On the total irregularity strength of disjoint union of arbitrary graphs

ON THE TOTAL IRREGULARITY STRENGTH
OF DISJOINT UNION OF ARBITRARY GRAPHS
R. RAMDANI, A.N.M. SALMAN, H. ASSIYATUM,
A. SEMANIČOVÁ-FEŇOVČÍKOVÁ and M. BAČA
Communicated by I. Tomescu
We deal with the modifications of the well-known irregular assignments, namely
vertex irregular total labelings, edge irregular total labelings and totally irregular total labelings of graphs. In the paper, we study the total vertex (edge)
irregularity strength and total irregularity strength for disjoint union of arbitrary
graphs and we establish the upper bounds for these invariants.
AMS 2010 Subject Classification: 05C78.
Key words: edge irregular total labeling, totally irregular total labeling, vertex
irregular total labeling.
1. INTRODUCTION
Let G be a connected, simple and undirected graph with vertex set V (G)
and edge set E(G). A total labeling f : V (G) ∪ E(G) → {1, 2, . . . , k} is called
a vertex irregular total k-labeling of G if every two distinct vertices x and y in
V (G) satisfy wtf (x) 6= wtf (y), where
X
wtf (x) = f (x) +
f (xz).
xz∈E(G)
The total vertex irregularity strength of G, denoted by tvs(G), is the minimum
k for which G has a vertex irregular total k-labeling. In [5] are given the bounds
for the total vertex irregularity strength of a graph G with minimum degree
δ(G) and maximum degree ∆(G) by the following form:
|V (G)| + δ(G)
(1)
≤ tvs(G) ≤ |V (G)| + ∆(G) − 2δ(G) + 1.
∆(G) + 1
Moreover, there is proved that tvs(G) ≤ |V (G)| − 1 − b(|V (G)| − 2)/(∆ + 1)c
for a graph with no component of order ≤ 2. Przybylo [19] proved that
tvs(G) < 32|V (G)|/δ(G)+8 in general and tvs(G) < 8|V (G)|/r+3 for r-regular
graphs. This was then improved in [4] that tvs(G) ≤ 3 d|V (G)|/δ(G)e + 1 ≤
MATH. REPORTS 18(68), 4 (2016), 469–482
470 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača
2
3|V (G)|/δ(G) + 4. Recently, Majerski and Przybylo in [12] based on a random
ordering of the vertices proved that if δ(G) ≥ n0.5 ln n then tvs(G) ≤ (2 +
o(1))|V (G)|/δ(G) + 4. The exact values of the total vertex irregularity strength
for several families of graphs can be found in [16, 17] and [18].
Furthermore, in [5] Bača, Jendrol’, Miller and Ryan defined the total
labeling ϕ to be an edge irregular total k-labeling of the graph G if for every
two different edges x1 x2 and x01 x02 of G one has wtϕ (x1 x2 ) 6= wtϕ (x01 x02 ), where
wtϕ (x1 x2 ) = ϕ(x1 ) + ϕ(x1 x2 ) + ϕ(x2 ).
The total edge irregularity strength of G, denoted by tes(G), is defined as the
minimum k for which G has an edge irregular total k-labeling.
In [5] is proved that for any graph G with a non-empty edge set E(G)
|E(G)| + 2
(2)
≤ tes(G) ≤ |E(G)|.
3
Ivančo and Jendrol’ [9] posed a conjecture that for arbitrary graph G different
from K5 ,
|E(G)| + 2
∆(G) + 1
tes(G) = max
,
.
3
2
This conjecture has been verified for complete graphs and complete bipartite
graphs in [10] and [11], for the Cartesian, categorical and strong products
of two paths in [1, 2, 14], for the categorical product of two cycles in [3], for
generalized Petersen graphs in [8], for generalized prisms in [6], for corona
product of a path with certain graphs in [15] and for large dense graphs with
(|E(G)| + 2)/3 ≤ (∆(G) + 1)/2 in [7].
