DOI : 10.17577/IJERTV2IS100899
- Open Access
- Total Downloads : 236
- Authors : V.Ramachandran, C.Sekar
- Paper ID : IJERTV2IS100899
- Volume & Issue : Volume 02, Issue 10 (October 2013)
- Published (First Online): 28-10-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
One Modulo N Gracefulness of n-Polygonal Snakes, C (t)n and pa,b
ONE MODULO N GRACEFULNESS OF
n
n
n-POLYGONAL SNAKES, C(t)
AND P
a,b
V.Ramachandran1 C.Sekar2
1 Department of Mathematics, P.S.R Engineering College (Afliated to Anna University Chennai), Sevalpatti, Sivakasi, Tamil Nadu, India.
2 Department of Mathematics, Aditanar College of Arts and Science (Afliated to MS University Tirunelveli), Tiruchendur, Tamil Nadu, India.
Abstract
| |
| |
{ }
{ }
A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set 0, 1, 2, . . . , q such that, when each edge xy is assigned the label f (x) f (y) , the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function from the vertex set of G to
{0, 1, N, (N + 1), 2N, (2N + 1), . . . , N (q 1), N (q 1) + 1} in such a way that (i) is 1 1 (ii)
induces a bijection from the edge set of G to {1, N + 1, 2N + 1, . . . , N (q 1) + 1} where
(t)
(t)
(uv) = |(u) (v)| . In this paper we prove that the n-Polygonal snakes , Cn and Pa,b are one
modulo N graceful for all positive integers N .
n
n
Keywords : Graceful, modulo N graceful, n -Polygonal snakes, C(t) and Pa,b .
AMS Subject Classifcation (2010):05C78
-
Introduction
S.W.Golomb [1] introduced graceful labelling.Odd gracefulness was introduced by R.B.Gnanajothi
[2] .C.Sekar [6] introduced one modulo three graceful labelling. In this paper we introduce the concept of one modulo N graceful where N is any positive integer.In the case N = 2 , the labelling is odd gracefuln
n
and in the case N = 1 the labelling is graceful.We prove that the n-Polygonal snakes, C(t)
and P
a,b
are one modulo N graceful for all positive integers N .
-
Main Results
Defnition 2.1. A graph G with q edges is said to be one modulo N graceful (where N is a positive integer) if there is a function from the vertex set of G to {0, 1, N, (N + 1), 2N, (2N + 1), . . . , N (q 1), N (q 1) + 1} in such a way that (i) is 1 1 (ii) induces a bijection from the edge set of G to {1, N + 1, 2N + 1, . . . , N (q 1) + 1} where (uv) = |(u) (v)| .
Defnition 2.2. Consider k copies of path Pn .An n -Polygonal snake containing k number of n- Polygons is obtained from a path v1, v2, . . . , vk by identifying the pendant vertices of ith copy of the path Pn with vi1 and vi for i = 1, 2, . . . , k.
n
n
Defnition 2.3. The one point union of t cycles of length n is denoted by C(t) .This graph has t(n1)+1
vertices and tn edges.
Defnition 2.4. Let u and v be two fxed vertices. We connect u and v by means of b internally disjoint paths of length a each. The resulting graph is denoted by Pa,b .
Theorem 2.5. n -Polygonal snakes for n 0(mod4) are one modulo N graceful for every positive integer N .
Proof: Let n = 4r, r 1 .Let there be k polygons.This graph has 4rk (k 1) vertices and 4rk
(j) (4r)
(j) (4r)
(1)
edges. For 1 j k let u , j = 1, 2, . . . 4r be the vertices of i th polygon. Identify u with u ,
u
u
(4r) 2
i 1 2
3
3
with u(1) , and so on.
Defne
(u(1)
2i1
) = (u(4r)
2(i1
) = 4Nr(i 1) for i = 1, 2, 3, 4 . . . ,
(u(1)) = (u(4r)
) = 4Nrk (N 1) 2Nr + N 4Nr(i 1) for i = 1, 2, 3, 4 . . . ,
2i
(u(j)
2i1
(u(j)
2i1
(u(j)
2i1
(2i1
N(j2)
N(j2)
) = 4Nrk (N 1) 2 4Nr(i 1) for j = 2, 4, 6, 8 . . . , 4r 2 and i=1,2,3,4. . .
