On Odd Sum Graphs. S.Arockiaraj. Department of Mathematics. Mepco Schlenk Engineering College, Sivakasi , Tamilnadu, India. P.

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1 International J.Math. Combin. Vol.4(0), -8 On Odd Sum Graphs S.Arockiaraj Department of Mathematics Mepco Schlenk Engineering College, Sivakasi , Tamilnadu, India P.Mahalakshmi Department of Mathematics Kamaraj College of Engineering and Technology, Virudhunagar , Tamilnadu, India psarockiaraj@gmail.com, mahajai@gmail.com Abstract: An injective function f : V (G) {0,,,..., q} is an odd sum labeling if the induced edge labeling f defined by f (uv) = f(u)+f(v), for all uv E(G), is bijective and f (E(G)) = {,,,..., q }. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property for the graphs paths P p, cycles C p, C p K, the ladder P P p, P m nk, the balloon graph P n(c p), quadrilateral snake Q n,[p m; C n], (P m; Q ), T (n) p, H n mk, bistar graph and cyclic ladder P C p. Key Words: Labeling, odd sum labeling, odd sum graph. AMS(00): 0C8. Introduction Throughout this paper, by a graph we mean a finite, undirected simple graph. Let G(V, E) be a graph with p vertices and q edges. For notations and terminology we follow []. Path on p vertices is denoted by P p and a cycle on p vertices is denoted by C p whose length is p. If m number of pendant vertices are attached at each vertex of G, then the resultant graph obtained from G is the graph G mk. When m =, G K is the corona of G. The bistar graph B m,n is the graph obtained from K by identifying the central vertices of K,m and K,n at the end vertices of K respectively. The graph P P p is the ladder and P C p is the cyclic ladder. The balloon of a graph G, P n (G) is the graph obtained from G by identifying an end vertex of P n at a vertex of G. Let v be a fixed vertex of G. The graph [P m ; G] is obtained from m copies of G and the path P m : u u... u m by identifying u i with the vertex v of the i th copy of G, for i m. The graph (P m ; G) is obtained from m copies of G and the path P m : u u...u m by joining u i with the vertex v of the i th copy of G by means of an edge, for i m []. The cube graph Q is P C 4. A quadrilateral snake is obtained from a path by Received September 6, 0, Accepted November 6, 0.

2 60 S.Arockiaraj and P.Mahalakshmi identifying each edge of the path with an edge of the cycle C 4. The graph T p (n) is a tree formed from n copies of path on p vertices by joining an edge uu between every pair of consecutive paths where u is a vertex in i th copy of the path and u is the corresponding vertex in the (i + ) th copy of the path. In [], an odd edge labeling of a graph is defined as follows: A labeling f : V (G) {0,,,..., p } is called an odd edge labeling of G if for the edge labeling f + on E(G) defined by f + (uv) = f(u) + f(v) for any edge uv E(G), for a connected graph G, the edge labeling is not necessarily injective. In [], the concept of pair sum labeling was introduced. An injective function f : V (G) {±, ±,..., ±p} is said to be a pair sum labeling if the induced edge function f e : E(G) Z {0} defined by f e (uv) = f(u) + f(v) is one-one and f e (E(G)) is either of the form {±k, ±k,...,± kq } or {±k, ±k,..., ±k q } { kq+ } according as q is even or odd. A graph with a pair sum labeling defined on it is called a pair sum graph. In [6], the concept of mean labeling was introduced. An injective function f : V (G) {0,,,..., q} is said to be a mean labeling if the induced edge labeling f defined by f(u) + f(v) if f(u) + f(v) is even, f (uv) = f(u) + f(v) + if f(u) + f(v) is odd is injective and f (E(G)) = {,,..., q}. A graph G is said to be odd mean if there exists an injective function f from V (G) to {0,,,,..., q } such that the induced map f from E(G) {,,,..., q } defined by f(u) + f(v) if f(u) + f(v) is even, f (uv) = f(u) + f(v) + if f(u) + f(v) is odd is a bijection [6]. Motivated by these, we introduce a new concept called odd sum labeling. An injective function f : V (G) {0,,,..., q} is an odd sum labeling if the induced edge labeling f defined by f (uv) = f(u)+f(v), for all uv E(G), is bijective and f (E(G)) = {,,,..., q }. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property for the graphs paths P p, cycles C p, C p K, the ladder P P p, P m nk, the balloon graph P n (C p ), quadrilateral snake Q n, [P m ; C n ], (P m ; Q ), T p (n), H n mk, bistar graph and cyclic ladder P C p.. Main Results Observation. Every graph having an odd cycle is not an odd sum graph. Proof If a graph has a cycle of odd length, then at least one edge uv on the cycle such that f(u) and f(v) are of same suit and hence its induced edge label f (uv) is even. Proposition. Every path P p, p is an odd sum graph.