Combining previous modifications of the irregularity strength, Marzuki,
Salman and Miller [13] introduced a new irregular total k-labeling of a graph G
called totally irregular total k-labeling, which is required to be at the same time
vertex irregular total and also edge irregular total. The minimum k for which
a graph G has a totally irregular total k-labeling is called the total irregularity
strength of G and it is denoted by ts(G). In [13] there is proved that for every
graph G,
(3)
ts(G) ≥ max{tes(G), tvs(G)}.
Ramdani and Salman in [20] determined the exact values of the total irregularity strength for several Cartesian product graphs. Namely, they proved that
for n ≥ 3, ts(Pn P2 ) = n, ts(Cn P2 ) = n + 1 and ts(Sn P2 ) = n + 1.
2. MAIN RESULTS
S
Let m
i=1 Gi denote the disjoint union of graphs G1 , G2 , . . . , Gm , m ≥ 2.
Next theorem gives an upper bound of the total edge irregularity strength of
3
On the total irregularity strength of disjoint union of arbitrary graphs
471
the disjoint union of graphs.
Theorem 2.1. The total edge irregularity strength of disjoint union of
graphs G1 , G2 , . . . , Gm , m ≥ 2, is
!
m
m
[
X
m−1
tes
Gi ≤
tes(Gi ) −
.
2
i=1
i=1
Proof. Let Gi , i = 1, 2, . . . , m, be a graph of order pi and size qi . Let
tes(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be an edge irregular total
ti -labeling of Gi . We define t0 = 0. Let V (Gi ) = {via : a = 1, 2, . . . , pi }
and
{eix
qi }, for every i = 1, 2, . . . , m. Define a total
: x = 1, 2, . . . ,S
PmE(Gi ) =m−1
-labeling g of m
i=1 ti −
i=1 Gi as follows:
2
g(via ) = fi (via ) +
g(eix ) = fi (eix ) +
i−1
X
s=0
i−1
X
ts −
i
ts −
i−1 if via ∈ V (Gi ),
2
if eix ∈ E(Gi ),
2
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
Let eix = via vib be an edge in E(Gi ). We distinguish two cases.
Case 1. If i is odd then for edge-weight of eix = via vib under the labeling g we
have
wtg (eix ) = g(via ) + g(eix ) + g(vib )
=
+
fi (via ) +
fi (vib ) +
i−1
X
s=0
i−1
X
!
ts −
i−1
2
+
fi (eix ) +
i−1
X
!
ts −
i−1
2
s=0
!
ts −
i−1
2
= wtfi (eix ) + 3
i−1
X
!
ts −
i−1
2
.
s=0
s=0
P
i−1
i−1
Since, for a fixed i, wtfi (eix ) 6= wtfi (eiy ) for every x 6= y and 3
s=0 ts − 2
is a constant, thus also wtg (eix ) 6= wtg (eiy ).
Case 2. If i is even then for edge-weight of the edge eix = via vib under the
labeling g we get
wtg (eix ) = g(via ) + g(eix ) + g(vib )
!
i−1
X
i
= fi (via ) +
ts −
+
2
fi (eix ) +
s=0
+ fi (vib ) +
i−1
X
s=0
i
ts −
2
!
= wtfi (eix ) + 3
i−1
X
s=0
i−1
X
s=0
i
ts − + 1
2
i
ts −
2
!
+ 1.
!
472 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača
4
Again we can see that for a fixed i we have wtg (eix ) 6= wtg (eiy ) for every x 6= y.
Now we are going to show that wtg (eix ) < wtg (ei+1y ) for every x =
1, 2, . . . , qi and y = 1, 2, . . . , qi+1 . We distinguish two cases according to the
parity of i.
Case 1. If i is odd then
!
i−1
X
i−1
wtg (eix ) = wtfi (eix ) + 3
ts −
(4)
2
s=0
!
!
i−1
i
X
X
i−1
i−1
≤ 3ti + 3
ts −
=3
ts −
2
2
s=0
s=0
and
(5)
i
X
wtg (ei+1y ) = wtfi+1 (ei+1y ) + 3
s=0
≥3+3
i
X
s=0
i+1
ts −
2
i+1
ts −
2
!
+1
i
X
!
+1=3
s=0
i−1
ts −
2
!
+ 1.
Thus from (4) and (5) it follows that
!
