odd harmonious labeling of plus graphs

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 515-526 DOI: 10.7251/BIMVI1703515J Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p)

ODD HARMONIOUS LABELING OF PLUS GRAPHS P. Jeyanthi and S. Philo Abstract. A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f ∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f ∗ (uv) = f (u) + f (v) is a bijection. In this paper we prove that the plus graph P ln , open star of plus graph S(t.P ln ), path union of plus graph P ln , joining of Cm and plus graph P ln with a path, one point union of path of plus graph Pnt (t.n.P ln ) are odd harmonious graphs.

1. Introduction Throughout this paper by a graph we mean a finite, simple and undirected one. For standard terminology and notation we follow Harary [5]. A graph G = (V, E) with p vertices and q edges is called a (p, q) – graph. The graph labeling is an assignment of integers to the set of vertices or edges or both, subject to certain conditions have often been motivated by practical problems. An extensive survey of various graph labeling problems is available in [3]. Labeled graphs serves as useful mathematical models for many applications such as coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolution codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encoding of integers. Graham and Sloane [4] introduced harmonious labeling during their study of modular versions of additive bases problems stemming from error correcting codes. A graph G is said to be harmonious if there exists an injection f : V (G) → Zq such that the induced function f ∗ : E(G) → Zq defined by f ∗ (uv) = (f (u) + f (v)) (mod q) is a bijection and f is called harmonious labeling of G. The concept of odd harmonious labeling was due to Liang and Bai [6]. A graph G is said to be odd 2010 Mathematics Subject Classification. 05C78. Key words and phrases. Harmonious labeling; odd harmonious labeling; plus graph; open star of plus graph; one point union of path of plus graph. 515

516

P.JEYANTHI AND S.PHILO

harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f ∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f ∗ (uv) = f (u) + f (v) is a bijection. If f (V (G)) = {0, 1, 2, ....q} then f is called as strongly odd harmonious labeling and G is called as strongly odd harmonious graph.The odd harmoniousness of graph is useful for the solution of undetermined equations. Several results have been published on odd harmonious labeling. In [6], the following results have been proved: 1. If G is an odd harmonious graph, then G is a bipartite graph. Hence any graph that contains an odd cycle is not an odd harmonious. √ 2.If a (p, q) – graph G is odd harmonious, then 2 q 6 p 6 (2q − 1). 3.If G is an odd harmonious Eulerian graph with q edges, then q ≡ 0, 2(mod 4). We [7, 8] proved that the shadow and splitting of graph K2,n , Cn for n ≡ 0(mod 4) and the graph Hn,n are odd harmonious. Also, we proved that any two even cycles sharing a common vertex and a common edge are also odd harmonious graph. For further results, the interested reader can refer to [3, 6, 9, 10]. We use the following definitions in the subsequent section. Definition 1.1. [1] Let G be a graph and G1 , G2 , ....., Gn , n > 2 be n copies of graph G. Then the graph obtained by adding an edge from Gi to Gi+1 , (1 6 i 6 n − 1) is called path union of G. Definition 1.2. [12] Let G be a graph on n vertices. The graph obtained by replacing each vertex of the star K1,n by a copy of G is called a star of G denoted by G∗ . Definition 1.3. [1] Take Pn , Pn , ......, Pn−2 , Pn−4 , ...., P4 , P2 paths on n, n, ...., n − 2, n − 4, ....4, 2 vertices and arrange them centrally horizontal, where n ≡ 0(mod 2), n ̸= 2. A graph obtained by joining vertical vertices of given successive paths is called a double step grid graph of size n denoted by DStn . Definition 1.4. [1] Take P2 , P4 , ...., Pn−2 , Pn , Pn , Pn−2 , ...., P4 , P2 paths on 2, 4, ..., n − 2, n, n, n − 2, ....., 4, 2 vertices and arrange them centrally horizontal, where n ≡ 0(mod 2), n ̸= 2. A graph obtained by vertical vertices of given successive paths is known as a plus graph of n2 size n denoted by P ln . Obviously, |V (G)| = + n and |E(G)| = n2 . 2 The example for the plus graph P l6 is shown below (Figure 0):