3 On Odd Sum Graphs 6 Proof Let v, v,...,v p be the vertices of the path P p. The labeling f : V (G) {0,,,..., q} is defined as f(v i ) = i for i p and the induced edge label is f (v i v i+ ) = i, for i p. Then f is an odd sum labeling and hence P p is an odd sum graph Figure : An odd sum labeling of P 0. Proposition. Cycle C p is an odd sum graph only when p 0(mod 4) Figure : An odd sum labeling of C 4. Proof By Observation., C p is not an odd sum graph when p is odd. Suppose p = m, m and C p admits an odd sum labeling. Then f (uv) = (f(u) + f(v)). uv E(G) uv E(G) This implies that (4m ) = ( m) i where i is not a vertex label of C p. From this we have, i = m. If m is odd, then the number of even values is in excess of that of the number of odd values and they are to be assigned as vertex labels in C p. Thus if C p admits an odd sum labeling, then m should be even and hence p is a multiple of 4. Suppose p = 4m, m. Let v, v,..., v p be the vertices of the cycle C p. The labeling f : V (G) {0,,,..., 4m} is defined as follows. i, i m and i is odd, f(v i ) = i, i m and i is even, i, m + i 4m. The induced edge labels are obtained as follows. f i, i m, (v i v i+ ) = i +, m + i 4m and f (v 4m v ) = 4m +.

4 6 S.Arockiaraj and P.Mahalakshmi Hence f is an odd sum labeling of C p only when p 0(mod 4). Proposition.4 For each even integer p 4, C p K is an odd sum graph. Proof In C p K, let v, v,..., v p be the vertices on the cycle and let u i be the pendant vertex of v i at each i, i p. Case p = 4m, for m. The labeling f : V (C p K ) {0,,,..., 8m} is defined as follows. i, i m and i is odd, f(v i ) = i, m + i 4m and i is odd, i, i 4m and i is even and i, i 4m and i is odd, f(u i ) = i, i m and i is even, i, m + i 4m and i is even. The induced edge labels are obtained as follows. f 4i, i m, (v i v i+ ) = 4i +, m i 4m, f (v 4m v ) = 8m and f 4i, i m, (u i v i ) = 4i, m + i 4m. Thus f is an odd sum labeling of C p K. Hence C p K is an odd sum graph when p = 4m. Case p = 4m +, for m. The labeling f : V (C p K ) {0,,,..., 8m + 4} is defined as follows. i, i m + and i is odd, i, m + i 4m + and i is odd, f(v i ) = i, i m and i is even, i +, i = m +, i, m + 4 i 4m + and i is even and i, i m + and i is odd, i, i = m +, f(u i ) = i, m + i 4m + and i is odd, i, i m + and i is even, i, m + 4 i 4m + and i is even.