!
i
i
X
X
i−1
i−1
wtg (eix ) ≤ 3
ts −
<3
ts −
+ 1 ≤ wtg (ei+1y ).
2
2
s=0
s=0
Case 2. If i is even then
(6)
i−1
X
!
i
wtg (eix ) = wtfi (eix ) + 3
ts −
+1
2
s=0
!
!
i−1
i
X
X
i
i
+1=3
ts −
+1
≤ 3ti + 3
ts −
2
2
s=0
s=0
and
(7)
wtg (ei+1y ) = wtfi+1 (ei+1y ) + 3
i
X
s=0
i
ts −
2
!
≥3+3
i
X
s=0
i
ts −
2
!
.
So (6) and (7) give that
wtg (eix ) ≤ 3
i
X
s=0
i
ts −
2
!
+1<3
i
X
s=0
i
ts −
2
!
+ 3 ≤ wtg (ei+1y ).
S
Hence, in m
two edges
i=1 Gi under the labeling g, there are
the same
m−1of
Pno
m
edge-weight.
Therefore, g is an edge irregular total
-labeling
i=1 ti −
2
S
of m
G
.
This
concludes
the
proof.
i=1 i
5
On the total irregularity strength of disjoint union of arbitrary graphs
473
Next theorem provides an upper bound of the total vertex irregularity
strength for disjoint union of regular graphs.
Theorem 2.2. Let Gi , i = 1, 2, . . . , m, be an r-regular graph. Then
!
m
m
[
X
m−1
tvs
Gi ≤
tvs(Gi ) −
.
2
i=1
i=1
Proof. Let tvs(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be a
vertex irregular total ti -labeling of an r-regular graph Gi , i = 1, 2, . . . , m. We
define t0 = 0. Let V (Gi ) = {via : a = 1, 2, . . . , pi } andPE(Gi ) =
{eix :
m−1
m
t
−
x = 1, 2, . . . , qSi }, for every i = 1, 2, . . . , m. Define a total
i=1 i
2
m
labeling g of i=1 Gi in the following way:
g(via ) = fi (via ) +
g(eix ) = fi (eix ) +
i−1
X
s=0
i−1
X
ts −
i
ts −
i−1 if via ∈ V (Gi ),
2
if eix ∈ E(Gi ),
2
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
Let eia1 , eia2 , . . . , eiar be edges incident with the vertex via .
Case 1. If i is odd then for vertex-weights under the labeling g we have
wtg (via ) = g(via ) +
=
r
X
g(eiah )
h=1
i−1
X
fi (via ) +
s=0
i−1
ts −
2
i−1
X
!
+
r
X
fi (eiah ) +
i−1
X
s=0
h=1
i−1
ts −
2
!
!
i−1
.
2
s=0
P
i−1
i−1
It is easy to see that (r + 1)
t
−
is a constant for a fixed i and
s=0 s
2
since wtfi (via ) 6= wtfi (vib ) for every a 6= b, thus also wtg (via ) 6= wtg (vib ), for
i = 1, 2, . . . , m.
Case 2. If i is even then for vertex-weights we get
= wtfi (via ) + (r + 1)
wtg (via ) = g(via ) +
=
r
X
ts −
g(eiah )
h=1
i−1
X
fi (via ) +
s=0
i
ts −
2
!
+
r
X
h=1
fi (eiah ) +
i−1
X
s=0
i
ts − + 1
2
!
474 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača
i−1
X
= wtfi (via ) + (r + 1)
s=0
i
ts −
2
6
!
+ r.
Again we can see that wtg (via ) 6= wtg (vib ) for every a =
6 b and fixed i, i =
1, 2, . . . , m.
Next we show that wtg (via ) < wtg (vi+1b ) for every a ∈ {1, 2, . . . , pi } and
b ∈ {1, 2, . . . , pi+1 }.
Case 1. If i is odd then
(8)
i−1
X
!
i−1
wtg (via ) = wtfi (via ) + (r + 1)
ts −
2
s=0
!