2
2
) = N + N(j3) + 4Nr(i 1) for j = 3, 5, 7, 9 . . . , 2r 1 and i=1,2,3,4. . .
2
2
) = 2N + N(j3) + 4Nr(i 1) for j = 2r + 1, 2r + 3, . . . , 4r 1 and i=1,2,3,4. . .
2i
2i
2
2
(u(j)) = N (2r + 1) + N(j2) + 4Nr(i 1) for j = 2, 4, 6, 8 . . . , 4r 2 and i=1,2,3,4. . .
2i
2i
2
2
(u(j)) = 4Nrk (N 1) 2Nr N(j3) 4Nr(i 1) for j = 3, 5, 7 . . . , 2r 1 and i=1,2,3,4. . .
2i
2i
2
2
(u(j)) = 4Nrk(N 1)2NrN N(j3) 4Nr(i1) for j = 2r + 1, 2r + 3, . . . , 4r 1 and i=1,2,3,4. . .
Clearly is 1 1 and defnes a one modulo N graceful labelling of the n -Polygonal snakes for
n 0(mod4) .
Example 2.6. Graceful labelling of the 12 -Polygonal snake. (No.of polygons = 5 )
58 4 9
51 46 16 21
39 34 28
2 57
53 10
14 45
41 22
26 33
59 5
8 50
47 17
20 38 35 29
1 56 54
11 13
44 42
23 25 32
60 6 7
49 48
18 19
37 36 30
0 55 12 43 24 31
Example 2.7. Odd graceful labelling of the 8 -Polygonal snake. (No.of polygons = 9 )
141 6
12 131
125 22
28 115
109 38 44 99
93 54 60
83 77 70
2 139
135 14
18
123
119
30
34 107
103 46
50
87 62
91 66 75
143 8
10 129
127 240
26 113
111 40
42 97
95 56
58 81 79 72
0 341 40 301 80 261 120 221 160 73
Example 2.8. One modulo 4 graceful labelling of the 4 -Polygonal snake. (No.of polygons = 4 )
61 8
12 49
45 24 28 33
0 57 16 41 32
Theorem 2.9. n -Polygonal snakes for n 2(mod4) are one modulo N graceful for every positive integer N if the number of polygons is even .
Proof: Let n = 4r + 2 and let there be k = 2s polygon.This graph has 4rk (k 1) vertices and
(j) (4r)
(j) (4r)
4rk edges. For 1 j k let u , j = 1, 2, . . . 4r be the vertices of i th polygon. Identify u with
u(1) , u(4r)
i 1
with u(1) , and so on.
2 2 3
Defne
(u(1)
2i1
) = (u(4r+2)) = N (4r + 3)(i 1) for i = 1, 2, 3, 4 . . . , s
2(i1
(u(1)) = (u(4r+2)) = N (4r + 2)k (N 1) 4Nr N (4r + 1)(i 1) for i = 1, 2, 3, 4 . . . , s
2i
(u(j)
2i1
(u(j)
2i1
(2i1
2
2
) = N + N(j3) + N (4r + 3)(i 1) for j = 3, 5, 7, 9 . . . , 4r + 1 and i=1,2,3,4. . . ,s
N(j2)
N(j2)
) = N (4r+2)k(N 1) 2 N (4r+1)(i1) for j = 2, 4, 6, 8 . . . , 4r and i=1,2,3,4. . . ,s
2i
2i
2
2
(u(j)) = N (4r+2)k(N 1)2NrN(j3) N (4r+1)(i1) for j = 3, 5, 7 . . . , 4r + 1 and i=1,2,3,4. . . ,s
2i
2i
2
2
(u(j)) = N (2r + 1) + N(j2) + N (4r + 3)(i 1) for j = 2, 4, 6, . . . , 2r and i=1,2,3,4. . . , s
2i
2i
2
2
(u(j)) = N (2r+1)+2N + N(j2)+N (4r+3)(i1) for j = 2r + 2, 2r + 4, . . . , 4r and i=1,2,3,4,. . . , s
Clearly is 1 1 and defnes a one modulo N graceful labelling of the n -Polygonal snakes for
n 0(mod2) .
