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1. International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org Volume 2 Issue 10ǁ October. 2013 ǁ PP.01-17
K-Even Edge-Graceful Labeling of Some Cycle Related Graphs
1
Dr. B. Gayathri, 2S. Kousalya Devi
1,2
M.Sc., M.Phil., Ph.D., Pgdca., M.C.A., Associate Professor And Research Advisor Post Graduate & Research
Department Of Mathematics Periyar E.V.R.College (Autonomous)
Tiruchirappalli-620 023. Tamilnadu. India.
ABSTRACT: In 1985, Lo[6] introduced the notion of edge-graceful graphs. In [4], Gayathri et al., introduced
the even edge-graceful graphs. In [8], Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang introduced the k-edgegraceful graphs.We introduced k-even edge-graceful graphs. In this paper, we investigate the k-even edgegracefulness of some cycle related graphs.
KEYWORDS: k-even edge-graceful labeling, k-even edge-graceful graphs. AMS MOS) subject classification:
05C78.
I. INTRODUCTION
All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense
of Harary[5]. The symbols V(G) and E(G) will denote the vertex set and edge set of a graph G. The cardinality
of the vertex set is called the order of G denoted by p. The cardinality of the edge set is called the size of G
denoted by q. A graph with p vertices and q edges is called a (p, q) graph.
In 1985, Lo[6] introduced the notion of edge-graceful graphs. In [4], Gayathri et al., introduced the
even edge-graceful graphs and further studied in. In [8], Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang
introduced the k-edge-graceful graphs. We have introduced k-even edge- graceful graphs.
Definition 1.1:
k-even edge-graceful labeling (k-EEGL) of a (p, q) graph G(V, E) is an injection f from E to {2k – 1, 2k,


2k + 1, ..., 2k + 2q – 2} such that the induced mapping f defined on V by f ( x) = (f(xy)) (mod 2s) taken
over all edges xy are distinct and even where s = max{p, q} and k is an integer greater than or equal to 1. A
graph G that admits k-even edge-graceful labeling is called a k-even edge-graceful graph (K-EEGG).
Remark 1.2:
1-even edge-graceful labeling is an even edge-graceful labeling.
The definition of k-edge-graceful and k-even edge-graceful are equivalent to one another in the case of
trees.
The edge-gracefulness and even edge-gracefulness of odd order trees are still open. The theory of 1-even
edge-graceful is completely different from that of k- even edge-graceful. For example, tree of order 4 is 2-even
edge-graceful but not 1-even edge-graceful. In this paper we investigate the k-even edge-gracefulness of some
cycle related graphs. Throughout this paper, we assume that k is a positive integer greater than or equal to 1.
2. Prior Results:
1.Theorem : If a (p, q) graph G is k-even edge-graceful with all edges labeled with even numbers and p  q
then G is k-edge-graceful.
2.Theorem : If a (p, q) graph G is k-even edge-graceful in which all edges are labeled with even numbers and
p  q then q(q  2k  1) 
Further q(q + 2k – 1) 
p( p  1)
(mod p) .
2
0(mod p) if p is odd

p
 2 (mod p) if p is even

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2. K-Even Edge-Graceful Labeling Of…
3.Theorem : If a (p, q) graph G is k-even edge-graceful in which all edges are labeled with even numbers and p
 q then p  0, 1 or 3 (mod 4).
4.Theorem : If a (p, q) graph G is a k-even edge-graceful tree of odd order then
p
(l – 1) + 1
2
k=
where l is any odd positive integer and hence k  1(mod p).
5. Observation : We observe that any tree of odd order p has the sum of the labels congruent to 0 (mod p).
6. Theorem :
If a (p, q) graph G is a k-even edge-graceful tree of even order with
p  0 (mod 4) then k =
k
Further
p
(2l – 1) + 1 where l is any positive integer.
4
p4
 4 (mod p) if l is odd


 3 p  4 (mod p) if l is even
 4

2. MAIN RESULTS
Definition 2.1
Let Cn denote the cycle of length n. Then the join of {e} with any one vertex of Cn is denoted by Cn 
{e}. In this graph, p = q = n + 1.
Theorem 2.2
The
kz
graph
Cn