!
i−1
i
X
X
i−1
i−1
≤ (r + 1)ti + (r + 1)
= (r + 1)
ts −
ts −
2
2
s=0
s=0
and
(9)
i
X
!
i+1
ts −
wtg (vi+1b ) = wtfi+1 (vi+1b ) + (r + 1)
+r
2
s=0
!
i
X
i+1
≥ (r + 1) · 1 + (r + 1)
ts −
+r
2
s=0
!
!
i
i
X
X
i−1
i−1
= (r + 1)
ts −
+ r > (r + 1)
ts −
.
2
2
s=0
s=0
According to (8) and (9) we have that
wtg (via ) ≤ (r + 1)
i
X
s=0
i−1
ts −
2
!
< wtg (vi+1b ).
Case 2. If i is even then
(10)
i−1
X
!
i
wtg (via ) = wtfi (via ) + (r + 1)
ts −
+r
2
s=0
!
i−1
X
i
≤ (r + 1)ti + (r + 1)
ts −
+r
2
s=0
!
i
X
i
= (r + 1)
ts −
+r
2
s=0
7
On the total irregularity strength of disjoint union of arbitrary graphs
475
and
(11)
wtg (vi+1b ) = wtfi+1 (vi+1b ) + (r + 1)
i
X
s=0
≥ (r + 1) · 1 + (r + 1)
i
X
s=0
i
ts −
2
!
> (r + 1)
i
ts −
2
i
X
s=0
!
i
ts −
2
!
+ r.
From (10) and (11) it follows that
wtg (via ) ≤ (r + 1)
i
X
s=0
i
ts −
2
!
+ r < wtg (vi+1b ).
labeling
g has the
Sm required properties of vertex irregular total
Pm Thus, the
m−1
-labeling of i=1 Gi . i=1 ti −
2
Using Theorems 2.1 and 2.2 we obtain the following result.
Theorem 2.3. Let Gi , i = 1, 2, . . . , m, be an r-regular graph. Then
!
m
m
[
X
m−1
ts
Gi ≤
ts(Gi ) −
.
2
i=1
i=1
Proof. Let ts(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be a totally
irregular total ti -labeling of an r-regular graph Gi , for every i = 1, 2, . . . , m.
We put t0 = 0. Let V (Gi ) = {via : a = 1, 2, . . . , pi } and
E(Gi ) =
{eix :
m−1
Pm
x = 1, 2, . . . , qSi }, for every i = 1, 2, . . . , m. Define a total
i=1 ti −
2
labeling g of m
G
as
follows:
i=1 i
g(via ) = fi (via ) +
g(eix ) = fi (eix ) +
i−1
X
s=0
i−1
X
ts −
i
ts −
i−1 2
2
if via ∈ V (Gi ),
if eix ∈ E(Gi ),
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
It follows from Theorems S
2.1 and 2.2 that under the labeling g for every
m
0
0
two different edges e and e of S
i=1 Gi there is wtg (e) 6= wtg (e ) and for every
m
two distinct vertices x and y P
of i=1 Gi there
wtg (y). Therefore,
is wtg (x) 6= S
m
m
m−1
g is a totally irregular total
t
−
-labeling
of
i=1 i
i=1 Gi .
2
Next theorem gives the exact value of the total edge irregularity strength
for disjoint union of graphs Gi , i = 1, 2, . . . , m, satisfying some certain conditions.
476 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača
8
Theorem 2.4. Let Gi , i = 1, 2, . . . , m, be a graph. If there is an edge irregular total (tes(Gi ))-labeling of Gi such that the edge-weight function wtfi (eix ) :
E(Gi ) → {3, 4, . . . , 3tes(Gi ) − 1} is a bijection for every i = 1, 2, . . . , m, then
!
m
m
[
X
tes
Gi =
tes(Gi ) − m + 1.
i=1
i=1
Proof. Let Gi , i = 1, 2, . . . , m, be a graph of order pi and size qi . Let
tes(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be an edge irregular total
ti -labeling of Gi such that the edge-weight function wtfi is a bijection from
E(Gi ) to {3, 4, . . . , 3ti − 1}, for every i = 1, 2, . . . , m. For purposes of the proof
let t0 = 0. Thus the size of Gi is qi = 3ti − 3, i = 1, 2 . . . , m. Therefore,
!