Example 2.10. One modulo 8 graceful labelling of the 10 -Polygonal snake. (No.of polygons = 4 )
16 297
273
72 104
225
201
160
305 24 48
265 233
112 136
193
8 289
281
80 96
217
209
168
313 32
40 257
241 120
128
185
Example 2.11. One modulo 5 graceful labelling of the 10 -Polygonal snake. (No.of polygons = 6 )
10 286
271
45 65
241
226
100
120
196
181
155
291 15
30 266
246 70
85 221
201
125
140
176
5 281
276 50
60 236
231
105
115
191
186
160
296 20
25 261
251
75 80
216
206
130
135
171
0 256 55 211 110 166 165
Example 2.12. Graceful labelling of the 6 -Polygonal snake. (No.of polygons = 4 )
1 23
22 6
8 18 17 13
24 2 3 21 19 9 10 16
0 20 7 15 14
n
n
Theorem 2.13. Let C(t)
denote the one point union of t cycles of length n . C(t)
is one modulo N
n
n
graceful when n = 4, 8, t > 2 and n = 6, t even and t 4 for every positive integer N > 1 .
Proof: Case (i) n = 4, t > 2
(j)
(j)
For 1 i t . Let ui , j = 1, 2, 3, 4 be the vertices of the i th cycle with the one point identifcation
of u(1), u(1), . . . , u(1) at u0 .
1 2 t
Defne
(u0) = 0
i
i
2
2
(u(j)) = 4Nt (N 1) N (j2) 2N (i 1) for j = 2, 4 and i=1,2,3,4. . . ,t
i
i
(u(3)) = 4Nt 2N 4N (i 1) for i = 1, 2, 3, 4 . . . , t
(t)
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
when n = 4, t > 2 .
4
4
Example 2.14. One modulo 10 graceful labelling of C(6)
20
60
141
131
121
231
220
151
161
0 211
221
100 171
181
191
201
180
140
4
4
Example 2.15. Odd graceful labelling of C(4)
4
17
31
28
19
21
12 23
29
0 27
20
25
Case (ii) n = 8, t > 2
(j)
(j)
For 1 i t . Let ui , j = 1, 2, . . . , 8 be the vertices of the i th cycle with the one point identifcation
of u(1), u(1), . . . , u(1) at u0 .
1 2 t
Defne
(u0) = 0
(ui ) =
(j)
(j)
(ui ) =
8Nt (N 1) 2N (i 1) N (j2) if i = 1, 2, 3, . . . , t and j = 2, 8
6Nt (N 1) 2N (i 1)
6Nt (N 1) 2N (i 1)
6
N ( 4)j
6
N ( 4)j
if i = 1, 2, 3, . . . , t and j = 4, 6
if i = 1, 2, 3, . . . , t and j = 4, 6
2N (j3)
2N (j3)
2
2
6Nt 2N 4N (i 1) if i = 1, 2, 3, . . . , t and j = 5
6Nt 2N 4N (i 1) if i = 1, 2, 3, . . . , t and j = 5
4Nt N 4N (i 1) 4 if i = 1, 2, 3, . . . , t and j = 3, 7
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
when n = 8, t > 2 .
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
when n = 8, t > 2 .
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
when n = 8, t > 2 .
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
when n = 8, t > 2 .
8
8
Example 2.16. Odd graceful labelling of C(4)
20 47 44
33 30
2
45
49 63
35
26
6 0 61
51
59
53 22
10 43
55 57
37 18
14
41 36
28 39
8
8
Example 2.17. One modulo 7 graceful labelling of C(3)
56
85
7
120
77
112
113
92
21
134
127
162
0
63
155
141
35
148
49
99
106
84
Case (iii) n = 6, t is even t 4 let t = 2s
(j)
(j)
For 1 i 2s . Let ui , j = 1, 2, . . . , 6 be the vertices of the i th cycle with the one point identifcation
of u(1), u(1), . . . , u(1) at u0 .
