Chapter 3: The results on edge-odd gracefulness of star...

52

CHAPTER III

SOME RESULTS ON EDGE-ODD GRACEFULNESS

OF STAR RELATED GRAPHS

Introduction: Maheo [1980] proved that the books of the form B2m (S2 □ Sn)

are graceful and conjectured that the books B4m+1 were graceful. Choudum

and Kishore [1988] verified that all 5-stars are Skolem graceful. Lee, Quach,

and Wang [1988] showed that the disjoint union of the path Pn

Lee and Wui [1991] obtained that the disjoint union of 2 or 3 stars is

Skolem-graceful if and only if at least one star has even size.

and the star of

size m is Skolem-graceful if and only if n = 2 and m is even or n ≥ 3 and m ≥

1.

In considering graceful labeling of the disjoint unions of two or three

stars with e edges, Yang and Wang [1994] got the result that Sm ∪ Sn is

graceful if and only if m or n is even and that Sm ∪ Sn ∪ Sk

is graceful if and

only if at least one of m, n, or k is even (m > 1, n > 1, k > 1).

Seoud and Youssef analyzed that the join of any two stars is graceful.

Choudum and Kishore [1996] determined that the disjoint union of k copies of

53

the star K1,2p is Skolem graceful if k ≤ 4p+ 1 and the disjoint union of any

number of copies of K1,2

is Skolem graceful. Chen, Liu, and Yeh [1997] found

that firecrackers are graceful.

3.1 Section – Previous works on star related graph:

Eldergill [1997] generalized the result on stars by showing that the

graph obtained by joining one end point from each of any number of paths of

equal length is odd-graceful. Kathiresan [2000] got that ladders and graphs

obtained from them by subdividing each step exactly once are odd-graceful.

Sekar [2002] provided that the graph obtained by identifying an

endpoint of a star with a vertex of a cycle is graceful. Wu [2002] found that

if G is a graceful graph with n edges and n+1vertices then join of G and any

star are graceful. Youssef [2003] investigated that for all n ≥ 2, Pn ∪ Sm

is

Skolem - graceful if and only if n = 3 or n = 2 and m is even.

Sethuraman and Selvaraju [2001] contributed that paths and stars are

the only graphs for which every supersubdivision is graceful. The conjecture

that every supersubdivision of a star is graceful was proved by Kathiresan and

Amutha in [2004]. Sethuraman and Jesintha [2005 and 2009] gave results

54

that all banana trees and extended banana trees are graceful. Gao [2007] made

a result that the union of any number of stars are odd graceful. Lee, Wang, and

Nowak [1988-2008] got the result that double star S(m, n) is super

edge-graceful if and only if m and n are both odd.

A.Solairaju and K.Chitra in [2008 – 2010] obtained edge – odd graceful

labeling for the star related graphs given below:

(1) Sn ∪ Sm when n and m are even; (2) Sn ∪ Sm ∪ Sr when n, m and r

are even; (3) P3 ∪ Sn, P4 ∪ Sn and C4 ∪ Sn when n is even; (4) bi-star Bn,n

when n is odd; (5) ⟨K1, n : 2⟩ for odd n; (6) double star K1, n, n for n ≥ 2; (7)

triple star K1, n, n, n

for n ≥ 3.

This chapter contains the results on edge-odd graceful labeling for the star

related graphs such as book graph (S2□Sn), m-star Sm,n and the graph JE3,n

.

3.2 Section - Edge odd gracefulness of the connected graph S2□Sn for

n≥3:

55

Here the edge-odd graceful labeling is verified for the connected graph

S2 □ Sn

, for n ≥ 3 and the results are furnished with example

for each case.

3.2.1 Definition: The book graph S2 □ Sn (or Cartesian product of the star

graphs S2 and Sn) is a connected graph obtained by adding ‘n’ number of C4

with one edge. It has 2n vertices and (3n – 2) edges.

3.2.2 Theorem: The connected graph S2□Sn

Proof: The figure (3.01) is the Cartesian product graph S

, for n ≥ 3, is edge – odd graceful.

