BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 303-4874, ISSN (o) 303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(017), 515-56 DOI: 10.751/BIMVI1703515J Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-579 (o), ISSN 1986-51X (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,,, q 1} such that the induced function f : E(G) {1, 3,, q 1} defined by f (uv) = f(u) + f(v) is a bijection. In this paper we prove that the plus graph P l n, open star of plus graph S(t.P l n ), path union of plus graph P l n, joining of C m and plus graph P l n with a path, one point union of path of plus graph P t n(t.n.p l n) 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) Z q such that the induced function f : E(G) Z q 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 010 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,,, q 1} such that the induced function f : E(G) {1, 3,, q 1} defined by f (uv) = f(u) + f(v) is a bijection. If f(v (G)) = {0, 1,,...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..if a (p, q) graph G is odd harmonious, then q p (q 1). 3.If G is an odd harmonious Eulerian graph with q edges, then q 0, (mod 4). We [7, 8] proved that the shadow and splitting of graph K,n, C n for n 0(mod 4) and the graph H n,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 G 1, G,..., G n, n be n copies of graph G. Then the graph obtained by adding an edge from G i to G i+1, (1 i n 1) is called path union of G. Definition 1.. [1] Let G be a graph on n vertices. The graph obtained by replacing each vertex of the star K 1,n by a copy of G is called a star of G denoted by G. Definition 1.3. [1] Take P n, P n,..., P n, P n 4,..., P 4, P paths on n, n,..., n, n 4,...4, vertices and arrange them centrally horizontal, where n 0(mod ), n. A graph obtained by joining vertical vertices of given successive paths is called a double step grid graph of size n denoted by DS tn. Definition 1.4. [1] Take P, P 4,..., P n, P n, P n, P n,..., P 4, P paths on, 4,..., n, n, n, n,..., 4, vertices and arrange them centrally horizontal, where n 0(mod ), n. A graph obtained by vertical vertices of given successive paths is known as a plus graph of size n denoted by P l n. Obviously, V (G) = n + n and E(G) = n. The example for the plus graph P l 6 is shown below (Figure 0):
ODD HARMONIOUS LABELING OF PLUS GRAPHS 517 u 1,1 u 1, u,1 u, u,3 u,4 u 3,1 u3, u 3,3 u3,4 u3,5 u 3,6 u 4,1 u 4, u 4,3 u 4,4 u 4,5 u 4,6 u 5,1 u 5, u 5,3 u5,4 u 6,1 u6,. Main Results In this section we prove that the plus graph P l n, open star of plus graph S(t.P l n ), path union of plus graph P l n, joining of C m and plus graph P l n with a path of arbitrary length, one point union of path of plus graph P t n(t.n.p l n ) are odd harmonious graphs. Theorem.1. A plus graph P l n is odd harmonious, where n 0(mod ), n. Proof. Let G = P l n be any graph of size n, n 0(mod ), n. We denote each vertices of first row as u 1,j (1 j ) and nd row as u,j (1 j 4). ( n ) th ( n ) th Similarly 1 row as u n 1,j (1 j n ) and row as un,j ( n ) th (1 j n) and + 1 row as u n +1,j (1 j n). Similarly (n 1) th row as u n 1,j (1 j 4) and n th row as u n,j (1 j ). Note that the number of vertices in G = P l n is V (G) = n + n and E(G) = n. We define a labeling f : V (G) { 0, 1,,, n 1 } as follows: f(u i,j) = nj i + j 1, j = 1, 3, 5,, n 1 and i = j + 1, j + 3,, n. f(u i,j) = n(j ) + n i + j + 1, j =, 4,, n and i = j, j +,, n. f(u i,j ) = nj + in n n + j, i = n + 1, n +,, n and j =, 4,, n [i n ]. f(u i,j ) = nj + in n 3n + j, i = n + 1, n +,, n and j = 1, 3, 5,, n [i n 1].