!
m
m
m
m
m
X
[
X
X
X
Gi =
qi =
(3ti − 3) =
3ti − 3m = 3
ti − m .
E
i=1
i=1
i=1
i=1
i=1
From (2) it follows that
  P

m
m
!  S
+2
m
m
3
t
−m
+2
E
G
i
i
X
[
i=1
i=1




ti − m + 1.
(12) tes
Gi ≥ 
3
3
=
=
 
 i=1

i=1
Pm
SmFor the converse, we define a suitable total ( i=1 ti − m + 1)-labeling g
of i=1 Gi as follows:
g(zia ) = fi (zia ) +
i−1
X
ts − i + 1
if zia ∈ V (Gi ) ∪ E(Gi ),
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
Let eix = via vib be an edge in E(Gi ) for via , vib ∈ V (Gi ). For edge-weight
of an edge eix = via vib in E(Gi ) under the labeling g we have
wtg (eix ) =g(via ) + g(eix ) + g(vib )
= fi (via ) +
+
i−1
X
!
ts − i + 1
s=0
i−1
X
fi (vib ) +
+
fi (eix ) +
i−1
X
!
ts − i + 1
s=0
!
ts − i + 1
= wtfi (eix ) + 3
i−1
X
s=0
Since for fixed i is wtfi (eix ) 6= wtfi (eiy ) for every x 6= y and 3
is also a constant, thus we get wtg (eix ) 6= wtg (eiy ).
!
ts − i + 1 .
s=0
P
i−1
s=0 ts
−i+1
9
On the total irregularity strength of disjoint union of arbitrary graphs
477
Now we show that wtg (eix ) < wtg (ei+1y ), for every x = 1, 2, . . . , qi , y =
1, 2, . . . , qi+1 and i = 1, 2, . . . , m − 1. So
!
i−1
X
(13)
wtg (eix ) = wtfi (eix ) + 3
ts − i + 1
s=0
i−1
X
≤ 3ti − 1 + 3
!
ts − i + 1
=3
s=0
=3
i
X
i
X
!
ts − i + 1
−1
s=0
!
ts − i
+2
s=0
and
(14)
i
X
wtg (ei+1y ) = wtfi+1 (ei+1y ) + 3
!
ts − i
≥3+3
s=0
>3
i
X
i
X
!
ts − i
s=0
!
ts − i
+ 2.
s=0
From (13) and (14) it follows that
wtg (eix ) ≤ 3
i
X
!
ts − i
+ 2 < wtg (ei+1y ).
s=0
P
We can see that all vertex and edge labels are at most m
i=1 ti − m + 1
and the edge-weights are distinct for all
pairs
of
distinct
edges.
Therefore,
the
Pm
Sm
labeling g is an edge irregular total ( i=1 ti − m + 1)-labeling of i=1 Gi . It
proves that
!
m
m
[
X
(15)
tes
Gi ≤
ti − m + 1.
i=1
i=1
Inequalities (12) and (15) imply the assertion.
The exact value of the total vertex irregularity strength for disjoint union
of arbitrary r-regular graphs Gi , i = 1, 2, . . . , m, satisfying some certain conditions determines the following theorem.
Theorem 2.5. Let Gi , i = 1, 2, . . . , m, be an r-regular graph. If there
is a vertex irregular total (tvs(Gi ))-labeling of Gi such that the vertex-weight
function wtfi (via ) : V (Gi ) → {r + 1, r + 2, . . . , (r + 1)tvs(Gi ) − 1} is a bijection
for every i = 1, 2, . . . , m, then
!
m
m
[
X
tvs
Gi =
tvs(Gi ) − m + 1.
i=1
i=1
478 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača 10
Proof. Let Gi , i = 1, 2, . . . , m, be an r-regular graph of order pi . Let
tvs(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be a vertex irregular
total ti -labeling of Gi such that the vertex-weight function wtfi is a bijection
from V (Gi ) to {r + 1, r + 2, . . . , (r + 1)ti − 1}, for every i = 1, 2, . . . , m. Thus
pi = (r + 1)(ti − 1). Therefore,
!
m
m
m
m
X
[
X
X
Gi =
pi =
(r + 1)(ti − 1) = (r + 1)(
ti − m).