{e}
p
p

 mod  , where 0  z   1 ,
2
2

of
even
order
is
for
all
n  1 (mod 4) and n  1.
Proof
Let {v1, v2, ..., vn, v} be the vertices of Cn  {e}, the edges ei =
and en+1 =
k-even-edge-graceful
 v2 , v  (see Figure 1).
vn-1
 vi , vi1  for 1  i  n – 1; e =  vn , v1 
n
vn
en-1
en
...
v1
..
e1
.
e2
v3
en+1
v2
v
Figure 1: Cn  {e} with ordinary labeling
First, we label the edges as follows:
For k  1, 1  i  n and i is odd,
f  ei 
= 2k + i – 2.
When i is even, we label the edges as follows:
For
p2
p

,
k  z  mod  , 0  z 
4
2

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3. K-Even Edge-Graceful Labeling Of…
For
For
For
n  4z  7

2k  n  i  2 for 1  i 

2
f  ei  = 
n  4z  7
 2k  n  i
for
 i  n.

2

p2
p
k
 mod  ,
4 
2
f  ei  = 2k + n + i.
k
p6
p
 mod  ,
4 
2
f  ei  = 2k + n + i – 2.
p  p  10
p

k  z  mod  ,
 z   1,
2
4
2

3n  4 z  9

2k  n  i  2 for 1  i 

2
f  ei  = 
3n  4 z  9
 2k  n  i
for
 i  n.

2

 2k
when k  0  mod p 
f  en1  = 
2k  2n  2 z  2 when k  z  mod p  and 1 z  p  1.
Then the induced vertex labels are as follows:
p2
p

 mod  , 0  z 
4
2

n  4z  7

n  4 z  2i  5 for 1  i 

2
f   vi  = 
 4 z  n  2i  5 for n  4 z  7  i  n.

2

p+2
p
Case 2: k 
 mod 
4 
2

f  v1  = 2n
f   vi  = 2i – 2
;
Case 1: k  z
Case 3:
k
f   vi 
Case 4:
for 2  i  n.
p+6
p
 mod 
4 
2
= 2i
for 1  i  n.
p  p  10
p

k  z  mod  ,
 z  1
2
4
2

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4. K-Even Edge-Graceful Labeling Of…
3n  4 z  9

4 z  n  2i  7 for 1  i <

2
f   vi  = 
4 z  3n  2i  7 for 3n  4 z  9  i  n.

2

p
p

For k  z  mod  , 0  z 
 1,
2
2

f   vn1  = 0.
Therefore,
labels
Cn

0z
are
{e}
of
f  V 
= {0, 2, 4, ..., 2s – 2}, where s = max{p, q} = n + 1. So, it follows that the vertex
all
even
distinct
order
is
and
even.
k-even-edge-graceful
for
Hence,
all
k
the
graph

p

 mod  ,
2

where
z
p
 1 , n  1 (mod 4) and n  1. 
2
For example, consider the graph C13  {e}. Here n = 13;
s = max {p, q} = 14; 2s = 28. The 21-EEGL of C13  {e} is given in Figure 2.
51
65
67
6
4
49
53
8
2
10
41
26
56
12
61
0
55
24
14
47
22
16
20
59
43
18
57
45
Figure 2: 21-EEGL of C13  {e}
For example, consider the graph C17  {e}. Here n = 17;
s = max {p, q} = 18; 2s = 36. The 5-EEGL of C17  {e} is given in Figure 3.
23
41
21
28
26
30
43
32
25
24
39
34
22
19
9
20
2
36
37
0
29
18
4
17
16
35
6
14
15
8
12
10
33
11
31
13
Figure 3: 5-EEGL of C17  {e}
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5. K-Even Edge-Graceful Labeling Of…
Definition 2.3 [8]
For p  4, a cycle (of order p) with one chord is a simple graph obtained from a p-cycle by adding a
chord. Let the p-cycle be v1v2 ... vpv1. Without loss of generality, we assume that the chord joins v1 with any one vi,
where 3  i  p – 1. This graph is denoted by Cp(i). For example
cycle
by
q = p + 1.
adding
a
chord
between
the
C p  5
vertices
means a graph obtained from a p-
v1
and
v5.
In
this
graph,
Theorem 2.4
The graph Cn(i), (n > 4), 3  i  n – 1, cycle with one chord of odd order is k-even-edge-graceful for all
kz
q
q

