of double wheel graphs
P. Jeyanthi
Govindammal Aditanar College for Women, India
and
A. Sudha
Wavoo Wajeeha Women’s College of Arts & Science, India
Received : October 2015. Accepted : March 2016
Abstract
An edge irregular total k-labeling f : V ∪ E → {1, 2, 3, . . . , k}
of a graph G = (V, E) is a labeling of vertices and edges of G in
such a way that for any two different edges uv and u0 v 0 their weights
f (u) + f (uv) + f (v) and f (u0 ) + f (u0 v 0 ) + f (v 0 ) are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for
which the graph G has an edge irregular total k-labeling. In this paper,
we determine the total edge irregularity strength of disjoint union of
p isomorphic double wheel graphs and disjoint union of p consecutive
non-isomorphic double wheel graphs.
Keywords: Irregularity strength; total edge irregularity strength; edge
irregular total labeling, disjoint union of double wheel graphs.
AMS Classification (2010): 05C78.
252
P. Jeyanthi and A. Sudha
1. Introduction
The graphs in this paper are simple, finite and undirected. In [2] Bača et
al. defined the notion of edge irregular total k-labeling of a graph G as a
funtion φ : V ∪ E → {1, 2, . . . , k} such that the edge weights wtφ (uv) =
φ(u) + φ(uv) + φ(v) are distinct for all the edges. That is wtφ (uv) 6=
wtφ (u0 v 0 ) for every pair of edges uv, u0 v0 ∈ E. The minimum k for which
the graph G has an edge irregular total k-labeling is called the total edge
irregularity strength of G, tes(G). They found a lower bound for the total
edge irregularity strength of a graph as
(1.1)
⎧
⎨ » (|E(G)| + 2) ¼
tes(G) ≥ max
⎩
3
,
»
⎫
¼⎬
(∆(G) + 1)
⎭
2
where ∆(G) is the maximum degree of G. Ivančo and Jendroˇl [3] posed
the following conjecture.
Conjecture:1.1 [3] Let G be an arbitrary graph different from K5 . Then
(1.2)
⎧
⎨ » (|E(G)| + 2) ¼
tes(G) = max
⎩
3
,
»
⎫
¼
(∆(G) + 1) ⎬
⎭
2
Conjecture(1.1) has been verified by several authors for several families
of graphs. Motivated by the results in [1,4,5,6] we determine the total edge
irregularity strength of the disjoint union of double wheel graphs. A double
wheel graph DWn of size n can be composed of 2Cn + K1 , n ≥ 3,that is it
consists of two cycles of size n, where all the vertices of the two cycles are
connected to a common hub.
2. Main Results
In this section, first we determine the total edge irregularity strength of
the disjoint union of p isomorphic double wheel graphs with the vertex
set V (pDWn ) = {vj , vij , uji : 1 ≤ i ≤ n, 1 ≤ j ≤ p} and the edge set
j
=
{vj vij , vj uji , vij vi+1
, uji uji+1
:
E(pDWn )
1 ≤ i ≤ n, 1 ≤ j ≤ p} where the subscript i is taken modulo n.
Lemma 2.1. tes(2DWn ) =
l
8n+2
3
m
, n ≥ 3.
Total edge irregularity strength of disjoint union of double ...
Since|E(2DWn )| = 8n, tes(2DWn ) ≥
Proof.
l
m
l
8n+2
3
m
253
by (1.1). Let
. To prove the reverse inequality we define f : V ∪ E →
k = 8n+2
3
{1, 2, 3, . . . , k} as follows:
f (v1 ) = 1;
f (v2 ) = n;
f (vi1 ) = i, 1 ≤ i ≤ n;
f (vi2 ) = k, 1 ≤ i ≤ n;
f (u1i ) = n, 1 ≤ i ≤ n;
f (u2i ) = k, 1 ≤ i ≤ n;
f (v1 vi1 ) = 1, 1 ≤ i ≤ n;
f (v1 u1i ) = 1 + i, 1 ≤ i ≤ n;
1 ) = 2n + 1 − i, 1 ≤ i ≤ n;
f (vi1 vi+1
f (u1i u1i+1 ) = n + 2 + i, 1 ≤ i ≤ n;
f (v2 vi2 ) = 3n + 2 − k + i, 1 ≤ i ≤ n;
f (v2 u2i ) = 4n + 2 − k + i, 1 ≤ i ≤ n;
2 ) = 6n + 2 − 2k + i, 1 ≤ i ≤ n;
f (vi2 vi+1
2
f (ui u2i+1 ) = 7n + 2 − 2k + i, 1 ≤ i ≤ n.