518 P.JEYANTHI AND S.PHILO The induced edge labels are f (u i,ju i,j+1) = nj 4i + j + 1, j = 1,, 3,, n 1 and i = 1,,..., n. f (u i,ju i,j+1) = nj + 4in n 4n + j 1, j = 1,, 3,, n and i = n + 1, n +,, n. f (u i,ju i+1,j+1) = nj 4i + j 1, i = 1,,, n and j = 1,,, n. f (u i,j u i+1,j ) = nj + in n n i + j 1, i = n and j = 1,,, n. f (u i,ju i+1,j 1) = nj + 4in n 4n + j 3, i = n + 1, n +,, n 1 and j =, 3,, n 1. In the view of above defined labeling pattern, the plus graph P l n is odd harmonious, where n 0(mod ), n. The odd harmonious labeling of P l 6 is shown in Figure 1. u 1,1 u 1, 4 7 u,1 u, u,3 u,4 5 16 19 u 3,1 u 3, u 3,3 u 3,4 u u 3,6 3,5 0 3 14 17 8 31 u 4,1 1 u 5,1 u 4, u 4,4 u 4,3 u 4,5 1 15 6 9 40 u 5, u 5,3 13 4 7 38 u 5,4 u 4,6 u 6,1 u 6, 5 36 FIGURE 1: Odd harmonious labeling of P l 6. Theorem.. Path union of finite copies of the plus graph P l n is odd harmonious, where n 0(mod ), n. Proof. Let G be a path union of r copies of the plus graph pl n, where n 0(mod ), n. Let u k,i,j ( i = 1,,, n and j = 1,,, i and i = n + 1,, n and j = 1,,, n i + ) be the vertices of the kth copy of P l n, k = 1,,, r, where the vertices of k th copy of P l n is p = n + n and the
ODD HARMONIOUS LABELING OF PLUS GRAPHS 519 edges of the k th copy of P l n is q = n. Join the vertices u k, n +1,n to u k+1, n +1,1 for k = 1,,, r 1 by an edge. ( ) n Note that in the graph G, the vertices V (G) = + n r and E(G) = rn + r 1. We define the labeling f : V (G) { 0, 1,,, [rn + r 1] 1 } as follows: f(u k,i,j ) = n(j 1) + n i + j 1 + n(n + 1)(k 1), j = 1, 3, 5,, n 1 and i = j + 1, j + 3,, n, k = 1,,, r. f(u k,i,j ) = n(j ) + n i + j + 1 + [n(n 1) + ](k 1), j =, 4, 6,, n and i = j, j +,, n, k = 1,,, r. f(u k,i,j ) = nj + in n 3n + j + [n(n 1) + ](k 1), j = 1, 3, 5,, n 1 and i = n + 1, n +,, n, k = 1,,, r. f(u k,i,j ) = nj + in n n + j + n(n + 1)(k 1), j =, 4, 6,, n and i = n + 1, n +,, n, k = 1,,, r. The induced edge labels are f (u k,i,j u k,i,j+1 ) = nj + n k 4i n + j + k 1, j = 1,, 3,, n 1 and i = 1,,, n, k = 1,,, r. f (u k,i,j u k,i,j+1 ) = nj + n k + 4in 4n 4n + j + k 3, j = 1,, 3,, n 1 and i = n + 1,, n, k = 1,,, r. f (u k,i,j u k,i+1,j ) = nj + n k + in 3n n i + j + k 3, j = 1,,, n and i = n, k = 1,,, r. f (u k,i,j u k,i+1,j 1 ) = nj + n k + 4in 4n 4n + j + k 5, j =, 3,, n 1 and i = n + 1,, n 1, k = 1,,, r. f (u k,i,j u k,i+1,j+1 ) = nj + n k 4i n + j + k 3, j = 1,, 3,, n and i = 1,,, n, k = 1,,, r. f (u k,i,n u k+1,i,1 ) = 4in + n k n 4n + k 1, j =, 3,, n 1 and i = n + 1, k = 1,,, r 1. In the view of above defined labeling pattern, path union of finite copies of the plus graph P l n is odd harmonious, where n 0(mod ), n. The odd harmonious labeling of path union of 3 copies of the plus graph P l 4 is shown in Figure.