ODD HARMONIOUS LABELING OF PLUS GRAPHS

u1,1

t

u3,1

t

u4,1

t

u1,2

t

u2,1

t

tu2,2

tu2,3

t 2,4

tu3,2

tu3,3

tu3,4

tu3,5

tu3,6

tu4,2

tu4,3

tu4,4

tu4,5

tu4,6

tu5,2

t u5,3

t u5,4

t

t

u5,1 t

u6,1

517

u

u6,2

2. Main Results In this section we prove that the plus graph P ln , open star of plus graph S(t.P ln ), path union of plus graph P ln , joining of Cm and plus graph P ln with a path of arbitrary length, one point union of path of plus graph Pnt (t.n.P ln ) are odd harmonious graphs. Theorem 2.1. A plus graph P ln is odd harmonious, where n ≡ 0(mod 2), n ̸= 2. Proof. Let G = P ln be any graph of size n, n ≡ 0(mod 2), n ̸= 2. We denote each vertices of first row as u1,j (1 6 j 6 2) and 2nd row as u2,j (1 6 j 6 4). (n )th ( n )th Similarly −1 row as u n2 −1,j (1 6 j 6 n − 2) and row as un,j 2 2 (n )th (1 6 j 6 n) and +1 row as u n2 +1,j (1 6 j 6 n). Similarly (n − 1)th row 2 as un−1,j (1 6 j 6 4) and nth row as un,j (1 6 j 6 2). Note that the number of n2 vertices in G = P ln is |V (G)| = + n and |E(G)| = n2 . 2 { } We define a labeling f : V (G) → 0, 1, 2, · · · , 2n2 − 1 as follows: j+1 j+3 n , ,··· , . 2 2 2 j j+2 n f (ui,j ) = n(j − 2) + n − 2i + j + 1, j = 2, 4, · · · , n and i = , ,··· , . 2 2 2 n n f (ui,j ) = nj + 2in − n2 − 2n + j − 2 , i = + 1, + 2, · · · , n and 2 2 j = 2, 4, · · · , n − [2i − n − 2]. n n f (ui,j ) = nj + 2in − n2 − 3n + j, i = + 1, + 2, · · · , n and 2 2 j = 1, 3, 5, · · · , n − [2i − n − 1].

f (ui,j ) = nj − 2i + j − 1 ,

j = 1, 3, 5, · · · , n − 1 and i =

518


The induced edge labels are f ∗ (ui,j ui,j+1 ) = 2nj − 4i + 2j + 1,

j = 1, 2, 3, · · · , n − 1 and i = 1, 2, ...,

n . 2

f ∗ (ui,j ui,j+1 ) = 2nj + 4in − 2n2 − 4n + 2j − 1, j = 1, 2, 3, · · · , n and n n i = + 1, + 2, · · · , n. 2 2 n−2 ∗ f (ui,j ui+1,j+1 ) = 2nj − 4i + 2j − 1, i = 1, 2, · · · , and j = 1, 2, · · · , n − 2. 2 n and j = 1, 2, · · · , n. f ∗ (ui,j ui+1,j ) = 2nj + 2in − n2 − n − 2i + 2j − 1, i = 2 n n f ∗ (ui,j ui+1,j−1 ) = 2nj + 4in − 2n2 − 4n + 2j − 3, i = + 1, + 2, · · · , n − 1 and 2 2 j = 2, 3, · · · , n − 1.

In the view of above defined labeling pattern, the plus graph P ln is odd harmonious, where n ≡ 0(mod 2), n ̸= 2. The odd harmonious labeling of P l6 is shown in Figure 1. u1,1 t

4

2

0

u3,1 t

u4,1 t 1

u1,2 t7

u2,1 t

t u2,2

tu3,2

u t 3,3

u t 3,4

14

17

tu4,3

5

3

tu4,2

13

16

u2,4 t 19

t u3,5

u3,6 t

28

31

tu4,4

tu4,5

15

26

29

t u5,2

tu5,3

12

u5,1 t

t u2,3

24

u6,1 t

25

27

t u4,6 40

t u5,4 38

t u6,2 36

FIGURE 1: Odd harmonious labeling of P l6 .