 mod  , where 0  z   1 .
2
2

Proof
Let
ei =
 vi , vi1 
{v1,
v2,
v3,
...,
vi,
for 1  i  n – 1; en =
vi+1,
 vn , v1 
...,
vn}
and en+1 =
be
the
vertices
of
Cn(i),
the
edges
 v1 , vi  , 3  i  n – 1 (see Figure 4). The chord
connecting the vertex v1 with vi, (i  3) is shown in Figure 4. For this graph, p = n and q = n + 1.
v1
vn
en
e1
v2
en-1
e2
v3
vn-1
e3
en-2
en+1
vn-2
v4
e4
vi+2
..
.
.
..
ei+2
ei+1
ei
vi+1
vi
Figure 4: Cn(i) with ordinary labeling
First, we label the edges as follows:
For k  1 and 1  i  n,
f  ei 
=
f  en1 
=
when i is odd
 2k  i  2

2k  n  i  2 when i is even.
 2k
when k  0  mod q 

2k  2n  2 z  2 when k  z  mod q  , 1  z  q  1.
Then the induced vertex labels are as follows:
Case I: n  1 (mod 4) and n  1
q+2
q

 mod  , 0  z 
4
2

n  4z  7

 4 z  n  2i  5 for 1  i 

2
f   vi  = 
4 z  n  2i  7 for n  4 z  7  i  n.

2

Case 1: k  z
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6. K-Even Edge-Graceful Labeling Of…
Case 2:
k
f   vi 
q6
q
 mod 
4 
2
= 2i
for 1  i  n.
q  q  10
q

k  z  mod  ,
 z  1
2
4
2

3n  4 z  9

4 z  n  2i  7 for 1  i 

2
f   vi  = 
4 z  3n  2i  9 for 3n  4 z  9  i  n.

2

Case 3:
Case II: n  3 (mod 4) and n  3
q
q

k  z  mod  , 0  z 
2
4

n  4z  7

 4 z  n  2i  5 for 1  i 

2
f   vi  = 
4 z  n  2i  7 for n  4 z  7  i  n.

2

Case 1:
Case 2:
k
f   vi 
Case 3:
q4
q
 mod 
4 
2
= 2i – 2 for 1  i  n.
q  q 8
q

k  z  mod  ,
 z  1
2 4
2

f   vi 
Therefore,
=
f  V 
3n  4 z  9

4 z  n  2i  7 for 1  i 


2

4 z  3n  2i  9 for 3n  4 z  9  i  n.

2

 {0, 2, 4, ..., 2s – 2}, where s = max {p, q} = n + 1. So, it follows that the vertex labels
are all distinct and even. Hence, the graph
Cn(i), (n > 4), 3  i  n – 1, cycle with one chord of odd order is k-even-edge-graceful for all k  z
0z
q
 1. 
2
For example, consider the graph
q

 mod  ,
2

C9  5 . Here n = 9; s = max {p, q} = 10; 2s = 20.
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7. K-Even Edge-Graceful Labeling Of…
The 15-even-edge-graceful labeling of
C9  5
is given in Figure 5.
6
29
37
2
8
39
45
10
0
40
31
35
12
18
41
43
33
14
16
Figure 5: 15-EEGL of C9(5)
For example, consider the graph C11(6). Here n = 11; s = max {p, q} = 12; 2s = 24. The 11-EEGL of C11(6) is given
in Figure 6.
0
4
31
21
6
41
22
33
29
8
24
20
23
39
18
10
35
27
16
12
25
37
14
Figure 6: 11-EEGL of C11(6)
Theorem 2.5
The graph
for all
Cn  i  , (n  4), 3  i  n – 1, cycle with one chord of even order is k-even-edge-graceful
k  z  mod q  , where 0  z  q – 1 and n  0  mod 4  .
Proof
Let the vertices and edges be defined as in Theorem 6.3.2.
First, we label the edges as follows:
For k  1,
f  e1 
= 2k + 1
f  ei 
f  e2  = 2k – 1
= 2k + 2i – 3
;
for
3i 
n
 3.
2
n
 3  i  n,
2
2k  2i  1 when i is odd
f  ei  = 
2k  2i  3 when i is even.
For k  1 and
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8. K-Even Edge-Graceful Labeling Of…
f  en1 
=
 2k
when k  0  mod q 