5 On Odd Sum Graphs 6 The induced edge labels are obtained as follows. 4i, i m, f 4i +, i = m +, (v i v i+ ) = 4i +, i = m +, 4i +, m + i 4m +, f (v 4m+ v ) = 8m + and 4i, i m +, f 4i, i = m +, (u i v i ) = 4i, i = m + 4i, m + 4 i 4m +. Thus f is an odd sum labeling of C p K. Hence C p K is an odd sum graph Figure : An odd sum labeling of C 4 K.

6 64 S.Arockiaraj and P.Mahalakshmi Figure 4: An odd sum labeling of C K. Proposition. For every positive integer p, the ladder P P p is an odd sum graph. Proof Let u, u,..., u p and v, v,..., v p be the vertices of the two copies of P p. The labeling f : V (P P p ) {0,,,..., p } is defined as follows. f(u i ) = i, for i p and f(v i ) = i, for i p. The induced edge labels are obtained as follows. f (u i u i+ ) = 6i, for i p, f (v i v i+ ) = 6i, for i p and f (u i v i ) = 6i, for i p. Thus f is an odd sum labeling of P P p. Hence P P p is an odd sum graph Figure : An odd sum labeling of P P.

7 On Odd Sum Graphs 6 Proposition.6 The graph P m nk is an odd sum graph if either m is an even positive integer and n is any positive integer or m is an odd positive integer and n =,. Proof In P m nk, let u, u,..., u m be the vertices on the path and {u i,j : j n} be the pendant vertices attached at u i, i m. Case m is even. The labeling f : V (P m nk ) {0,,,..., m(n + ) } is defined as follows. For i m, (n + )(i ), i is odd, f(u i ) = (n + )i, i is even. For i m and j n, (n + )(i ) + j, i is odd, f(u i,j ) = (n + )(i ) + j, i is even. The induced edge labels are obtained as follows. f (u i u i,j ) = (n + )(i ) + j, for i m and j n and f (u i u i+ ) = (n + )i, for i m. Thus f is an odd sum labeling of P m nk Figure 6: An odd sum labeling of P 6 K. Case m is odd. If P m nk has an odd sum labeling f when m is odd, then f is a bijection from V (P m nk ) to the set {0,,,..., m(n+) }. Since the number of even integers in this set is either equal to or one excess to the number of odd integers in this set, n should be less than or equal to. In case of m is odd and n =,, the labeling f : V (P m nk ) {0,,,..., m(n+) } is defined as follows. For i m, (n + )(i ) +, i is odd, f(u i ) = (n + )i, i is even.

8 66 S.Arockiaraj and P.Mahalakshmi For i m and j n, (n + )(i ) + (j ), i is odd, f(u i,j ) = (n + )(i ) + j +, i is even. The induced edge labels are obtained as follows. For i m and j n, f (u i u i,j ) = (n + )(i ) + j. For i m, Thus f is an odd sum labeling of P m nk. f (u i u i+ ) = (n + )i Figure : An odd sum labeling of P K Figure 8: An odd sum labeling of P K. Proposition. The graph P n (C p ) is an odd sum graph if either p 0(mod 4) or p (mod 4) and n (mod ). Proof Let u, u,..., u p be the vertices of C p and v, v,...,v n be the vertices of the path P n and u p be identified with v in P n (C p ). Case p 0(mod 4). Let p = 4m, m. The labeling f : V (P n (C p )) {0,,,..., 4m + n } is defined as follows. i, i 4m and i is odd, f(u i ) = i, i m and i is even, i, m + i 4m and i is even, f(v i ) = 4m + i, i n.

9 On Odd Sum Graphs 6 The induced edge labels are obtained as follows. f i, i m, (u i u i+ ) = i +, m + i 4m. f (u u 4m ) = 4m + and Thus f is an odd sum labeling of P n (C p ). f (v i v i+ ) = 8m + i, i n Figure : An odd sum labeling of P (C 8 ). Case p (mod 4). Let p = 4m +, m. Subcase. n 0(mod ). The labeling f : V (P n (C p )) {0,,,..., 4m + n + } is defined as follows. f(u ) = 4m +, i, i m +, f(u i ) = i, m + 4 i 4m + and i is even, i, m + 4 i 4m + and i is odd and 4m + i +, i n and i (mod ), 4m + i, i n and i (mod ), f(v i ) = 4m + i +, i n and i 0(mod ), 4m + n +, i = n, 4m + n, i = n.