Theorem 2.2. Path union of finite copies of the plus graph P ln is odd harmonious, where n ≡ 0(mod 2), n ̸= 2. Proof. Let G be a path union of r copies of the plus graph pln , where n ≡ 0(mod 2), n ̸= 2. Let uk,i,j (∀i = 1, 2, · · · , n2 and ∀j = 1, 2, · · · , 2i and ∀i = n th 2 + 1, · · · , n and ∀j = 1, 2, · · · , 2n − 2i + 2) be the vertices of the k copy of n2 th P ln , ∀k = 1, 2, · · · , r, where the vertices of k copy of P ln is p = 2 + n and the


519

edges of the k th copy of P ln is q = n2 . Join the vertices uk, n2 +1,n to uk+1, n2 +1,1 for k = 1, 2, · · · , r − 1 by an edge. ( 2 ) n Note that in the graph G, the vertices |V (G)| = + n r and |E(G)| = 2 2 rn + r − 1. We define the labeling { } f : V (G) → 0, 1, 2, · · · , 2[rn2 + r − 1] − 1 as follows: f (uk,i,j ) = n(j − 1) + n − 2i + j − 1 + n(n + 1)(k − 1), j = 1, 3, 5, · · · , n − 1 and j+1 j+3 n i= , , · · · , , k = 1, 2, · · · , r. 2 2 2 f (uk,i,j ) = n(j − 2) + n − 2i + j + 1 + [n(n − 1) + 2](k − 1), j = 2, 4, 6, · · · , n and j j+2 n i= , , · · · , , k = 1, 2, · · · , r. 2 2 2 f (uk,i,j ) = nj + 2in − n2 − 3n + j + [n(n − 1) + 2](k − 1), j = 1, 3, 5, · · · , n − 1 and n n i = + 1, + 2, · · · , n, k = 1, 2, · · · , r. 2 2 f (uk,i,j ) = nj + 2in − n2 − 2n + j − 2 + n(n + 1)(k − 1), j = 2, 4, 6, · · · , n and n n i = + 1, + 2, · · · , n, k = 1, 2, · · · , r. 2 2

The induced edge labels are f ∗ (uk,i,j uk,i,j+1 ) = 2nj + 2n2 k − 4i − 2n2 + 2j + 2k − 1, j = 1, 2, 3, · · · , n − 1 and n i = 1, 2, · · · , , k = 1, 2, · · · , r. 2 f ∗ (uk,i,j uk,i,j+1 ) = 2nj + 2n2 k + 4in − 4n2 − 4n + 2j + 2k − 3, j = 1, 2, 3, · · · , n − 1 n and i = + 1, · · · , n, k = 1, 2, · · · , r. 2 f ∗ (uk,i,j uk,i+1,j ) = 2nj + 2n2 k + 2in − 3n2 − n − 2i + 2j + 2k − 3, j = 1, 2, · · · , n and n i = , k = 1, 2, · · · , r. 2 f ∗ (uk,i,j uk,i+1,j−1 ) = 2nj + 2n2 k + 4in − 4n2 − 4n + 2j + 2k − 5, j = 2, 3, · · · , n − 1 n and i = + 1, · · · , n − 1, k = 1, 2, · · · , r. 2 ∗ f (uk,i,j uk,i+1,j+1 ) = 2nj + 2n2 k − 4i − 2n2 + 2j + 2k − 3, j = 1, 2, 3, · · · , n − 2 and n−2 i = 1, 2, · · · , , k = 1, 2, · · · , r. 2 n f ∗ (uk,i,n uk+1,i,1 ) = 4in + 2n2 k − 2n2 − 4n + 2k − 1, j = 2, 3, · · · , n − 1 and i = + 1, 2 k = 1, 2, · · · , r − 1.