1 2 t
Defne
(u0) = 0
(u(j)) =
i
(u(j)) =
i
6Nt (N 1) 2N (i 1) N (j2)
4Nt N 4N (i 1)
4N (j1)
4Nt N 4N (i 1)
4N (j1)
if i = 1, 2, 3, . . . , 2s and j = 2, 6
2 if i = 1, 2, 3, . . . , 2s and j = 3, 5
2 if i = 1, 2, 3, . . . , 2s and j = 3, 5
2N (j3) 4
4N (j2)
4Nt (4N 1) 2
4N (j2)
4Nt (4N 1) 2
if i = 2, 4, 6, . . . , 2s and j = 4
if i = 2, 4, 6, . . . , 2s and j = 4
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
(t)
Clearly is 1 1 and defnes a one modulo N graceful labelling of Cn
4Nt (N 1) 2 if i = 1, 3, 5, . . . , 2s 1 and j = 4
when n = 6, t is even
when n = 6, t is even
when n = 6, t is even
when n = 6, t is even
and t 4 .
6
6
Example 2.18. One modulo 4 graceful labelling of C(6)
49
61 20
28
12
101
105 97
4
92
93
141
109
113
0 137 84
133
36 117 121
44
65 52
129 76
125
60 68
81
1
6
6
Example 2.19. One modulo 5 graceful labelling of C(4)
41 75 76
5
65
81 116
15
86
91
25
56 35
0
96 101
111
106
55
45 61
Theorem 2.20. Pa,b for all a 2 and for all odd b is one modulo N graceful for every positive integer
N . Here Pm is a path of length m 1 .
(
(
Proof: Let b = 2r + 1, r 1 Defne
X(t) = 1 if t r
0 if t > r
Def ine
(
(
(u) = 0
Na(2r+1)
if a is even
(v) =
2
N (ab 1)
2
+ 1 if a is odd
For j = 1, 3, 5, . . .
j
j
(v(i)) = N (a(2r + 1) 1) + 1 N (i 1) (2r + 1)(j 1) if i = 1, 2, 3, . . . , 2r + 1
For j = 2, 4, 6, . . .
j
j
}
}
{ } {
{ } {
(v(i)) = X(i) 2N (r + 1) + N (i 1) + (2r + 1)(j 2) + (1 X(i)) N + N (i 1) + (2r + 1)(j 2) if i = 1, 2, 3, . . . , 2r + 1
Clearly is 1 1 and defnes a one modulo N graceful labelling of Pa,b for all a 2 and for all odd b .
Example 2.21. One modulo 3 graceful labelling of P6,5
88 18 73 33 58
85
21
70
36 55
82
9
67
24 52
79
12
64
27 49
76
15
61
30 46
85
21
70
36 55
82
9
67
24 52
79
12
64
27 49
76
15
61
30 46
0 45
Example 2.22. One modulo 4 graceful labelling of P5,7
137 32
109 60
133 36 105 64
129 40 101 68
125 16 97 44
0 121 20 93 48 69
117 24 89 52
113 28 85 56
xample 2.23. Graceful labelling of P7,7
49 8
48 9
42 15 35 22
41 16 34 23
47 10 40 17 33 24
0 46
45
44
4 39
5 38
6 37
11 32 18 25
12 31 19
13 30 20
43 7 36 14 29 21
Theorem 2.24. P4,b for all b 2 is one modulo N graceful for every positive integer N .
Proof:
Defne
(u) = 0
(v) = 2Nb
j
j
(v(i)) = N (ab 1) + 1 b(j 1) N (i 1) for i = 1, 2, . . . , b and j = 1, 3
2
2
(v(i)) = 2Nb N 2N (i 1) for i = 1, 2, . . . , b
Clearly is 1 1 and defnes a one modulo N graceful labelling of P4,b for all b 2
Example 2.25. One modulo 3 graceful labelling of P4,9
106 51 79
103 45 76
100 39 73
97 33 70
94 27 67
0
54
91 21 64
88 15 61
85 9 58
82 3 55
Example 2.26. One modulo 5 graceful labelling of P4,6
116 55 86
111 45 81
106 35 76
0
60
101 25 71
96 15 66
91 5 61
Example 2.27. Graceful labelling of P4,4
16 7 12
15 5 11
0 8
14 3 10
13 2 9
Theorem 2.28. P2,b for all b 2 is one modulo N graceful for every positive integer N .