50 P.JEYANTHI AND S.PHILO 5 19 4 33 0 3 10 13 0 17 30 7 40 31 50 41 1 8 9 11 16 18 15 8 3 5 36 38 9 48 37 39 56 58 FIGURE : Odd harmonious lableing path union of 3 copies of P l 4 Theorem.3. Open star of plus graph S(t.P l n ) is odd harmonious, where n 0(mod ), n and if t is odd. Proof. Let G = S(t.P l n ) be a graph obtained by replacing all vertices of K 1,t except the apex vertex of K 1,t by the graph P l n. Let u 0 be the apex vertex of K 1,t. That is it is the central vertex of the graph G. Let u k,i,j ( i = 1,,, n and j = 1,,, i and i = n + 1,, n and j = 1,,, n i + ) be the vertices of the k th copy of P l n in G, k = 1,,, t, where the vertices of k th copy of P l n is p = n + n and the edges of the kth copy of P l n is q = n. Join the vertices u k, n +1,1 with the vertex u 0 by an edge, k = 1,,, t. In graph G, V (G) = t( n + n) + 1 and E(G) = t(n + 1). We define the labeling f : V (G) { 0, 1,,, t(n + 1) 1 } as follows: f(u 0 ) = 0. f(u k,i,j ) = jnt + (j + 3)t it 4(k 1), i = j + 1, j + 3,, n, j = 1, 3,, n 1, k = 1,,, t. f(u k,i,j ) = (j 1)nt + jt it + 1 + (k 1), i = j, j +,, n, j =, 4,, n, k = 1,,, t. f(u k,i,j ) = jnt + int n t 3nt + (j 1)t + 1 + (k 1), i = n + 1, n +,, n, j = 1, 3,, n [i n 1], k = 1,,, t. f(u k,i,j ) = jnt + int n t nt + (j + )t 4(k 1), i = n + 1, n +,, n, j =, 4,, n [i n ], k = 1,,, t. The induced edge labels are f (u k,i,j u k,i,j+1 ) = jnt + jt + 4t 4it 1 (k 1), i = 1,,, n, j = 1,,, n 1, k = 1,,, t. f (u k,i,j u k,i,j+1 ) = jnt+4int n t 4nt+jt+t 1 (k 1), i = n +1, n +,, n, j = 1,,, n 1, k = 1,,, t.
ODD HARMONIOUS LABELING OF PLUS GRAPHS 51 f (u k,i,j u k,i+1,j+1 ) = jnt + jt + t 4it 1 (k 1), i = 1,,, n, j = 1,,, n, k = 1,,, t. f (u k,i,j u k,i+1,j 1 ) = jnt + 4int n t 4nt + jt 1 (k 1), i = n + 1,, n 1, j =, 3,, n 1, k = 1,,, t. f (u k,i,j u k,i+1,j ) = jnt+int n t nt+jt it+t 1 (k 1), i = n, j = 1,,, n, k = 1,,, t. f (u k, n +1,1 u 0 ) = 1 + (k 1), k = 1,,, t. In the view of above defined labeling pattern, open star of plus graph S(t.P l n ) is odd harmonious. The odd harmonious labeling of S(5.P l 4 ) is shown in Figure 3. 14 4 13 3 64 63 8 18 11 1 58 41 1 68 51 98 53 3 54 43 94 61 108 0 1 19 9 4 49 104 9 5 69 59 8 9 10 15 5 50 45 16 6 17 7 0 46 47 5 60 65 55 90 7 56 67 57 86 100 96 FIGURE 3: Odd harmonious labeling of S(5.P l 4 ). Theorem.4. A graph obtained by joining C m, m 0(mod 4) and a plus graph P l n,n 0(mod ), n with a path of arbitrary length t is odd harmonious. Proof. Let G be a graph obtained by joining a cycle C m, m 0(mod 4) and a plus graph P l n with P t, a path of length t on t + 1 vertices. Let u 1, u,, u m be the vertices of the cycle C m and v 1 = u m, v,, v t+1 be the vertices of path P t of t length and w i,j, ( i = 1,,, n and j = 1,,, i and i = n + 1,, n and j = 1,,, n i+) be the vertices of the plus graph P l n, where v t+1 = w n +1,1 if t is odd and v t+1 = w n,1 if t is even.
5 P.JEYANTHI AND S.PHILO In G, V (G) = m + (t 1) + n + n and E(G) = m + t + n. We define the labeling f : V (G) { 0, 1,,, (m + t + n ) 1 } as follows: f(u i ) = i 1, 1 i m {. i + 1 if i is odd m f(u i ) = + 1 i m. i 1 if i is even, { m + i if i is odd f(v i) = 1 i t + 1. m + i if i is even, If t is even f(w i,j) = nj i + j 1 + m + t 1, j = 1, 3,, n 1 and i = j + 1, j + 3,, n. f(w i,j) = n(j ) + n i + j + 1 + m + t + 1, j =, 4,, n and i = j, j +,, n. f(w i,j ) = nj + in n n + j + m + t 1, i = n + 1, n +,, n, j =, 4,, n [i n ]. f(w i,j ) = nj+in n 3n+j+m+t+1, i = n +1, n +,, n, j = 1, 3,, n [i n 1]. If t is odd f(w i,j) = nj i + j 1 + m + t, j = 1, 3,, n 1 and i = j + 1, j + 3,, n. f(w i,j) = n(j ) + n i + j + 1 + m + t, j =, 4,, n and i = j, j +,, n. f(w i,j ) = nj+in n n+j +m+t, i = n +1, n +,, n, j =, 4,, n [i n ]. f(w i,j ) = nj +in n 3n+j +m+t, i = n +1, n +,, n, j = 1, 3,, n [i n 1]. The induced edge labels are f (u i u i+1 ) = i 1, 1 i m 1. f (u m u m +1) = m + 1. f (u i u i+1) = i + 1, f (u m u 1 ) = m 1. m + 1 i m 1. f (v i v i+1 ) = m + i 1, 1 i t. f (v t w n,1) = m + t 1, if t is even. f (v t w n +1,1) = m + t 1, if t is odd. f (w i,j w i,j+1 ) = nj 4i + j + m + t + 1, i = 1,,, n and j = 1,, 3,, n 1. f (w i,j w i,j+1 ) = nj + 4in + j n 4n + m + t 1, i = n + 1, n +,, n and j = 1,, 3,, n 1.