In the view of above defined labeling pattern, path union of finite copies of the plus graph P ln is odd harmonious, where n ≡ 0(mod 2), n ̸= 2. The odd harmonious labeling of path union of 3 copies of the plus graph P l4 is shown in Figure 2.

520


2

0

t

1

t

8 9

t

t5

t3

t10

t

t11

22

t

13

t

t 18

t

t

16

20

t

t17 t30

t

15 t

42

19

28

t25 t38

t

t

23

36

t33

t31

t50

t41

t

t39

t58

t

t

40

t27

t

t

t

29

t

48

37

56

FIGURE 2: Odd harmonious lableing path union of 3 copies of P l4

Theorem 2.3. Open star of plus graph S(t.P ln ) is odd harmonious, where n ≡ 0(mod 2), n ̸= 2 and if t is odd. Proof. Let G = S(t.P ln ) be a graph obtained by replacing all vertices of K1,t except the apex vertex of K1,t by the graph P ln . Let u0 be the apex vertex of n K1,t . That is it is the central vertex of the graph G. Let uk,i,j (∀i = 1, 2, · · · , 2 n and ∀j = 1, 2, · · · , 2i and ∀i = + 1, · · · , n and ∀j = 1, 2, · · · , 2n − 2i + 2) be 2 the vertices of the k th copy of P ln in G, ∀k = 1, 2, · · · , t, where the vertices of k th n2 copy of P ln is p = + n and the edges of the k th copy of P ln is q = n2 . Join the 2 vertices uk, n2 +1,1 with the vertex u0 by an edge, ∀k = 1, 2, · · · , t. n2 In graph G, |V (G)| = t( + n) + 1 and |E(G)| = t(n2 + 1). 2 { } We define the labeling f : V (G) → 0, 1, 2, · · · , 2t(n2 + 1) − 1 as follows: f (u0 ) = 0. f (uk,i,j ) = jnt + (j + 3)t − 2it − 2 − 4(k − 1), i = k = 1, 2, · · · , t.

n j+1 j+3 , , · · · , , j = 1, 3, · · · , n − 1, 2 2 2

j j+2 n f (uk,i,j ) = (j − 1)nt + jt − 2it + 1 + 2(k − 1), i = , , · · · , , j = 2, 4, · · · , n, 2 2 2 k = 1, 2, · · · , t. n n f (uk,i,j ) = jnt + 2int − n2 t − 3nt + (j − 1)t + 1 + 2(k − 1), i = + 1, + 2, · · · , n, 2 2 j = 1, 3, · · · , n − [2i − n − 1], k = 1, 2, · · · , t. n n f (uk,i,j ) = jnt + 2int − n2 t − 2nt + (j + 2)t − 2 − 4(k − 1), i = + 1, + 2, · · · , n, 2 2 j = 2, 4, · · · , n − [2i − n − 2], k = 1, 2, · · · , t.

The induced edge labels are n f (uk,i,j uk,i,j+1 ) = 2jnt + 2jt + 4t − 4it − 1 − 2(k − 1), i = 1, 2, · · · , , j = 1, 2, · · · , n − 1, 2 k = 1, 2, · · · , t. n n f ∗ (uk,i,j uk,i,j+1 ) = 2jnt+4int−2n2 t−4nt+2jt+2t−1−2(k −1), i = +1, +2, · · · , n, 2 2 j = 1, 2, · · · , n − 1, k = 1, 2, · · · , t. ∗


521

n−2 f ∗ (uk,i,j uk,i+1,j+1 ) = 2jnt + 2jt + 2t − 4it − 1 − 2(k − 1), i = 1, 2, · · · , , j = 2 1, 2, · · · , n − 2, k = 1, 2, · · · , t. n f ∗ (uk,i,j uk,i+1,j−1 ) = 2jnt + 4int − 2n2 t − 4nt + 2jt − 1 − 2(k − 1), i = + 1, · · · , n − 1, 2 j = 2, 3, · · · , n − 1, k = 1, 2, · · · , t. n f ∗ (uk,i,j uk,i+1,j ) = 2jnt+2int−n2 t−nt+2jt−2it+2t−1−2(k−1), i = , j = 1, 2, · · · , n, 2 k = 1, 2, · · · , t. f ∗ (uk, n2 +1,1 u0 ) = 1 + 2(k − 1), k = 1, 2, · · · , t.