We observe that,
wt(v1 vi1 ) = 2 + i, 1 ≤ i ≤ n;
wt(v1 u1i ) = n + 2 + i, 1 ≤ i ≤ n;
1 ) = 2n + 2 + i, 1 ≤ i ≤ n;
wt(vi1 vi+1
1
wt(ui u1i+1 ) = 3n + 2 + i, 1 ≤ i ≤ n;
wt(v2 vi2 ) = 4n + 2 + i, 1 ≤ i ≤ n;
wt(v2 u2i ) = 5n + 2 + i, 1 ≤ i ≤ n;
2 ) = 6n + 2 + i, 1 ≤ i ≤ n;
wt(vi2 vi+1
wt(u2i u2i+1 ) = 7n + 2 + i, 1 ≤ i ≤ n. 2
Theorem 2.2. Let n ≥ 3 and p ≥ 3 be two integers. Then the total edge
irregularity
l
m strength of disjoint union of p isomorphic double wheel graphs
4pn+2
.
is
3
Proof.
l
Since |E(pDWn )| = 4pn, by (1.1) we have tes(pWn ) ≥
4pn+2
3
m
l
4pn+2
3
m
.
. To prove the reverse inequality, we define the total edge
Let k =
irregular k-labeling f for 1 ≤ i ≤ n and 1 ≤ j ≤ p as follows:
f (vj ) = f (vij ) = f (uji ) = min{(j − 1)2n + 1, k}.
For 1 ≤ j ≤ p and (j − 1)2n + 1 ≤ k,
f (vj vij ) = i;
f (vj uji ) = n + i;
254
P. Jeyanthi and A. Sudha
j
) = 2n + i;
f (vij vi+1
j j
f (ui ui+1 ) = 3n + i.
For 1 ≤ j ≤ p and (j − 1)2n + 1 > k,
f (vj vij ) = 4(j − 1)n + 2 − 2k + i;
f (vj uji ) = 4(j − 1)n + 2 − 2k + n + i;
j
) = 4(j − 1)n + 2 − 2k + 2n + i;
f (vij vi+1
j j
f (ui ui+1 ) = 4(j − 1)n + 2 − 2k + 3n + i.
We observe that,
wt(vj vij ) = 4(j − 1)n + 2 + i, 1 ≤ i ≤ n, 1 ≤ j ≤ p;
wt(vj uji ) = 4(j − 1)n + 2 + n + i, 1 ≤ i ≤ n, 1 ≤ j ≤ p;
j
) = 4(j − 1)n + 2 + 2n + i, 1 ≤ i ≤ n, 1 ≤ j ≤ p;
wt(vij vi+1
j j
wt(ui ui+1 ) = 4(j − 1)n + 2 + 3n + i, 1 ≤ i ≤ n, 1 ≤ j ≤ p .
It can be easily verified that all the vertex and edge labels are at most
k and the weights of the edges are pair-wise distinct. Thus the resulting
total labeling is the edge irregular total k-labeling. Figure 1 illustrates the
edge irregular total labeling of the disjoint union of 4 copies of double wheel
graphs. 2
Total edge irregularity strength of disjoint union of double ...
255
Figure 1 : Total irregularity strength of disjoint union of 4 isomorphic
double Wheel graphs. tes(4DW6 ) = 33.
256
P. Jeyanthi and A. Sudha
Now we determine the total edge irregularity strength of the disjoint
= ni + 1, i ≥
wheel
union of p consecutive (ni+1 Ã
! 1) non-isomorphic double
Ã
!
graphs with the vertex set V
where V
Ã
p
S
!
p
S
j=1
{vj , vij , uji
DWnj =
j=1
j
, uji uji+1
{vj vij , vj uji , vij vi+1
taken modulo n.