2□Sn with 2n vertices

and (3n-2) edges, with some arbitrary labeling to its vertices and edges. It is

proved that the graph S2 □ Sn

Case (i): n is odd

, for n ≥ 3, is edge – odd graceful by taking two

cases such as n is odd and n is even.

e

e3n-3

e3n-5

n+3 e

en+5

n+1

e

1 e2 e3 e4 en -1 en

e

en+2

n +4 en +6

e

3n-4

e

3n -2

56

Figure 3.01: Edge – odd graceful Graph S2□Sn

for n odd

Define f: E(G) → {1, 3, …, 2q-1} by

f(ei

) = (2i-1), for i = 1, 2, …, (3n -2) …. Rule (1)

Then the induced vertex label f+: V(G) → {0, 1, 2, …, (2k-1)} is

defined by f+

Thus the map f and the induced map f

(v) ≡ Σf(uv) mod (2k), where this sum run over all edges

through v. …. Rule (2)

+ provide labels as distinct odd

numbers for edges and also the labelings for vertex set has distinct values in

{0, 1, 2,…., (2k-1)}. Hence the graph S2□Sn

, for n is odd, is edge-odd

graceful.

Case (ii): n is even

Here it is proved that S2□Sn

i. n ≡ 2 (mod 6) and ii. n ≡ 0 (mod 6) and n ≡ 4 (mod 6)

is graceful by taking 2 sub cases for n such as

(i) n ≡ 2 (mod 6)

Here edges are given labeling with odd numbers as in the figure (3.02).

e

en -1

en -2

2 e e

3 1

57

e2n -1 e2n e2n +1 e2n +2 e3n -1 e3n -2

e

en

n +1 en +2

e

2n -3 e

2n -2


Define f: E(G) → {1, 3, …, 2q-1} by , for n = 2 (mod 6)

f(ei

i ≠ n + (n/2) -1 and 2n + (n/2) -1 …. Rule (3)

) = (2i-1), for i = 1, 2, …, (3n -2) and

f(e(n + n/2 -1)) = 5n – 3 and f(e(2n + n/2 -1)

Then the induced vertex label f

) = 3n – 3

+

f

: V(G) → {0, 1, 2, …, (2k-1)}is defined by

+

(v) ≡ Σf(uv) mod(2k), where this sum run over all edges through v. …. Rule (4)

Thus the map f and the induced map f+ provide labels as distinct odd



, for n ≡ 2 (mod 6), is edge-odd

graceful.

(ii) n ≡ 0, 4 (mod 6)

The edges are given labeling as in the figure (3.03).

e e

2n -1

e2n -2

n+2 e

en+3

n+1

e

1 e2 e3 e4 en -1 en

58

e

e2n

2n +1 e2n +2 e3n -3

e

3n -2


, for n = 0, 4 (mod 6)

Define f: E(G) → {1, 3, …, 2q-1} by

f(ei

For n ≡ 0 (mod 6)

) = (2i-1), for i = 1, 2, …, (3n -2) and i ≠ 1 and 2n -1

f(e1) = 1 and f(e2n -1

For n ≡ 4 (mod 6)

) = 4n – 3 …. Rule (5)

f(e1) = 4n - 3 and f(e2n -1

) = 1

Then the induced vertex label f+

f

: V(G) → {0, 1, 2, …, (2k-1)} is defined by

+

(v) ≡ Σ f(uv) mod (2k), where this sum run over all edges


Thus the map f and the induced map f+ provide labels as distinct odd



, for n ≡ 0, 4 (mod 6), is edge-odd

graceful.

3.2.4 Example: The connected graph S2□S7 is edge – odd graceful.

59

The following graph (figure 3.04) is the book graph with 14 vertices

and 19 edges, with some arbitrary edge-odd graceful labeling to its vertices

and edges based on rule (1) and rule (2).