In the view of above defined labeling pattern, open star of plus graph S(t.P ln ) is odd harmonious. The odd harmonious labeling of S(5.P l4 ) is shown in Figure 3. 24 s 14

s

s13 s64 s63

3 s 54s

s

28

s

18s 1

s

43

21 s

s23 s53 s

20 s

104

s

10s

94

s

s

s51 s

s

s

58

41

s15 s60 s65 s

s55 s

s

s

5 50

s11 s68 s61

45

0

s 25

100

90

s

108

98

16

s

12

s

2

9

s29

s27

6

s

s17 s56 s67

s

s19 s52 s69

s

s46 s57 s

s

s42 s59

7

s

s

49

s

s

92

47

s

96

86

82

FIGURE 3: Odd harmonious labeling of S(5.P l4 ).

Theorem 2.4. A graph obtained by joining Cm , m ≡ 0(mod 4) and a plus graph P ln ,n ≡ 0(mod 2), n ̸= 2 with a path of arbitrary length t is odd harmonious. Proof. Let G be a graph obtained by joining a cycle Cm , m ≡ 0(mod 4) and a plus graph P ln with Pt , a path of length t on t + 1 vertices. Let u1 , u2 , · · · , um be the vertices of the cycle Cm and v1 = um , v2 , · · · , vt+1 be the vertices of path Pt of t n n length and wi,j , (∀i = 1, 2, · · · , and ∀j = 1, 2, · · · , 2i and ∀i = + 1, · · · , n and 2 2 ∀j = 1, 2, · · · , 2n−2i+2) be the vertices of the plus graph P ln , where vt+1 = w n2 +1,1 if t is odd and vt+1 = w n2 ,1 if t is even.

522


n2 + n and |E(G)| = m + t + n2 . 2 { } We define the labeling f : V (G) → 0, 1, 2, · · · , 2(m + t + n2 ) − 1 as follows: In G , |V (G)| = m + (t − 1) +

f (ui ) = i − 1, 1 6 i 6 { i + 1 if i f (ui ) = i − 1 if i { m+i−2 f (vi ) = m+i

m . 2 is odd is even,

m + 1 6 i 6 m. 2

if i is odd if i is even,

1 6 i 6 t + 1.

If t is even j+1 j+3 n f (wi,j ) = nj − 2i + j − 1 + m + t − 1, j = 1, 3, · · · , n − 1 and i = , ,··· , . 2 2 2 j j+2 n ,··· , . f (wi,j ) = n(j − 2) + n − 2i + j + 1 + m + t + 1, j = 2, 4, · · · , n and i = , 2 2 2 n n f (wi,j ) = nj + 2in − n2 − 2n + j − 2 + m + t − 1, i = + 1, + 2, · · · , n, j = 2 2 2, 4, · · · , n − [2i − n − 2]. n n f (wi,j ) = nj+2in−n2 −3n+j+m+t+1, i = +1, +2, · · · , n, j = 1, 3, · · · , n−[2i−n−1]. 2 2 If t is odd j+1 j+3 n f (wi,j ) = nj − 2i + j − 1 + m + t, j = 1, 3, · · · , n − 1 and i = , ,··· , . 2 2 2 j j+2 n f (wi,j ) = n(j − 2) + n − 2i + j + 1 + m + t, j = 2, 4, · · · , n and i = , ,··· , . 2 2 2 n n 2 f (wi,j ) = nj+2in−n −2n+j−2+m+t, i = +1, +2, · · · , n, j = 2, 4, · · · , n−[2i−n−2]. 2 2 n n 2 f (wi,j ) = nj +2in−n −3n+j +m+t, i = +1, +2, · · · , n, j = 1, 3, · · · , n−[2i−n−1]. 2 2