10 68 S.Arockiaraj and P.Mahalakshmi The induced edge labels are obtained as follows. 4m +, i =, f (u i u i+ ) = i, i m +, i, m + i 4m +. f (u 4m+ u ) = 8m + and 8m + i +, i n 4 and i (mod ), f 8m + i +, i n 4 and i 0(mod ), (v i v i+ ) = 8m + i +, i n 4 and i (mod ), 8m + 4n i, n i n. Thus f is an odd sum labeling of P n (C p ) Figure 0: An odd sum labeling of P (C ). Subcase. n (mod ). The labeling f : V (P n (C p )) {0,,,..., 4m + n + } is defined as follows. f(u ) = 4m +, i, i m +, f(u i ) = i, m + 4 i 4m + and i is even, i, m + 4 i 4m + and i is odd, 4m + i +, i n and i (mod ), and f(v i ) = 4m + i, i n and i (mod ), 4m + i +, i n and i 0(mod ).

11 On Odd Sum Graphs 6 The induced edge labels are obtained as follows. 4m +, i =, f (u i u i+ ) = i, i m +, i, m + i 4m +, f (u 4m+ u ) = 8m + and 8m + i +, i n and i (mod ), f (v i v i+ ) = 8m + i +, i n and i (mod ), 8m + i +, i n and i 0(mod ). Thus f is an odd sum labeling of P n (C p ). Hence P n (C p ) is an odd sum graph Figure : An odd sum labeling of P (C 8 ). Proposition.8 [P m ; C n ] is an odd sum graph for n 0(mod 4) and any m. Proof In [P m ; C n ], let v, v,..., v m be the vertices on the path P m, v i,, v i,,..., v i,n be the vertices of the i th cycle C n, for i m and each vertex v i, of the i th cycle C n is identified with the vertex v i of the path P m, i m. Suppose n = 4t, t. The labeling f : V ([P m ; C n ]) {0,,,,..., m(n + ) } is defined as follows. For i m, (n + )(i ) + j, j t, i and j are odd, (n + )(i ) + j +, t + j 4t, i and j are odd, (n + )(i ) + j, j 4t, i is odd and j are even, f(v i,j ) = (n + )i j, j t, i is even and j is odd, (n + )i j, t + j 4t, i is even and j is odd, (n + )i j, j 4t, i is even and j is even.

12 0 S.Arockiaraj and P.Mahalakshmi For i m, the induced edge label is obtained as follows. (n + )(i ) + j, j t and i is odd, (n + )(i ) + j +, t j 4t and i is odd, f (v i,j v i,j+ ) = (n + )(i ) +, j = and i is even, (n + )(i ) + j, j t + and i is even, (n + )(i ) + j, t + j 4t and i is even and f (n + )(i ) + 4t, i is odd, (v i,4t v i, ) = (n + )(i ) + 8t, i is even. Thus f is an odd sum labeling of [P m ; C n ]. Hence [P m ; C n ] is an odd sum graph Figure : An odd sum labeling of [P ; C 8 ]. Proposition. Quadrilateral snake Q n is an odd sum graph for n. Proof The vertex set and edge set of the Quadrilateral snake Q n are V (Q n ) = {u i, v j, w j : i n +, j n} and E(Q n ) = {u i v i, v i w i, u i u i+, u i+ w i : i n} respectively. The labeling f : V (Q n ) {0,,,..., 4n} is defined as follows. 4i 4, i n + and i is odd, f(u i ) = 4i, i n and i is even, 4i, i n and i is odd, f(v i ) = 4i, i n and i is even 4i, i n and i is odd, and f(w i ) = 4i, i n and i is even. The induced edge labels are obtained as follows f (u i u i+ ) = 8i, i n, f (u i v i ) = 8i, i n, f (v i w i ) = 8i, i n, f (w i u i+ ) = 8i, i n.