ODD HARMONIOUS LABELING OF PLUS GRAPHS 53 f (w i,j w i+1,j+1 ) = nj 4i+j+m+t 1, i = 1,,, n and j = 1,, 3,, n. f (w i,j w i+1,j) = nj + in i + j n n + m + t 1, i = n and j = 1,, 3,, n. f (w i,j w i+1,j 1 ) = nj + 4in + j n 4n + m + t 3, i = n + 1, n +,, n 1 and j =, 3,, n 1. In the view of above defined labeling pattern, the cycle C m, m 0(mod 4) and a plus graph P l n with a path of arbitrary length t is odd harmonious. The odd harmonious labeling of cycle C 8 and a plus graph P l 4 with a path of arbitrary length t = 5 (odd) is shown in Figure 4. 1 3 0 6 7 8 5 10 9 1 11 15 18 13 16 3 4 14 1 9 6 31 FIGURE 4: Odd harmonious lableing of cycle C 8 and a plus graph pl 4 with a path of length t = 5. The odd harmonious labeling of cycle C 8 and a plus graph P l 6 with a path of arbitrary length t = 6 (even) is shown in Figure 5. 17 1 0 7 9 11 15 0 13 18 7 9 3 34 41 46 3 6 8 5 10 1 14 16 5 8 30 37 40 39 4 49 44 51 53 FIGURE 5: Odd harmonious lableing of cycle C 8 and a plus graph pl 4 with a path of length t = 6.
54 P.JEYANTHI AND S.PHILO Theorem.5. One point union for path of plus graph P t n(t.n.p l m ), m 0(mod ), m is odd harmonious if t is odd. Proof. Let G = Pn(t.n.P t l m ) be a graph obtained by replacing all vertices of Pn t except the central vertex by the graph P l m. That means G is the graph obtained by replacing all vertices of K 1,t except the apex vertex by the path union of n copies of the graph P l m. Let u 0 be the central vertex for the graph G with t branches. Let u s,k,i,j ( i = 1,,, m, j = 1,,, i, i = m + 1,, m, j = 1,,, m i + ) be the vertices of k th copy of path union of n copies of P l m lies in the s th branch of the graph G, s = 1,,, t. Join the vertices of u s,1, m +1,1 with u 0 by an edge. Also join the vertices u s,k, m +1,m to u s,k+1, m +1,1 for k = 1,,, n 1, s = 1,,, t by an edge. This graph G with V (G) = tn( m + m) + 1 and E(G) = tn(m + 1). We define the labeling f : V (G) { 0, 1,,, tn(m + 1) 1 } as follows: f(u 0) = 0. f(u s,k,i,j ) = jmt + (j + 3)t it 4(s 1) + (k 1)tm(m + 1), j = 1, 3,, m 1, i = j + 1, j + 3,, m, k = 1,,, n, s = 1,,, t. f(u s,k,i,j ) = (j 1)mt+jt it+1+(s 1)+(k 1)[tm(m+1) t(m 1)], j =, 4,, m, i = j, j +,, m, k = 1,,, n, s = 1,,, t. f(u s,k,i,j ) = jmt+imt m t 3mt+(j 1)t+1+(s 1)+(k 1)[tm(m+1) t(m 1)], i = m + 1, m +,, m, j = 1, 3,, m [i m 1], k = 1,,, n, s = 1,,, t. f(u s,k,i,j ) = jmt + imt m t mt + (j + )t 4(s 1) + (k 1)tm(m + 1), i = m + 1, m +,, m, j =, 4,, m [i m ], k = 1,,, n, s = 1,,, t. The induced edge labels are f (u s,k,i,j u s,k,i,j+1 ) = jmt + (j + 3)t + jt 4it + t (s 1) + (k 1)[tm(m + 1) t(m 1)] 1, i = 1,,, m, j = 1,,, m 1, s = 1,,, t, k = 1,,, n. f (u s,k,i,j u s,k,i,j+1 ) = jmt + 4imt m t 4mt + jt + t 1 (s 1) + (k 1)[tm(m + 1) t(m 1)] 1, i = m + 1,, m, j = 1,,, m 1, s = 1,,, t, k = 1,,, n. f (u s,k,i,j u s,k,i+1,j+1 ) = jmt+jt+t 4it 1 (s 1)+(k 1)[tm(m+1) t(m 1)] 1, i = 1,,, m, j = 1,,, m, s = 1,,, t, k = 1,,, n. f (u s,k,i,j u s,k,i+1,j ) = jmt + imt m t mt + jt it + t 1 (s 1) + (k 1)[tm(m + 1) t(m 1)] 1, i = m, j = 1,,, m, s = 1,,, t, k = 1,,, n. f (u s,k,i,j u s,k,i+1,j 1 ) = jmt + 4imt m t 4mt + jt 1 (s 1) + (k 1)[tm(m +