23 27 31 35 15 19

1 3

17

5 7 9 11 13

33 37

21 25 29

Figure 3.04: Edge – odd graceful Graph S2□S7

for odd n

3.2.5 Example: The connected graph S2□S8

The figure (3.05) is the book graph with 16 vertices and 22 edges, with

some arbitrary edge-odd graceful labeling to its vertices and edges based on

rule (3) and rule (4).

, for n ≡ 2 (mod 6) is edge – odd

graceful.

3 5 7 9 11 13 1 29 31 33 35 21 39 41 43

15 17 19

60

37 23 25 27


(n ≡ 2 (mod 6))

3.2.6 Example: The connected graph S2□S6, S2□S10

The (figure 3.06) is the book graph S

(n ≡ 0, 4 (mod 6)) is

edge – odd graceful based on rule (5) and rule (6).

2□S6

with 12 vertices and 16 edges, with

some arbitrary edge-odd graceful labeling to its vertices and edges.

15 17 19 21

13

1 3 5 7 9 11 23 25

27 29 31


, (n ≡ 0(mod 6))

The (figure 3.07) is the book graph S2□S10

with 20 vertices and 28 edges,

with some arbitrary edge-odd graceful labeling to its vertices and edges.

23 25 27 29 31 33 35 1 21 37 3 5 7 9 11 13 15 17 19

61

39 41 43

45 47 49 51 53 55


, (n ≡ 4 (mod 6))

3.3 Section - Edge – odd graceful labeling for the double star graph S2, n

Here the edge-odd graceful labeling is obtained for the double star

graph S

:

2, n

and the results are furnished with example.

3.3.1 Definition: Double star graph S2, n is a tree obtained from the star S1, n

by adding a new pendent edge to each of the existing n pendant vertices. It has

(2n+1) vertices and 2n edges.

3.3.2 Theorem: The connected graph S2, n

Proof: The figure (3.08) is the graph of S

is edge – odd graceful for all n >

1.

2, n

with (2n+1) vertices and

(2n) edges, with some arbitrary labeling for vertices and edges.

v4 v6 v e

8 4 e6 e8

v

2 e2 v3 v5 v7 v e

10 3 e5 e7 v9 e

v10

1 e1 e9 v11 e12 v12

62

e13 e11 e v

14 0 e2n-3 v

e14

2n-1 v2n-3 e2n-2

v .

2n-1

e .

2n v

2n-2

v

2n

Figure 3.08: Edge-odd gracefulness of S2, n

Define f: E(G) → {1, 3, …, 2q-1} by

f(ei

f(e

) = (2i - 1), for i = 2, 3, …, 2n-1

1) = 1 and f(e2n

f(e

) = 4n-1, if n ≡ 1, 2, 3 (mod 4) …. Rule

(1)

1) = 4n-1 and f(e2n

) = 1, if n ≡ 0 (mod 4). …. Rule (2)

Then the induced vertex label f+: V(G) → {0, 1, 2, …, (2k -1)} is

defined by f+

(v) ≡ Σf(uv)mod(2k), where this sum run over all edges


Thus the map f and the induced map f+ provide labels as odd numbers

for edges with all distinct and also the labeling for vertex set has distinct

values in {0, 1, 2,…., (2k-1)}. Hence the graph S2, n

is edge-odd graceful.

63

3.3.3 Example: The connected graph S2, 9

The graph of S

is edge – odd graceful.

2, 9

All the edges are labeled with distinct odd number as in rule (1). The labelings

for vertices and edges satisfy all conditions for edge-odd graceful.

with 19 vertices and 18 edges is given in figure (3.09).

11

7 15 3 19 5 9 13 17 23 1 21 29 25 27 33 31 35 Figure 3.09: Edge-odd gracefulness of S2, 9

(n ≡ 1, 2, 3 (mod 4))


The figure (3.10) is the graph of S

is edge – odd graceful.

2, 8

with 17 vertices and 16 edges. All the

edges are labeled with distinct odd number as in rule (2). The labelings for

vertices and edges satisfy all conditions for edge-odd graceful labeling.