2k  2n  2 z  2 when k  z  mod q  , 1  z  q  1.
Then the induced vertex labels are as follows:
Case 1: k  0 (mod q)
f   v1 
f   v3 
= 2n – 2
;
f   v2 
= 0.
= 2.
n

4i  8
for 4  i   3


2
f   vi  = 
4i  2n  6 for n  3  i  n.

2

q 1
Case 2: k  z (mod q), 1  z 
2

when i = 1, 2.
f  vi  = 4z + 4i – 8
For k  z (mod q), 1 
f   v3 
z
q 3
,
2
= 4z + 2
For 4  i  n,
f   vi 
For
=

4 z  4i  8

4 z  4i  2n  10



4 z  4i  2n  6

4 z  4i  4n  8

n
z3
2
n
n
for  z  3  i   3
2
2
n
for  3  i  n  z  2
2
for n  z  2  i  n.
for 4  i 
q 1
 mod q  ,
2
f   v3  = 0
k
n

4i  10
for 4  i   3


2
f   vi  = 
4i  2n  8 for n  3  i  n.

2

q +1
Case 3: k 
 mod q 
2
f   v1  = 2n
f   v2  = 2.
;
f   v3 
= 4.
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9. K-Even Edge-Graceful Labeling Of…
f   vi 
Case 4:

4i  6

= 0



4i  2n  4

k  z  mod q  ,
f   vi 
f   v3 
q+3
 z  q 1
2
= 4z + 4i – 2n – 10 when i = 1, 2.
= 4z – 2n.
For k  z (mod q),
f   vi 
n
2
2
n
when i   2
2
n
for
 3  i  n.
2
for 4  i <
q3
 z  q2,
2
4 z  4i  2n  10

4 z  4i  4n  12
= 


4 z  4i  4n  8

4 z  4i  6n  10


for 4  i  n  z  3
n
for n  z  3  i   3
2
n
3n
for  3  i 
z3
2
2
3n
for
 z  3  i  n.
2
For k  q – 1 (mod q),
f   vi 
n

4i  12
for 4  i   3


2
= 
4i  2n  10 for n  3  i  n.

2

Therefore,
f  V   0, 2, 4, ..., 2s  2 , where s = max{p, q} = n + 1. So, it follows that the
Cn  i  , (n  4), 3  i  n – 1, cycle will one chord of
vertex labels are all distinct and even. Hence, the graph
even order is k-even-edge-graceful for all k  z (mod q), where 0  z  q – 1 and n  0 (mod 4). 
For example, consider the graph C16(5), Here n = 16; s = max{p, q} = 17; 2s = 34.
The 18-EEGL of C16(5) is given in Figure 7.
0
30
4
37
6
35
65
39
68
67
12
41
26
16
61
43
22
20
63
45
18
24
57
14
47
59
49
53
10
51
2
28
32
Figure 7: 18-EEGL of C16(5)
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9 | Page

10. K-Even Edge-Graceful Labeling Of…
Definition 2. 6
The crown Cn  K1 is the graph obtained from the cycle Cn by attaching pendant edge at each vertex of
the cycle and is denoted by
Cn . In this graph,
p = q = 2n.
Theorem 2.7
The crown graph
where