The induced edge labels are f ∗ (ui ui+1 ) = 2i − 1, 1 6 i 6 um f ∗ (u m +1 ) = m + 1. 2 2 f ∗ (ui ui+1 ) = 2i + 1, f ∗ (um u1 ) = m − 1.

m − 1. 2

m + 1 6 i 6 m − 1. 2

f ∗ (vi vi+1 ) = 2m + 2i − 1, 1 6 i 6 t. f ∗ (vt w n2 ,1 ) = 2m + 2t − 1, if t is even. f ∗ (vt w n2 +1,1 ) = 2m + 2t − 1, if t is odd. f ∗ (wi,j wi,j+1 ) = 2nj − 4i + 2j + 2m + 2t + 1, i = 1, 2, · · · ,

n 2

and j = 1, 2, 3, · · · , n − 1.

n n f ∗ (wi,j wi,j+1 ) = 2nj + 4in + 2j − 2n2 − 4n + 2m + 2t − 1, i = + 1, + 2, · · · , n 2 2 and j = 1, 2, 3, · · · , n − 1.


523

n−2 and j = 1, 2, 3, · · · , n−2. 2 n f ∗ (wi,j wi+1,j ) = 2nj + 2in − 2i + 2j − n2 − n + 2m + 2t − 1, i = and j = 1, 2, 3, · · · , n. 2 n n ∗ 2 f (wi,j wi+1,j−1 ) = 2nj + 4in + 2j − 2n − 4n + 2m + 2t − 3, i = + 1, + 2, · · · , n − 1 2 2 and j = 2, 3, · · · , n − 1. f ∗ (wi,j wi+1,j+1 ) = 2nj−4i+2j+2m+2t−1, i = 1, 2, · · · ,

In the view of above defined labeling pattern, the cycle Cm , m ≡ 0(mod 4) and a plus graph P ln with a path of arbitrary length t is odd harmonious. The odd harmonious labeling of cycle C8 and a plus graph P l4 with a path of arbitrary length t = 5 (odd) is shown in Figure 4. 0

s s

s

s

11

s9

7

1s

s

15 s 13

2

3s

s

8

s

s

10

5

s

s

14

12

s16

s23

s26

s

s24

s

s

s

21

22

s

s18

31

29

6

FIGURE 4: Odd harmonious lableing of cycle C8 and a plus graph pl4 with a path of length t = 5.

The odd harmonious labeling of cycle C8 and a plus graph P l6 with a path of arbitrary length t = 6 (even) is shown in Figure 5. 17 0

s

1s

s7

2s

s

3s

s

13

11 s

s

s

16

8

5

s

9s

s 10

s

12

s

14

s

s22

15

s

s20

s29

s34

18

s

s27

s32

s41

s

s30

s39

s 44 s 53

s

s42

s51

s

s49

25

s

28

37

40

s46

6

FIGURE 5: Odd harmonious lableing of cycle C8 and a plus graph pl4 with a path of length t = 6.

524


Theorem 2.5. One point union for path of plus graph Pnt (t.n.P lm ), m ≡ 0(mod 2), m ̸= 2 is odd harmonious if t is odd. Proof. Let G = Pnt (t.n.P lm ) be a graph obtained by replacing all vertices of except the central vertex by the graph P lm . That means G is the graph obtained by replacing all vertices of K1,t except the apex vertex by the path union of n copies of the graph P lm . Let u0 be the central vertex for the graph G with t branches. m m Let us,k,i,j (∀i = 1, 2, · · · , , ∀j = 1, 2, · · · , 2i, ∀i = + 1, · · · , m, ∀j = 2 2 1, 2, · · · , 2m − 2i + 2) be the vertices of k th copy of path union of n copies of P lm lies in the sth branch of the graph G, ∀s = 1, 2, · · · , t. Join the vertices of us,1, m2 +1,1 with u0 by an edge. Also join the vertices us,k, m2 +1,m to us,k+1, m2 +1,1 for k = 1, 2, · · · , n − 1, s = 1, 2, · · · , t by an edge. m2 This graph G with |V (G)| = tn( + m) + 1 and |E(G)| = tn(m2 + 1). 2{ } We define the labeling f : V (G) → 0, 1, 2, · · · , 2tn(m2 + 1) − 1 as follows: Pnt