64

7 11 15

3 19 5 9 13 23 31 17 21

25 27 29

1

Figure 3.10: Edge-odd gracefulness of S2, 8

(n≡ 0 (mod 4))

3.4 Section -The Edge – odd gracefulness of the connected graph Sm, n

Here the edge-odd graceful labeling is verified for the graph connected

graph S

:

m, n

, for all m, n > 1 and the results are furnished with example for

each case.

3.4.1 Definition: The m star graph Sm, n is a tree obtained from the double star

graph S2, n by merging a path P(m-2)

to each of the existing n pendant vertices.

It has (mn+1) vertices and mn edges.

3.4.2 Theorem: The connected graph Sm,n

Proof: The figure (3.11) is the graph of S

is edge–odd graceful for all m,

n>1.

m, n

with (mn+1) vertices and (mn)

edges, with some arbitrary labelings for all its vertices and edges.

65

e2 e3 em

e

1

em+2 em+3 e e

2m

m+1

e e

2m+1 2m+2 e2m+3 e3m

e .

(m-2)n+1

e .

(m-2)n+2 e(m-2)n+3 e(m-1)n

.

e e

(m-1)+1 (m-1)+2 e(m-1)n+3 e

mn

Figure 3.11: Edge-odd gracefulness of Sm, n

Clearly q = mn

Define f: E(G) → {1, 3, …, 2q-1} for various cases on n and m …. Rule (1)

Case i:

f(e

n is odd and m is even

i

) = (2i - 1), for i = 1, 2, 3, …, q

Case ii:

f(e

n is odd and m is odd

i

= m + i, for i = 2, 4, 6, …, (m-1).

) = i, for i = 1, 3, 5, …, m:

f(e(km+i)) = f(ei

) + 2mk, for i = 1, 2, …, m, k = 2, 3, …, n.

Case iii:

f(e

n is even, n ≡ 0 (mod 4)

i) = (2i - 1), for i = 2, 3, …, q – 1; f(e1) = 2q - 1 and f(eq

) = 1.

Case iv: n is even, n ≡ 2 (mod 4)

66

f(ei

f(e

) = (2i - 1), for i = 1, 2, 3, …, (n – 1)m;

(n-1)m+i

) = [2q-(2i-1)], for i = 1, 2, …, m

Then the induced vertex label f+

f

: V(G) → {0, 1, 2, …, (2k-1)} is defined by

+





values in {0, 1, 2, …., (2k-1)}. Hence the graph Sm, n


Now example for each case (i) to (iv) is furnished below:


The graph of S

is edge – odd graceful for

m ≡ 2 (mod 4) and n ≡ 1 (mod 4).

6, 5 with 31 vertices and 30 edges is given in figure (3.12). By

case (i) in theorem (3.4.2), the labelings for vertices and edges for edge-odd

graceful is given below.

67

3 5 7 9 11 1 15 17 19 13

21 23

25 27 29 31 33 35 37 39 41 43 45 47

49 51 53 55 57 59


3.4.4 Example: The connected graph S6, 7 is edge – odd graceful where

m ≡ 2 (mod 4) and n ≡ 3 (mod 4). The graph of S6, 7 with 43 vertices and

42 edges is given in figure (3.13). By case (i) in theorem (3.4.2), the labelings

for vertices and edges for edge-odd graceful is given below.

3 5 7 9 11 1 15 17 19 13

21 23

1 25 27 29 31 33 35 37

49 39 41 43 45 47

61 51 53 55 57 59 73 63 65 67 69 71 75 77 79 81 83


68


m ≡ 0 (mod 4) and n ≡ 1(mod 4). The graph of S8, 5

By case (i) in theorem (3.4.2), the labelings for vertices and edges for

edge-odd graceful is given below.

with 41 vertices and

40 edges is given in figure (3.14).