Cn ,  n  3
of even order is k-even-edge-graceful for all
p

k  z  mod  ,
3

p
 1 and n  0 (mod 3).
3
0 z
Proof
For this graph, p = q = 2n. Let v1, v2, v3, ..., vn and
C

n
'
'
'
v1' , v2 , v3 , ..., vn
be the vertices and pendant vertices of
respectively.
v1'
e1'
v1
e1
en
vn
'
vn
v2
e2'
v3
e3'
en'
en-1
e2
vn-1
vn' 1
v2'
'
en 1
v3'
e3
en-2
...
..
v4
.
e4'
..
v4'
.
Figure 8:
+
Cn
with ordinary labeling
The edges are defined by
ei =
 vi , vi1 
and
ei'
=
for 1  i  n – 1 ;
v , v 
i
'
i
en =
 vn , v1 
for 1  i  n (see Figure 8).
First, we label the edges as follows:
For k  1,
f  ei 
f  e1' 
f  ei' 
for 1  i  n
= 2k + 4i – 5
= 2k
for 2  i  n.
= 2k + 4(n – i + 1)
Then the induced vertex labels are as follows:
Case 1:
p

k  0  mod 
3

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11. K-Even Edge-Graceful Labeling Of…
f   vi 
=
4n  4i  10 when i  1, 2

for 3  i  n.
4i  10
p
p

k  z  mod  , 1  z   1 and z is odd
3
3

5  3z

6 z  4i  10
for 1  i  n 


2
f   vi  = 
6 z  4n  4i  10 for n  5  3z  i  n.

2

p
p

Case 3: k  z  mod  , 1  z 
 1 and z is even
3
3

6  3z

for 1  i  n 
6 z  4i  10

2
f   vi  = 
6 z  4n  4i  10 for n  6  3z  i  n.

2

Case 2:
The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex

Cn ,  n  3
labels are all distinct and even. Hence, the crown graph
for all
p

k  z  mod  ,
3

where
0 z
For example, consider the graph
graceful labeling of
C6
of even order is k-even-edge-graceful
p
 1 and n  0 (mod 3). 
3
C6 . Here p = q = 12; s = max{p, q} = 12; 2s = 24. The 2-even-edge-
is given in Figure 9.
4
4
23
8
8
6
3
19
12
12
24
10
2
0
7
22
15
14
18
20
20
11
16
16
Figure 9: 2-EEGL of
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+
C6
11 | Page

12. K-Even Edge-Graceful Labeling Of…
Theorem 2.8
The crown graph

Cn ,  n  4 
of even order is k-even-edge-graceful for all
k  z  mod p  ,
where 0  z  p – 1, n  1 (mod 3) and n  1.
Proof
Let the vertices and edges be defined as in Theorem 6.4.2. The edge labels are also same as in Theorem
6.4.2.
Then the induced vertex labels are as follows:
Case 1: k  0 (mod p)
f   vi 
=
4n  4i  10 when i  1, 2

for 3  i  n.
4i  10
When z is odd, the induced vertex labels are given below:
Case 2: k  z (mod p), 1  z 
p +1
3
5  3z

6 z  4i  10
for 1  i  n 


2
f   vi  = 
6 z  4n  4i  10 for n  5  3z  i  n.

2

p+4
2p+2
Case 3: k  z (mod p),
z
3
3
5  3z

6 z  4n  4i  10 for 1  i  2n  2

f   vi  = 
6 z  8n  4i  10 for 2n  5  3z  i  n.

2

2p+5
Case 4: k  z (mod p),
 z  p1
3
5  3z

6 z  8n  4i  10 for 1  i  3n+ 2

f   vi  = 
6 z  12n  4i  10 for 3n  5  3z  i  n.

2

When z is even, the induced vertex labels are given below:
Case 5:
k  z  modp  , 1  z 
f   vi 
Case 6:
=
p +1
3
6  3z

6 z  4i  10
for 1  i  n 


2

6 z  4n  4i  10 for n  6  3z  i  n.

2

k  z  mod p  ,
p+4
2p+2
z 
3
3
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12 | Page

13. K-Even Edge-Graceful Labeling Of…
6  3z

6 z  4n  4i  10 for 1  i  2n  2

f   vi  = 
6 z  8n  4i  10 for 2n  6  3z  i  n.