ODD HARMONIOUS LABELING OF PLUS GRAPHS 55 1) t(m 1)] 1, i = m +1,, m 1, j = 1,,, m 1, s = 1,,, t, k = 1,,, n. f (u 0 u s,1, m +1,1) = 1 + (s 1), s = 1,,, t. f (u s,k, m +1,m u s,k+1, m +1,1) = tm(m+1)+t 1 (s 1)+tm(m+1)(k 1) kt(m 1), s = 1,,, t, k = 1,,, n. In the view of above defined labeling pattern, P t n(t.n.p l m ) is odd harmonious. The odd harmonious labeling of P 3 3 (3.3.P l 4 ) is shown in Figure 6. 1 10 7 34 5 16 13 40 31 58 64 37 43 70 94 67 76 55 49 100 79 73 118 14 85 130 154 136 97 91 160 115 109 178 11 184 0 6 3 1 15 30 9 36 33 7 54 39 60 45 66 90 69 7 57 51 96 75 114 81 10 87 16 150 111 13 99 93 156 117 174 13 180 5 6 8 17 11 3 41 9 50 35 56 6 47 68 86 71 59 53 9 77 110 83 116 1 89 146 18 101 95 113 170 15 15 119 176 FIGURE 6:Odd harmonious labeling of P 3 3 (3.3.P l 4) References [1] V.J.Kaneria and H.M.Makadia. Graceful labeling of plus Graph. Inter. J. Current Research Sci. Tech., 1(3)(015), 15 0. [] V.J.Kaneria, Meera Meghpara and H.M.Makadia. Graceful labeling for open star of graphs. Inter. J. Math. Stat. Invent., (9)(014), 19-3. [3] J.A.Gallian. A dynamic Survey of Graph Labeling. The Electronics Journal of Combinatorics, 015, DS6.
56 P.JEYANTHI AND S.PHILO [4] R. L. Graham and N. J. A. Sloane. On Additive bases and Harmonious Graphs. SIAM J. Algebr. Disc. Meth., 4(1980), 38-404. [5] F. Harary. Graph Theory, Addison-Wesley, Massachusetts, 197. [6] Z.Liang and Z.Bai. On the Odd Harmonious Graphs with Applications. J. Appl. Math.Comput., 9 (009), 105-116. [7] P. Jeyanthi, S. Philo and Kiki A. Sugeng. Odd harmonious labeling of some new families of graphs., SUT J. Math., 51()(015), 53-65. [8] P.Jeyanthi and S.Philo. Odd Harmonious Labeling of Some Cycle Related Graphs. PROYECCIONES J. Math., 35(1)(016), 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)(013), 163-170. [10] S.K.Vaidya and N. H. Shah. Some New Odd Harmonious Graphs. Inter. J. Math. Soft Computing, 1(011), 9-16. [11] S.K.Vaidya and N. H. Shah. Odd Harmonious Labeling of Some Graphs. Inter. J.Math. Combin., 3(01), 105-11. [1] S.K.Vaidya, S.Srivastav, V.J.Kaneria and G.V.Ghodasara. Cordial and 3-equitable labeling of star of a cycle. Mathematics Today, 4(008), 54-64. Received by editors 0.1.016; Available online 17.04.017. P.Jeyanthi: Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur - 68 15, Tamil Nadu, India. E-mail address: jeyajeyanthi@rediffmail.com S.Philo: Department of Mathematics, PSN College of Engineering and Technology,, Melathediyoor, Tirunelveli, Tamil Nadu, India. E-mail address: lavernejudia@gmail.com