DWnj
and the edge set E
p
S
j=1
: 1 ≤ i ≤ nj , 1 ≤ j ≤ p} and E
DWnj
Ã
p
S
j=1
DWnj
: 1 ≤ i ≤ nj , 1 ≤ j ≤ p} where the subscript i is
Lemma 2.3. Let n1 ≥l 3 be an
m integer and n2 = n1 + 1. Then
8n1 +6
.
tes(DWn1 ∪ DWn2 ) =
3
l
m
Proof. Let k = 8n13+6 . Then by (1.1), tes(DWn1 ∪ Wn2 ) ≥ k. Now to
prove the reverse inequality, we define an edge irregular k-labeling of f as
follows:
f (v1 ) = 1;
f (vi1 ) = i, 1 ≤ i ≤ n1 ;
f (vi2 ) = f (u2i ) = k, 1 ≤ i ≤ n2 ;
f (u1i ) = n1 , 1 ≤ i ≤ n1 ;
f (v2 ) = n1 + 1;
f (v1 vi1 ) = 1, 1 ≤ i ≤ n1 ;
f (v1 u1i ) = 1 + i, 1 ≤ i ≤ n1 ;
f (v2 vi2 ) = 3n1 + 1 − k + i, 1 ≤ i ≤ n2 ;
1 ) = 2n + 1 − i, 1 ≤ i ≤ n ;
f (vi1 vi+1
1
1
1
f (ui u1i+1 ) = n1 + 2 + i, 1 ≤ i ≤ n1 ;
f (v2 u2i ) = 4n1 + 2 − k + i, 1 ≤ i ≤ n2 ;
2 ) = 6n + 4 − 2k + i, 1 ≤ i ≤ n ;
f (vi2 vi+1
1
2
2
f (ui u2i+1 ) = 7n1 + 5 − 2k + i, 1 ≤ i ≤ n2 .
We observe that,
for 1 ≤ i ≤ nj , j = 1, 2
wt(v1 vi1 ) = 2 + i;
wt(v1 u1i ) = n1 + 2 + i;
1 ) = 2n + 2 + i;
wt(vi1 vi+1
1
1
wt(ui u1i+1 ) = 3n1 + 2 + i;
wt(v2 vi2 ) = 4n1 + 2 + i;
wt(v2 u2i ) = 5n1 + 3 + i;
2 ) = 6n + 4 + i;
wt(vi2 vi+1
1
!
=
Total edge irregularity strength of disjoint union of double ...
257
wt(u2i u2i+1 ) = 7n1 + 5 + i.
It can be easily verified that all the vertex and edge labels are at most
k and the weights of the edges are pair-wise distinct. Thus the resulting
total labeling is the edge irregular total k-labeling. 2
Lemma 2.4. Let ln1 , n2 , n3m be integers and n1 ≥ 3. Then tes(DWn1 ∪
DWn2 ∪ DWn3 ) = 2(6n31 +7) .
Proof.
Since |E(DWn1 ∪
l DWn2 ∪ DW
m nl3 )| = 12(n
m 1 + 1) lby (1.1),m
12(n1 +1)+2)
2(6n1 +7)
tes(DWn1 ∪DWn2 ∪DWn3 ) ≥
=
.Let k = 2(6n31 +7) .