2

2p+5
Case 7: k  z  mod p  ,
z  p  1
3
6  3z

6 z  8n  4i  10 for 1  i  3n 


2
f   vi  = 
6 z  12n  4i  10 for 3n  6  3z  i  n.

2

The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex

Cn ,  n  4 
labels are all distinct and even. Hence, the crown graph
for all
of even order is k-even-edge-graceful
k  z  mod p  , where 0  z  p – 1, n  1 (mod 3) and n  1. 
For example, consider the graph
graceful labeling of
C

7
C7 .
Here p = q = 14; s = max {p, q} = 14; 2s = 28. The 5-even-edge-
is given in Figure 10.
10
10
24
9
33
14
20
14
0
29
18
18
34
6
13
16
4
25
30
2
17
12
8
21
22
26
26
22
Figure 10: 5-EEGL of
C 7+
Theorem 2.9
The crown graph

Cn , (n  5) of even order is k-even-edge-graceful for all k  z (mod p), where 0  z  p
– 1, n  2 (mod 3) and n  2.
Proof
Let the vertices and edges be defined as in Theorem 2.7. The edge labels are also same as in Theorem
2.7.
Then the induced vertex labels are as follows:
Case 1: k  0 (mod p)
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13 | Page

14. K-Even Edge-Graceful Labeling Of…
f   vi 
=
4n  4i  10 when i  1, 2

for 3  i  n.
4i  10
When z is odd, the induced vertex labels are given below:
Case 2: k  z (mod p), 1  z 
f   vi 
5  3z

6 z  4i  10
for 1  i  n 

2
= 

6 z  4n  4i  10 for n + 5  3z  i  n.

2

Case 3: k  z (mod p),
f

 vi 
p+5
3
z
2p +1
3

6 z  4n  4i  10
= 

 6 z  8n  4i  10


Case 4: k  z (mod p),
f   vi 
p+2
3
2p+4
3
for 1  i  2n 
for 2n 
5  3z
2
5  3z
 i  n.
2
zp–1

 6 z  8n  4i  10
= 

6 z  12n  4i  10


for 1  i  3n 
for 3n 
5  3z
2
5  3z
 i  n.
2
When z is even, the induced vertex labels are given below:
Case 5: k  z (mod p), 1  z 
f   vi 
6  3z

6 z  4i  10
for 1  i  n 


2
= 
6 z  4n  4i  10 for n  6  3z  i  n.

2

Case 6: k  z (mod p),
f

 vi 
f
 vi 
p+5
3
z
2p +1
3

6 z  4n  4i  10
= 

 6 z  8n  4i  10


Case 7: k  z (mod p),

p+2
3
2p+4
3
for 1  i  2n 
for 2n 
6  3z
2
6  3z
 i  n.
2
zp–1

6 z  8n  4i  10

= 

6 z  12n  4i  10


for 1  i  3n 
for 3n 
6  3z
2
6  3z
 i  n.
2
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14 | Page

15. K-Even Edge-Graceful Labeling Of…
The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex

Cn , (n  5) of even order is k-even-edge-graceful for
labels are all distinct and even. Hence, the crown graph
all k  z (mod p), where 0  z  p – 1, n  2 (mod 3) and n  2. 
For example, consider the graph
The 10-even-edge-graceful labeling of
C5 . Here p = q = 10; s = max {p, q} = 10; 2s = 20.
C5
is given in Figure 11.
0
20
4
14
35
16
19
24
36
18
10
23
31
6
2
27
28
32
12
8
Figure 11: 10-EEGL of
Definition 2.10
A graph
Hm G 
+
C5
is obtained from a graph G by replacing each edge with m parallel edges.
Theorem 2.11
The graph Hn(Pn), (n  2) of
k  z (mod p – 1), where 0  z  p – 2 and n is even.
Proof
even
order
Let {v1, v2, ..., vn} be the vertices of Hn(Pn). Let the edges eij of
eij = (vi, vi+1) for 1  i  n – 1, 1  j  n (see Figure 12).
v1
v2
v3
...
e21
e11
is
k-even-edge-graceful
H n  Pn 
...
for
all
be defined by
vn-1
vn
en-11
e12
e22
en-12
e13
e23
en-13
...
...
...
e1n
e2n
en-1n
Figure 12: Hn(Pn) with ordinary labeling
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16. K-Even Edge-Graceful Labeling Of…
For this graph, p = n; q = n(n – 1).
First, we label the edges as follows:
For k  1,
for 1  i  n  1, i is odd and 1 j  n
2k  ni  2 j  5
f  eij  = 
2k  2 j  n(i  1)  4 for 1  i  n  1, i is even and 1 j  n .
Then the induced vertex labels are as follows:
Case 1: k  0 (mod p – 1)
f   v1 
= 2n (n – 2).
f   vi 
=
f   vn 
= n (n – 4).
n  2n  2i  9  for 2  i  4