Proof:
Defne
(u) = 0
(v) = Nb
1
1
(v(i)) = 2Nb (N 1) N (i 1) for i = 1, 2, . . . , b
Clearly is 1 1 and defnes a one modulo N graceful labelling of P2,b for all b 2
Example 2.29. One modulo 6 graceful labelling of P2,8
91
85
79
73
0 48
67
61
55
49
Example 2.30. Graceful labelling of P2,6
12
11
10
0 6
9
8
7
Example 2.31. One modulo 8 graceful labelling of P2,10
153
145
137
129
121
0 80
113
105
97
89
81
Theorem 2.32. P4r1,4r for all r 1 is one modulo N graceful for every positive integer N .
Proof:
Defne
(
(
(u) = 0
N [6 + 16{ r21 }] + 1 if r is odd
r
r
(v) =
2
2
N [14 + 16{ 2 1}] + 1 if r is even
For i = 2, 3, 4, . . . , 4r
(v
(i)) = ( N (4r 1)4r (N 1) N (i 2) (4r 1)(j 1) if j = 1, 3, 5, . . . , 2r 1
2
2
N
N
N
j
j
N (4r 1)4r (2N 1) N (i 2) 2 (4r 1)(j 1) if j = 2r + 1, 2r + 3, 5, . . . , 4r 3
For j = 2, 3, 4, . . . , 2r
(v(i)) = 4Nr + N (i 2) + N (4r 1)(j 2) if i = 2, 4, . . . , 4r 2
j 2
For i = 2r + 1, 2r + 2, . . . , 4r
(v(i)) = 2Nr + N (i 2r 1) + N (4r 1)(j 2) if i = 2, 4, . . . , 4r 2
j
2
j
2
(v(i)) = ( 2Nr(4r 1) (N 1) + (j 1) if j = 1, 3, 5, . . . , 4r 3
1 2Nr(4r 1) N (j 2) if j = 2, 4, 6, . . . , 4r 2
Clearly is 1 1 and defnes a one modulo N graceful labelling of P4r1,4r for all a 2 and for all odd b .
Example 2.33. One modulo 3 graceful labelling of P7,8
82
166
163
81 85
24 145
27 142
78 88 75
45 121 66
48 118 69
160 30 139 51 115 72
0 157
154
151
12 136
15 133
18 130
33 112 54 91
36 109 57
39 106 60
148 21 127 42 103 63
Example 2.34. One modulo 5 graceful labelling of P3,4
26 25
56 20
0 31
51 10
46 15
Example 2.35. Graceful labelling of P3,4
6 5
12 4
0 7
11 2
10 3
Theorem 2.36. Pa,b for all even a 4 is one modulo N graceful and for all even b 4 for every positive integer N .
Proof: Case (i) Let a = 4r, r 1 Let b = 2m,
Def ine
(
(
x(t) = 1 if t m
(
(
0 if t > m
y(j) = 1 if j 1(mod 2)
0 if j 1(mod 2)
Def ine
(u) = N (r 1)
(
(
(v) = 4N rm N (r + 1)
(1)
N (2r1j) Nj
y(j){8Nrm (N 1) } + y(j + 1){Nr N } if j = 1, 2, 3, . . . , 2r 1
(1)
N (2r1j) Nj
y(j){8Nrm (N 1) } + y(j + 1){Nr N } if j = 1, 2, 3, . . . , 2r 1
2
2
(vj ) =
y(j){4Nrm (N 1)
N (j
2
2r1)
2
2
2
} + y(j + 1){4Nrm N
N (j 2r) 2
} if j = 2r, 2r + 1, . . . , 4r 1
For i = 2, 3, . . . , 2m
(i) ( 8Nrm (N 1) Nr N (i 2) N (j1)(2m1)
if j = 1, 3, . . . , 2r 1
(vj ) =
8Nrm (2N 1) Nr N (i 2)
2
N (j 1)(2
2
m1)
if j = 2r + 1, 2r + 3, . . . , 4r 1
j
j
2
2
(v(i)) = x(i){N (2m + r 1) + N (i 2) + N (j2)(2m1)} + (1 x(i)){N (m + r 1) +
N (j2)(2m1)
N (j2)(2m1)
N (i m 1) + 2 } if j = 1, 2, . . . , 4r 2
Clearly is 1 1 and defnes a one modulo N graceful labelling of Pa,b for for all even a 4 and for all even b 4 .