3
3
Now to prove the reverse inequality,we define an edge irregular k-labeling
f as follows:
f (v1 ) = 1;
f (vi1 ) = f (u1i ) = 1, 1 ≤ i ≤ n1 ;
f (v2 ) = f (vi2 ) = f (u2i ) = 2n + 1, 1 ≤ i ≤ n2 ;
f (v3 ) = 3n1 + 3;
f (u3i ) = f (vi3 ) = k, 1 ≤ i ≤ n3 ;
f (v1 vi1 ) = i, 1 ≤ i ≤ n1 ;
f (v1 u1i ) = n1 + i, 1 ≤ i ≤ n1 ;
f (v2 vi2 ) = i, 1 ≤ i ≤ n2 ;
f (v2 u2i ) = n1 + 1 + i, 1 ≤ i ≤ n2 ;
f (v3 vi3 ) = 5n1 + 3 − k + i, 1 ≤ i ≤ n3 ;
f (v3 u3i ) = 6n1 + 5 − k + i, 1 ≤ i ≤ n3 ;
1 ) = 2n + i, 1 ≤ i ≤ n ;
f (vi1 vi+1
1
1
1
f (ui u1i+1 ) = 3n1 + i, 1 ≤ i ≤ n1 ;
2 ) = 2n + 2 + i, 1 ≤ i ≤ n ;
f (vi2 vi+1
1
2
2
f (ui u2i+1 ) = 3n1 + 3 + i, 1 ≤ i ≤ n2 ;
3 ) = 10n + 10 − 2k + i, 1 ≤ i ≤ n ;
f (vi3 vi+1
1
3
3
f (ui u3i+1 ) = 11n1 + 12 − 2k + i.1 ≤ i ≤ n3 .
We observe that,
for1 ≤ i ≤ nj ; j = 1, 2, 3
wt(v1 vi1 ) = 2 + i;
wt(v1 u1i ) = n1 + 2 + i;
wt(v2 vi2 ) = 4n1 + 2 + i;
wt(v2 u2i ) = 5n1 + 3 + i;
wt(v3 vi3 ) = 8n1 + 6 + i;
258
P. Jeyanthi and A. Sudha
wt(v3 u3i ) = 9n1 + 8 + i;
1 ) = 2n + 2 + i;
wt(vi1 vi+1
1
wt(u1i u1i+1 ) = 3n1 + 2 + i;
2 ) = 6n + 4 + i;
wt(vi2 vi+1
1
wt(u2i u2i+1 ) = 7n1 + 5 + i;
3 ) = 10n + 10 + i;
wt(vi3 vi+1
1
3
wt(ui u3i+1 ) = 11n1 + 12 + i.
It can be easily verified that all the vertex and edge labels are at most k
and the weights of the edges are pair-wise distinct. Thus the resulting total
labeling is the edge irregular total k-labeling. 2
Theorem 2.5. The total edge irregularity strength of the disjoint
union ofm
l
p (p ≥ 4) consecutive non-isomorphic double wheel graphs is 2p(2n1 +p−1)+2
.
3
Proof.
l
Since |E
2p(2n1 +p−1)+2
3
m
Ã
p
S
j=1
DWnj
l
!
| = 2p(2n1 +p−1) by (1.1) tes
m
Ã
p
S
j=1
DWnj
!
. Let k = 2p(2n1 +p−1)+2
. To prove the reverse inequality
3
we define the total edge irregular k-labeling f for 1 ≤ i ≤ nj and 1 ≤ j ≤ p
as follows:
f (v1 ) = f (vi1 ) = f (u1i ) = 1, 1 ≤ i ≤ n1 ;
f (v1 vi1 ) = i, 1 ≤ i ≤ n1 ;
f (v1 u1i ) = n1 + i, 1 ≤ i ≤ n1 ;
1 ) = 2n + i, 1 ≤ i ≤ n ;
f (vi1 vi+1
1
1
1
f (ui u1i+1 ) = 3n1 + i, 1 ≤ i ≤ nn1 ;
o
P
n
)
+
1,
k
, 1 ≤ i ≤ nj and 2 ≤ j ≤
f (vj ) = f (vij ) = f (uji ) = min 2( j−1
s=1 s
p.
For 1 ≤ i ≤ nj and 2 ≤ i ≤ p, 2
f (vj vij ) = i;
f (vj uji ) = nj + i;
j
) = 2nj + i;
f (vij vi+1
j j
f (ui ui+1 ) = 3nj + i.
hP
We observe
that,
h ³
i
i
Pj−1
j
wt(vj vi ) = 2 2
n
)
+
1
+ i;
s
s=1
h ³P
j−1
wt(vj uji ) = 2 2
s=1 ns
´
i
j−1
s=1 ns
+ 1 + nj + i;
i
+ 1 ≤ k,
≥
Total edge irregularity strength of disjoint union of double ...