for 4  i  n  1.
n  2i  7 
Case 2: k  z (mod p – 1), 1  z  p – 2
f   v1 
Subcase (i):

= 2n (z – 1).
k  1 (mod p – 1)
f (vi )
= n(2i – 3) for 2  i  n – 1.
f  (vn )
= n[n + 2(z – 2)].
Subcase (ii):
p
2
n  2i  3  4n  z  1 for 2  i  n  2 z  3



n  2  i  2 z  n   5 for n  2 z  3  i  n  1.
 
k  z (mod p – 1), 2  z 
f   vi 
=
f  (vn )
= n[n + 2(z – 2)].
Subcase (iii):
p+2
 z  p2
2
n  2  i  2 z  n   5 for 2  i  2n  2 z  2

 

n  2  i  2n  2 z   3 for 2n  2 z  2  i  n  1.

 
k  z (mod p – 1),
f   vi 
=
f  (vn )
= n[2(z – 1) – n].
Therefore,
f  V 
 {0, 2, 4, ..., 2s – 2}, where s = max {p, q} = n(n – 1). So, it follows that the
vertex labels are all distinct and even. Hence, the graph Hn(Pn), (n  2) of even order is k-even-edge-graceful for
all k  z (mod p – 1), where 0  z  p – 2 and n is even. 
For example, consider the graph H8(P8). Here p = 8; q = 56;
s = max {p, q} = 56; 2s = 112. The 6-EEGL of H8(P8) is given in Figure 13.
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16 | Page

17. K-Even Edge-Graceful Labeling Of…
80
17
56
18
72
33
88
34
104
49
8
50
24
65
19
20
35
36
51
52
67
21
22
37
38
53
54
69
23
24
39
40
55
56
71
25
26
41
42
57
58
73
27
28
43
44
59
60
75
29
30
45
46
61
62
77
31
32
47
48
63
64
16
79
Figure 13: 6-EEGL of H8(P8)
REFERENCES
[1].
[2].
[3].
[4].
[5].
[6].
[7].
[8].
G.S. Bloom and S.W. Golomb, Applications of numbered undirected graphs, Proc. IEEE, 65(1977) 562 – 570.
G.S. Bloom and S.W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications, in Theory and Applications
of Graphs, Lecture Notes in Math., 642, Springer-Verlag, New York (1978) 53 – 65.
J.A. Gallian, A dynamic survey of graph labeling, The electronic journal of Combinatorics 16 (2009), #DS6.
B. Gayathri, M. Duraisamy, and M. Tamilselvi, Even edge graceful labeling of some cycle related graphs, Int. J. Math. Comput.
Sci., 2 (2007) 179 – 187.
F. Harary, “Graph Theory” – Addison Wesley, Reading Mass (1972).
S. Lo, On edge-graceful labelings of graphs, Congr. Numer., 50 (1985) 231 – 241.
W.C. Shiu, M.H. Ling, and R.M. Low, The edge-graceful spectra of connected bicyclic graphs without pendant, JCMCC, 66
(2008) 171 – 185.
Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang, On the edge-graceful spectra of cycles with one chord and dumbbell graphs,
Congressus Numerantium, 170 (2004) 171 – 183.
www.ijesi.org
17 | Page