V
i=1
i=1
i=1
i=1
From (1) it follows that
m

! 
P
m
m
(r+1)
ti −m +r
X
[
i=1
=
ti − m + 1.
(16)
tvs
Gi ≥ 
r+1


 i=1

i=1
Sm
Pm
i) ≤
i=1 ti − m + 1 we define a total
SG
Pm For proving that tvs ( i=1
G
in
the
following way:
( i=1 ti − m + 1)-labeling g of m
i
i=1
g(zia ) = fi (zia ) +
i−1
X
ts − i + 1
if zia ∈ V (Gi ) ∪ E(Gi ),
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
Observe that if eia1 , eia2 , . . . , eiar are edges incident with the vertex via ∈
V (Gi ) then under the total labeling g the vertex via , i = 1, 2, . . . , m, receives
the weight
r
X
wtg (via ) =g(via ) +
g(eiah )
h=1
i−1
X
= fi (via ) +
=fi (via ) +
!
ts − i + 1
+
s=0
i−1
X
r
X
s=0
h=1
fi (eiah ) +
h=1
ts − i + 1 +
= wtfi (via ) + (r + 1)
r
X
i−1
X
fi (eiah ) + r
i−1
X
s=0
i−1
X
!
ts − i + 1
!
ts − i + 1
s=0
!
ts − i + 1 .
s=0
P
i−1
Since, for a fixed i, the expression (r + 1)
t
−
i
+
1
is a constant and
s=0 s
wtfi (via ) 6= wtfi (vib ) for every a 6= b, thus also wtg (via ) 6= wtg (vib ).
Next we show that wtg (via ) < wg (vi+1b ) for every a = 1, 2, . . . , pi , b =
1, 2, . . . , pi+1 and i = 1, 2, . . . , m − 1. It means
!
i−1
X
(17)
wtg (via ) = wtfi (via ) + (r + 1)
ts − i + 1
s=0
11
On the total irregularity strength of disjoint union of arbitrary graphs
i−1
X
≤ (r + 1)ti − 1 + (r + 1)
479
!
ts − i + 1
s=0
= (r + 1)
i
X
!
− 1.
ts − i + 1
s=0
and
(18)
wtg (vi+1b ) = wtfi+1 (vi+1b ) + (r + 1)
i
X
!
ts − i
s=0
≥ (r + 1) + (r + 1)
i
X
!
ts − i
s=0
= (r + 1)
i
X

!
ts − i + 1
> (r + 1) 
s=0
i
X

tp − i + 1 − 1.
p=1
According to inequalities (17) and (18) we have that wtg (via ) < wg (vi+1b ).
P
We can see that all vertex and edge labels are at most m
i=1 ti − m +
1 and the vertex-weights are different for all pairs of distinct vertices. In
fact,
the required properties of vertex irregular total
Pm the total labeling g has S
( i=1 ti − m + 1)-labeling of m
i=1 Gi and thus
!
m
m
[
X
(19)
tvs
Gi ≤
ti − m + 1.
i=1
i=1
Combining (16) and (19) we obtain the equality.
Using Theorems 2.4 and 2.5, we have the following result.
Theorem 2.6. Let Gi , i = 1, 2, . . . , m, be an r-regular graph. If there
is a totally irregular total (ts(Gi ))-labeling of Gi such that the edge-weight
function wtfi (eix ) : E(Gi ) → {3, 4, . . . , 3ts(Gi ) − 1} is a bijection for every
i = 1, 2, . . . , m and the vertex-weight function wtfi (via ) : V (Gi ) → {r + 1, r +
2, . . . , (r + 1)ts(Gi ) − 1} is a bijection for every i = 1, 2, . . . , m, then
!
m
m
[
X
ts
Gi =
ts(Gi ) − m + 1.
i=1
i=1
Proof. Let Gi , i = 1, 2, . . . , m, be an r-regular graph of order pi . Let
ts(Gi ) = ti and fi : V (Gi ) ∪ E(Gi ) → {1, 2, . . . , ti } be a totally irregular total
ti -labeling of Gi , for every i = 1, 2, . . . , m. Let t0 = 0.