h ³P
j−1
´
i
j
wt(vij vi+1
)=2 2
s=1 ns + 1 + 2nj + i;
wt(uji uji+1 ) = 2 2
s=1 ns + 1 + 3nj + i.
h ³P
j−1
´
For 1 ≤ i ≤ nj and 2 ≤ i ≤ p, 2
h ³
259
i
hP
´
j−1
s=1 ns
i
i
+ 1 > k,
Pj−2
ns + 1 − 2k + i;
f (vj vij ) = 4nj−1 + 2 2
h ³P s=1 ´
i
j−2
f (vj uji ) = 4nj−i + 2 2
n
+
1
+ nj − 2k + i;
s
s=1
h ³P
´
i
j j
j−2
n
+
1
+ 2nj − 2k + i;
f (vi vi+1 ) = 4nj−i + 2 2
s
h ³Ps=1 ´
i
j j
j−2
f (ui ui+1 ) = 4nj−i + 2 2
s=1 ns + 1 + 3nj − 2k + i.
We observe that, h ³
i
Pj−2 ´
wt(vj vij ) = 4nj−1 + 2 2
n
+
1
+ i;
s
s=1
h ³P
j−2
wt(vj uji ) = 4nj−i + 2 2
´
i
s=1 ns + 1 + nj + i;
h ³P
´
i
j j
j−2
wt(vi vi+1 ) = 4nj−i + 2 2
s=1 ns + 1 + 2nj + i;
h ³P
´
i
j−2
wt(uji uji+1 ) = 4nj−i + 2 2
n
+
1
+ 3nj + i.
s=1 s
It can be easily verified that all the vertex and edge labels are at most k
and the weights of the edges are pair-wise distinct .Thus the resulting total
labeling is the edge irregular k-labeling. Figure 2 illustrates the edge irregular total labelings of the disjoint union of 4 Consecutive non-isomorphic
double wheel graphs DW3 ∪ DW4 ∪ DW5 ∪ DW6 . 2
260
P. Jeyanthi and A. Sudha
Figure 2 : Total edge irregularity strength of disjoint union of 4
non-isomorphic double Wheel graphs.
tes(DW3 ∪ DW4 ∪ DW5 ∪ DW6 ) = 25.
Total edge irregularity strength of disjoint union of double ...
261
3. Conclusion
In this paper we determine the total edge irregularity strength of the disjoint union of p isomorphic double wheel graphs and disjoint union of p
consecutive non isomorphic double wheel graphs. We conclude this paper
by stating the following open problem.
Open problem:
For m ≥ 2,find the exact value of the total edge irregularity strength of
a disjoint union of m arbitrary double wheel graphs.
References
[1] A. Ahmad and M.Bača, and Muhammad Numan, On irregularity
strength of the disjoint union of friendship graphs, Electronic Journal of Graph Theory and Applications, 11 (2), pp. 100—108, (2013).
[2] M.Bača,S.Jendroˇl, M. Miller and J. Ryan, On irregular total labellings, Discrete Math., 307, pp. 1378-1388, (2007).
[3] J.Ivančo,S.Jendroˇl, Total edge irregularity strength of trees, Discussiones Math. Graph Theory, 26, pp. 449-456, (2006).
[4] M.K.Siddiqui,A.Ahmad,M.F.Nadeem,Y.Bashir, Total edge irregularity
strength of the disjoint union of sun graphs, International Journal of
Mathematics and Soft Computing 3 (1), pp. 21-27, (2013).
[5] P. Jeyanthi and A. Sudha, Total Edge Irregularity Strength of Disjoint
Union of Wheel Graphs, Electron. Notes in Discrete Math., 48, pp.
175-182, (2015).
P. Jeyanthi
Research Centre,
Department of Mathematics
Govindammal Aditanar College for Women
Tiruchendur-628 215, Tamil Nadu,
India
e-mail: jeyajeyanthi@rediffmail.com
262
P. Jeyanthi and A. Sudha
and
A. Sudha
Department of Mathematics
Wavoo Wajeeha Women’s College of Arts & Science,
Kayalpatnam -628 204,Tamil Nadu, India
e-mail: [email protected]