69

3 5 7 9 11 13 15 1 19 21 23 25 27 29 31 17

33 35 37 39 41 43 45 47 49

51 53 55 57 59 61 63 65 67 69 71 73 75 77 79

Figure 3.14: Edge-odd gracefulness of S

8, 5


m ≡ 0 (mod 4) and n ≡ 3(mod 4). The graph of S8, 7 with 57 vertices and

56 edges is given in figure (3.15). By case (i) in theorem (3.4.2), the labelings


3 5 7 9 11 13 15

1 17 19 21 23 33

25 27 29 31

35 37 39 41 43 45 47 49 65 51 53 55 57 59 61 63 67 69 71 73 75 77 79 81

97

83 85 87 89 91 93 95

99 101 103 105 107 109 111


8, 7

70

From theorem (3.4.2), for m is odd and n is odd, Sm, n



m ≡ 3 (mod 4) and n ≡ 1(mod 4). The graph of S7, 5

with 36 vertices and

35 edges is given in figure (3.16). By case (ii) in theorem (3.4.2), the labelings


9 3 11 5 13 7 1 23 17 25 15

19 27 21

29 37 31 39 33 41 35 43 51 45 53 47 55 49

57 65 59 67 61 69 63


7, 5





71

9 3 11 5 13 7

1 23 17 25

15 19 27 21

37 31 39 33 41 35 29 43 51 45 53 47 55 49 57

65 59 67 61 69 63 71 79 73 81 75 83 77 85 93 87 95 89 97 91






11 3 13

5 15 7 17 9

1 29 21 31 23 33 25 35 27 19 37 47 39 49 41 51 43 53 45 55 65 57 67 59 69 61 71 63 73

83 75 85 77 87 79 89 81


72



63 edges is given in figure (3.19). By case (ii) in theorem (3.4.2), the

labelings for vertices and edges for edge-odd graceful is given below.

11 3 13 5 15 7 17 9 1 29 21 31 19

23 33 25 35 27

37 47 39 49 41 51 43 53 45 1 55 65 57 67 59 69 61 71 63 73 91 83 75 85 77 87 79 89 81

109 101 93 103 95 105 97 107 99

119 111 121 113 123 115 125 117


From theorem (3.4.2), for n is even and n = 0 (mod 4), Sm, n

is edge-odd

graceful.


m ≡ 2 (mod 4) and n ≡ 0 (mod 4). The graph of S6, 8 with 49 vertices and 48

edges is given in figure (3.20). By case (iii) in theorem (3.4.2), the labelings


73

3 5 7 9 11

95 15 17 19

13 27 29 31 33 35

21 23

25 39 41 43 45 47 37 49 51 53 55 57 59

61 63 65 67 69 71

73 75 77 79 81 83 85 87 89 91 93 1


6, 8


m ≡ 0 (mod 4) and n ≡ 0 (mod 4). The graph of S4, 8

with 33 vertices and 32

edges is given in figure (3.21).

By case (iii) in theorem (3.4.2), the labelings for vertices and edges for

edge-odd graceful is given below.

74

63 11 13 15

3 5 7

9 19 21 23 17 27 29 31 25 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 1


4, 8


m ≡ 3 (mod 4) and n ≡ 0(mod 4). The graph of S7, 8 with 57 vertices and 56

edges is given in figure (3.22). By case (iii) in theorem (3.4.2), the labelings


3 5 7 9 11 13

111 17 19

21 23 25 27

15 31 33 35 37 39 41 29 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 85 73 75 77 79 81 83 99 87 89 91 93 95 97 101 103 105 107 109 1


75

From theorem (3.4.2), Sm,n

is edge – odd graceful for n is even and

n ≡ 2(mod 4).