f (u0 ) = 0. f (us,k,i,j ) = jmt + (j + 3)t − 2it − 2 − 4(s − 1) + (k − 1)tm(m + 1), j = 1, 3, · · · , m − 1, j+1 j+3 m i= , , · · · , , k = 1, 2, · · · , n, s = 1, 2, · · · , t. 2 2 2 f (us,k,i,j ) = (j−1)mt+jt−2it+1+2(s−1)+(k−1)[tm(m+1)−2t(m−1)], j = 2, 4, · · · , m, j j+2 m , , · · · , , k = 1, 2, · · · , n, s = 1, 2, · · · , t. 2 2 2

i=

f (us,k,i,j ) = jmt+2imt−m2 t−3mt+(j −1)t+1+2(s−1)+(k −1)[tm(m+1)−2t(m−1)], i=

m m + 1, + 2, · · · , m, j = 1, 3, · · · , m − [2i − m − 1], k = 1, 2, · · · , n, s = 1, 2, · · · , t. 2 2

f (us,k,i,j ) = jmt + 2imt − m2 t − 2mt + (j + 2)t − 2 − 4(s − 1) + (k − 1)tm(m + 1), m m i= + 1, + 2, · · · , m, j = 2, 4, · · · , m − [2i − m − 2], k = 1, 2, · · · , n, s = 1, 2, · · · , t. 2 2

The induced edge labels are

f ∗ (us,k,i,j us,k,i,j+1 ) = 2jmt + (j + 3)t + jt − 4it + t − 2(s − 1) + (k − 1)[2tm(m + 1) − m 2t(m − 1)] − 1, i = 1, 2, · · · , , j = 1, 2, · · · , m − 1, s = 1, 2, · · · , t, k = 1, 2, · · · , n. 2 f ∗ (us,k,i,j us,k,i,j+1 ) = 2jmt + 4imt − 2m2 t − 4mt + 2jt + 2t − 1 − 2(s − 1) + (k − m + 1, · · · , m, j = 1, 2, · · · , m − 1, s = 1, 2, · · · , t, 1)[2tm(m + 1) − 2t(m − 1)] − 1, i = 2 k = 1, 2, · · · , n. f ∗ (us,k,i,j us,k,i+1,j+1 ) = 2jmt+2jt+2t−4it−1−2(s−1)+(k−1)[2tm(m+1)−2t(m−1)]−1, i = 1, 2, · · · ,

m−2 , j = 1, 2, · · · , m − 2, s = 1, 2, · · · , t, k = 1, 2, · · · , n. 2

f ∗ (us,k,i,j us,k,i+1,j ) = 2jmt + 2imt − m2 t − mt + 2jt − 2it + 2t − 1 − 2(s − 1) + (k − m 1)[2tm(m + 1) − 2t(m − 1)] − 1, i = , j = 1, 2, · · · , m, s = 1, 2, · · · , t, k = 1, 2, · · · , n. 2 f ∗ (us,k,i,j us,k,i+1,j−1 ) = 2jmt + 4imt − 2m2 t − 4mt + 2jt − 1 − 2(s − 1) + (k − 1)[2tm(m +


1)−2t(m−1)]−1, i =

525

m +1, · · · , m−1, j = 1, 2, · · · , m−1, s = 1, 2, · · · , t, k = 1, 2, · · · , n. 2

f ∗ (u0 us,1, m +1,1 ) = 1 + 2(s − 1), s = 1, 2, · · · , t. 2 f ∗ (us,k, m +1,m us,k+1, m +1,1 ) = tm(m+1)+2t−1−2(s−1)+tm(m+1)(2k−1)−2kt(m−1), 2 2 s = 1, 2, · · · , t, k = 1, 2, · · · , n.