480 R. Ramdani, A.N.M. Salman, H. Assiyatum, A. Semaničová-Feňovčı́ková, M. Bača 12
As wtfi (eix ) : E(Gi ) → {3, 4, . . . , 3ts(Gi ) − 1} is a bijection, for every
i = 1, 2, . . . , m, then from (12) we have that
!
m
m
[
X
(20)
tes
Gi ≥
ti − m + 1
i=1
i=1
and from (3), we get
(21)
ts
m
[
m
[
!
≥ tes
Gi
i=1
!
Gi
.
i=1
Thus (20) and (21) give
(22)
m
[
ts
!
≥
Gi
i=1
m
X
ti − m + 1.
i=1
As wtfi (via ) : V (Gi ) → {r + 1, r + 2, . . . , (r + 1)ts(Gi ) − 1} is a bijection,
for every i = 1, 2, . . . , m, then from (16) we get
!
m
m
[
X
(23)
tvs
Gi ≥
ti − m + 1
i=1
i=1
and from (3) we have that
(24)
ts
m
[
!
Gi
≥ tvs
i=1
m
[
!
Gi
.
i=1
Hence from (23) and (24) it follows
!
m
m
[
X
(25)
ts
Gi ≥
ti − m + 1.
i=1
i=1
P
Sm
For the converse, we define a total ( m
i=1 ti − m + 1)-labeling g of i=1 Gi
as follows:
i−1
X
g(zia ) = fi (zia ) +
ts − i + 1 if zia ∈ V (Gi ) ∪ E(Gi ),
s=0
where a = 1, 2, . . . , pi , x = 1, 2, . . . , qi and i = 1, 2, . . . , m.
From Theorem S
2.4 and 2.5 it follows that the S
labeling g isP
a totally irregm
m
ular total labeling of i=1 Gi which proves that ts ( i=1 Gi ) ≤ m
i=1 ti − m + 1.
Combining with the lower bound (25), we conclude that
!
m
m
m
[
X
X
ts
Gi =
ti − m + 1 =
ts(Gi ) − m + 1. i=1
i=1
i=1
As a consequence of Theorem 2.6 we obtain the exact value of the total
irregularity strength for disjoint union of prisms Dni = Cni P2 , i = 1, 2, . . . , m.
13
On the total irregularity strength of disjoint union of arbitrary graphs
481
Corollary 2.7. Let m ≥ 2, ni ≥ 3, i = 1, 2, . . . , m. Then the total
irregularity strength for disjoint union of prisms Dni = Cni P2 is
!
m
m
[
X
ts
Dni =
ni + 1.
i=1
i=1
Proof. Ramdani and Salman [20] constructed a totally irregular total
(ts(Dn ))-labeling f : V (Dn ) ∪ E(Dn ) → {1, 2, . . . , ts(Dn )} of Dn such that the
edge-weight and vertex-weight functions satisfy the assumptions of Theorem
2.6 and they proved that ts(Dn ) = ts(Cn P2 ) = n + 1. Hence from Theorem
2.6 for disjoint union of prism Dni , i = 1, 2, . . . , m, it follows that
!
m
m
X
[
ts(Dni ) − m + 1.
(26)
ts
Dni =
i=1
i=1
As ts(Dni ) = ni + 1, for i = 1, 2, . . . , m, then from (26) we get
!
m
m
m
m
X
X
X
[
ts
ts(Dni ) − m + 1 =
(ni + 1) − m + 1 =
ni + 1.
Dni =
i=1
i=1
i=1
i=1
Acknowledgments. The research for this article was supported by APVV-15-0116.
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Received 23 February 2015
Institut Teknologi Bandung,
Department of Mathematics,
Bandung, Indonesia
[email protected]
[email protected]
Institut Teknologi Bandung,
Department of Mathematics,
Bandung, Indonesia
Universitas Islam Negeri,
The Faculty of Sciences and Technologies,
Department of Mathematics,
Bandung, Indonesia
[email protected]
Technical University,
Department of Applied Mathematics and Informatics,
Košice, Slovak Republic
[email protected]
[email protected]