30 edges is given in figure (3.23). By case (iv) in theorem (3.4.2), the


3 5 7 9 1 11 13 15 17 21

19

31 23 25 27 29 41 33 35 37 39 59 43 45 47 49 57 55 53 51


5, 6


m ≡ 2 (mod 4) and n ≡ 2 (mod 4). The graph of S6, 6 with 37 vertices and

36 edges is given in figure (3.24). By case (iv) in theorem (3.4.2), the


76

3 5 7 9 11 1 15 17 19 13

21 23

25 27 29 31 33 35 37 49 39 41 43 45 47 71 51 53 55 57 59 69 67 65 63 61


6, 6

3.5 Section - Edge–odd gracefulness of JE3, n

In this section the edge-odd graceful labeling is verified for the

connected graph JE

:

3, n and the results are furnished with example for

each

case.

3.5.1 Theorem: The graph JE3, n

Proof: The figure (3.25) is the graph of JE

is edge–odd graceful.

3, n with (2n+4) vertices and

(2n+3) edges, with some arbitrary labelings for all its vertices and edges. The

edge-odd graceful labeling is found for the graph JE3, n

Case i: n is even and n ≡ 0, 2 (mod4)

in two cases.

The edge-odd graceful labeling for this case is furnished in figure (3.25).

77

e

n+1

e1 e e

n+2 2n+1 e2

e

3 e e

n+3 2n+2

e . . .

2n+3 en-1 e

. . . 2n-1

e

n

e

2n

Figure 3.25: Edge-odd graceful Graph JE3, n

(n ≡ 0, 2 (mod4))

Now define f: E(G) →{1, 3, …, (2k-1)} by

For n ≡ 0 (mod4)

f(ei

) = (2i - 1), for i = 1, 2, 3, …, (2n+3); …. Rule (1)

For n ≡ 2 (mod4) …. Rule (2)

f(ei

f(e

) = (2i - 1), for i = 1, 2, 3, …, (2n+3) and i ≠ (n-1), (2n-1);

2n-1) = (2n - 3); f(en-1

) = (4n - 3).


defined by f+

Case ii: n is odd and n ≡ 1, 3 (mod4)

(v) ≡Σf(uv) mod (2k), where this sum run over all edges


The edge-odd graceful labeling for this case is furnished in figure (3.26).

78

e

2

e1 e e

4 2n+1 e

e3

2n+2 e5 e e

6 2n-3

e . . .

2n+3

e . . .

2n-1 e

2n-2

e

2n

Figure 3.26: Edge-odd graceful Graph JE3, n

, for n ≡ 1, 3 (mod4)

Now define f: E(G) →{1, 3, …, (2k-1)} by

For n ≡ 1 (mod4)

f(ei

) = (2i - 1), for i = 1, 2, 3, …, (2n+3); …. Rule (4)

For n ≡ 3 (mod4) …. Rule (5)

f(ei

f(e

) = (2i - 1), for i = 1, 2, 3, …, (2n+3) and i ≠ (2n-2), (2n-3);

2n-3) = (4n - 5); f(e2n-2

) = (4n - 7).


defined by f+



79



values in {0, 1, 2, …., (2k-1)}. Hence the graph JE3, n


3.5.2 Example: The graph JE3, 4

The figure (3.27) is the graph of JE

is edge–odd graceful, where n is even and

n ≡ 0 (mod 4).

3, 4


with edge–odd graceful labeling for all its vertices and edges.

9 17 1 11 3 19 5 13 21 7 15

Figure 3.27: Edge-odd graceful Graph JE3, 4

(n ≡ 0 (mod4))



is edge–odd graceful, where n is even and

n ≡ 2 (mod4).

3, 6 with 16 vertices and 15 edges,

with edge–odd graceful labelings for all its vertices and edges.

80

13

25 3

1 15

5 17 27 7 21 29

19 11

9

23


(n ≡ 2 (mod4))



is edge–odd graceful, where n is odd and

n ≡ 1 (mod4).

3, 5



3

1 7 21 5 23 9 11 13 25 15

17

19


(n ≡ 1 (mod4))

81



is edge–odd graceful, where n is odd and

n ≡ 3 (mod4).

3, 7



3

1 7 5 11 29 9

31 13 15 17

33 19

23

25 21

27

Figure 3.30: Edge-odd graceful Graph JE3,7

(n ≡ 3 (mod 4))

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