In the view of above defined labeling pattern, Pnt (t.n.P lm ) is odd harmonious.

The odd harmonious labeling of P33 (3.3.P l4 ) is shown in Figure 6. 16

t

13

t

t7

t40 t37

t

t

t

10

1

34

t

25

0

6

t

t

t

t

t

t

t

t

t

33

9

30

t

2

t

t

t

t

t49 t100 t79

t

t

t

t

t

43

64

t

72

t

t

66

t

t

54

8

t

t

68 62

t35 t

47

t t

29

56

t

50

t

90

t

t11 t32 41 t

t

t

124

t

t91 t160 t121

t

t

t115 t

t

t

85

109

57

t

t75 t t

t t

71

t

87

t

t

t117 t 180

t

t174

t

t

120

150

111

t

128

t77 t 116

t

t93 t156 t123

59

110

178

126

114

t

184

132 99

t

t53 t92 t83 86

154

118

t51 t96 t

69

17

t

73

97

t

130

81

45

60

t

27

26

94

67

t39

t

t

15 36

136

55

t

70

58

t

5

31

12

3

76

t

t

122

t

89

101

t95 t152 t125 t

t119 t176

t

t

146

113

170

FIGURE 6:Odd harmonious labeling of P33 (3.3.P l4 )

References [1] V.J.Kaneria and H.M.Makadia. Graceful labeling of plus Graph. Inter. J. Current Research Sci. Tech., 1(3)(2015), 15–20. [2] V.J.Kaneria, Meera Meghpara and H.M.Makadia. Graceful labeling for open star of graphs. Inter. J. Math. Stat. Invent., 2(9)(2014), 19-23. [3] J.A.Gallian. A dynamic Survey of Graph Labeling. The Electronics Journal of Combinatorics, 2015, DS6.

526


[4] R. L. Graham and N. J. A. Sloane. On Additive bases and Harmonious Graphs. SIAM J. Algebr. Disc. Meth., 4(1980), 382-404. [5] F. Harary. Graph Theory, Addison-Wesley, Massachusetts, 1972. [6] Z.Liang and Z.Bai. On the Odd Harmonious Graphs with Applications. J. Appl. Math.Comput., 29 (2009), 105-116. [7] P. Jeyanthi , S. Philo and Kiki A. Sugeng. Odd harmonious labeling of some new families of graphs., SUT J. Math., 51(2)(2015), 53-65. [8] P.Jeyanthi and S.Philo. Odd Harmonious Labeling of Some Cycle Related Graphs. PROYECCIONES J. Math., 35(1)(2016), 85-98. [9] P. Selvaraju, P. Balaganesan and J.Renuka. Odd Harmonious Labeling of Some Path Related Graphs. Inter. J. Math. Sci. Engg. Appls., 7(III)(2013), 163-170. [10] S.K.Vaidya and N. H. Shah. Some New Odd Harmonious Graphs. Inter. J. Math. Soft Computing, 1(2011), 9-16. [11] S.K.Vaidya and N. H. Shah. Odd Harmonious Labeling of Some Graphs. Inter. J.Math. Combin., 3(2012), 105-112. [12] S.K.Vaidya, S.Srivastav, V.J.Kaneria and G.V.Ghodasara. Cordial and 3-equitable labeling of star of a cycle. Mathematics Today, 24(2008), 54-64. Received by editors 20.12.2016; Available online 17.04.2017.

P.Jeyanthi: Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur - 628 215, Tamil Nadu, India. E-mail address: [email protected] S.Philo: Department of Mathematics, PSN College of Engineering and Technology,, Melathediyoor, Tirunelveli, Tamil Nadu, India. E-mail address: [email protected]