Example 2.37. One modulo 5 graceful labelling of P8,6
231 0
226 35
236
201
115
60
116
171
110
85
121
146
221 40 196 65 166 90 141
5 105
216 20 191 45 161 70 136
211 25 186 50 156 75 131
206 30 181 55 151 80 126
Example 2.38. One modulo 7 graceful labelling of P4,4
106 49 50
99 28 71
0 42
92 14 64
85 21 57
Example 2.39. Graceful labelling of P12,4
46 1
47 0
48 23
24 22
25 21 26
45 6
42 9
39 12
35 15
32 18 29
2 20
44 4
41 7
38 10
34 13
31 16 28
43 5
40 8
37 11
33 14
30 17 27
Case (ii) Let a = 4r + 2, r 1 Let b = 2m,
(
(
Def ine
x(t) = 1 if t m
(
(
0 if t > m
y(j) = 1 if j 1(mod 2)
0 if j 1(mod 2)
Def ine
(u) = Nr
(v) = N (4r + 2)m Nr
j
j
2
2
2
2
(v(1)) = y(j){2N (4r + 2)m (N 1) N (2r+1j) } + y(j + 1){ N (2rj)} if j = 1, 2, 3, . . . , 2r + 1 .
(v(1)) = y(j){N (4r+2)m+1+ N (j2r3)}+y(j+1){N (4r+2)m N (j2r2)} if j = 2r + 2, 2r + 3, . . . , 4r + 1
j 2 2
For i = 2, 3, . . . , 2m
j
j
2
2
(v(i)) = 2Nm(4r + 2) (N 1) N (r + 1) N (i 2) N (j1)(2m1) if j = 1, 3, . . . , 4r + 1 .
j
j
2
2
(v(i)) = x(i){N (r + 2m) + N (i 2) + N (j2)(2m1)} + (1 x(i)){N (r + m) +
N (j2)(2m1)
N (j2)(2m1)
N (i m 1) + 2 } if j = 2, 4, . . . , 2r
j
j
2
2
(v(i)) = x(i){N (r + 2m + 1) + N (i 2) + N (j2)(2m1)} + (1 x(i)){N (r + m + 1) +
N (j2)(2m1)
N (j2)(2m1)
N (i m 1) + 2 } if j = 2r + 2, 2r + 4, . . . , 4r
Clearly is 1 1 and defnes a one modulo N graceful labelling of Pa,b for for all even a 4 and for all even b 4 .
Example 2.40. One modulo 4 graceful labelling of P6,4
89 0 93 48 49
85 20 73 36 61
4 44
81 12 69 28 57
77 16 65 32 53
Example 2.41. One modulo 5 graceful labelling of P10,4
186 5
181 30
191 0
166 45
196
151
100
65
101 95
136 80
106
121
10 90
176 20
171 25
161 35
156 40
146 55
141 60
131 70
126 75
116
111
Example 2.42. Graceful labelling of P6,6
35 0 36
18 19
34 7 29
13 24
33 8 28 14 23
1 17
32 4 27 10 22
31 5 26 11 21
30 6 25
12 20
References
-
S.W.Golomb, How to number a graph in Graph theoy and computing R.C. Read, ed., Academic press, New york (1972)23-27.
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R.B.Gnanajothi, Topics in Graph theory, Ph.D. Thesis, Madurai Kamaraj University, 1991.
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Joseph A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics,
18 (2011), #DS6.
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A.Rosa, On certain valuations of the vertices of a graph, Theory of graphs.(International Symposium, Rome July 1966)Gordom and Breach, N.Y and Dunod paris(1967)349-355.
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V.Ramachandran and C.Sekar, One modulo N gracefulness of Acyclic graphs, Ultra Scientist of Physical Sciences, Vol.25 No (3) 2013 Dec (Accepted)
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C.Sekar, Studies in Graph theory, Ph.D. Thesis, Madurai Kamaraj University, 2002.
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V.Swaminathan and C.Sekar, Gracefulness of Pa,b Ars Combinatoria,June 2004.
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