13 On Odd Sum Graphs Thus f is an odd sum labeling of Q n. Hence the Quadrilateral snake Q n is an odd sum graph for n Figure : An odd sum labeling of Q. Proposition.0 (P m ; Q ) is an odd sum graph for any positive integer m. Proof Let v i,j, j 8 be the vertices in the i th copy of Q, i m and u, u,..., u m be the vertices on the path P m. {u i u i+ : i m } {u i v i, : i m} {v i, v i,, v i, v i,4, v i, v i,6, v i, v i,, v i, v i,, v i, v i,4, v i, v i,8, v i,4 v i,, v i, v i,6, v i, v i,8, v i,6 v i,, v i, v i,8 : i m} be the edge set of (P m ; Q ). The labeling f : V [(P m ; Q )] {0,,,..., 4m } is defined as follows: For i m, 4(i ), i is odd, f(u i ) = 4i, i is even. For i m and i is odd, f(v i,j ) = For i m and i is even, f(v i,j ) = 4i, j =, 4i + j, j, 4i, j = 4, 4i, j =, 4i 8 + j, 6 j, 4i 4, j = 8. 4i, j =, 4i j, j, 4i, j = 4, 4i 0, j =, 4i j, 6 j, 4i, j = 8. The induced edge label of (P m ; Q ) is obtained as follows:

14 S.Arockiaraj and P.Mahalakshmi For i m, For i m, f (u i u i+ ) = 8i. f 8i, (u i v i, ) = 8i, i is odd, i is even. For i m and i is odd For i m and i is even f (v i, v i, ) = 8i f (v i, v i, ) = 8i f (v i, v i,4 ) = 8i, f (v i, v i,4 ) = 8i f (v i, v i,6 ) = 8i f (v i, v i,6 ) = 8i f (v i, v i, ) = 8i f (v i, v i, ) = 8i f (v i, v i, ) = 8i f (v i, v i, ) = 8i f (v i, v i,4 ) = 8i f (v i, v i,4 ) = 8i f (v i, v i,8 ) = 8i f (v i, v i,8 ) = 8i f (v i,4 v i, ) = 8i f (v i,4 v i, ) = 8i f (v i, v i,6 ) = 8i f (v i, v i,6 ) = 8i f (v i, v i,8 ) = 8i f (v i, v i,8 ) = 8i, f (v i,6 v i, ) = 8i f (v i,6 v i, ) = 8i f (v i, v i,8 ) = 8i f (v i, v i,8 ) = 8i Thus f is an odd sum labeling of (P m ; Q ). Hence (P m ; Q ) is an odd sum graph Figure 4: An odd sum labeling of (P 4 ; Q ). Proposition. For all positive integers p and n, the graph T (n) p is an odd sum graph. Proof Let v (j) i, i p be the vertices of the j th copy of the path on p vertices, j n. The graph T p (n) is formed by adding an edge v (j) i v (j+) i between j th and (j + ) th copy of the path at some i, i p., The labeling f : V (G) {0,,,..., np } is defined as follows:

15 On Odd Sum Graphs For j n and i p, f(v (j) i ) = p(j ) + i, pj i, j is odd, j is even. The induced edge labeling is obtained as follows: For j n and i p, f (v (j) i v (j) i+ ) = p(j ) + i, j is odd, pj i, j is even and f (v (j) i v (j+) i ) = pj. Thus f is an odd sum labeling of the graph T p (n). Hence T p (n) is an odd sum graph Figure : An odd sum labeling of T () 8. Proposition. The graph H n mk is an odd sum graph for all positive integers m and n. Proof Let u, u,..., u n and v, v,..., v n be the vertices on the path of length n. Let x i,k and y i,k, k m, be the pendant vertices at u i and v i respectively, for i n. Define f : V (H n mk ) {0,,,..., n(m + ) } as follows:

16 4 S.Arockiaraj and P.Mahalakshmi For i n, i + m(i ), i is odd, f(u i ) = i(m + ), i is even and f(u i ) + n(m + ) + m, i is odd and n is odd, f(v i ) = f(u i ) + n(m + ) m +, i is even and n is odd, f(u i ) + n(m + ), n is even. For i n and k m, (m + )(i ) + k, i is odd, f(x i,k ) = (m + )(i ) + k +, i is even and f(x i,k ) + n(m + ) m +, i is odd and n is odd, f(y i,k ) = f(x i,k ) + n(m + ) + m, i is even and n is odd, f(x i,k ) + n(m + ), n is even. The induced edge labels are obtained as follows: For i n, f (u i u i+ ) = i(m + ) and f (v i v i+ ) = f (u i u i+ ) + n(m + ). For i n and k m, f (u i x i,k ) = (m + )(i ) + k and f (v i y i,k ) = f (u i x i,k ) + n(m + ). When n is odd, ( ) f u n+v n+ = n(m + ). When n is even, f ( u n + v n ) = n(m + ). Thus f is an odd sum labeling of H n mk. Hence H n mk is an odd sum graph for all positive integers m and n.

17 On Odd Sum Graphs Figure 6: An odd sum labeling of H 4 K Figure : An odd sum labeling of H 4K.

18 6 S.Arockiaraj and P.Mahalakshmi Corollary. For any positive integer m, the bistar graph B(m, m) is an odd sum graph. Proof By taking n = in Proposition., the result follows. Proposition.4 For any even integer p 4, the cyclic ladder P C p is an odd sum graph. Proof Let u, u,..., u p and v, v,..., v p be the vertices of the inner and outer cycle which are joined by the edges {u i v i : i p}. Case p = 4m, m Figure 8: An odd sum labeling of P C 4. The labeling f : V (P C p ) {0,,,..., m} is defined as follows: i, i m and i is odd, f(u i ) = i +, i 4m and i is even, i +, m + i 4m and i is odd, f(u 4m ) = and 8k + i, i 4m and i is odd, f(v i ) = 8k + i, i m and i is even, 8k + i, m + i 4m and i is even.

19 On Odd Sum Graphs The induced edge labeling is obtained as follows. i +, i m, f (u i u i+ ) = i +, m i 4m, i +, i = 4m, f (u u 4m ) =, f 6m + i, i m (v i v i+ ) = 6m + i +, m + i 4m, f (v v 4m ) = 0m +, f 8m + i, i m, (u i v i ) = 8m + i +, m + i 4m and f (u 4m v 4m ) = m +. Thus f is an odd sum labeling of P C p. Hence P C p is an odd sum graph when p = 4m. Case p = 4m +, m Figure : An odd sum labeling of P C. The labeling f : V (P C p ) {0,,,..., m} is defined as follows: i, i m +, f(u i ) = i +, m + i 4m + and i is odd,

20 8 S.Arockiaraj and P.Mahalakshmi i, m + 4 i 4m and i is even, f(u i ) = i, i = 4m + and i, i m +, i +, m + i 4m and i is even, f(v i ) = i, m + i 4m + and i is odd, i, i = 4m +. The induced edge labels are given as 6i, i m +, f (u i u i+ ) = 6i +, m + i 4m, 6i +, i = 4m +, f (u u 4m+ ) = m +, 6i, i m, f (v i v i+ ) = 6i +, m + i 4m, 6i +, i = 4m +, f (v v 4m+ ) = m + and f 6i, i m +, (u i v i ) = 6i, m + i 4m +. Thus f is an odd sum labeling of P C p. Whence P C p is an odd sum graph if p = 4m +. References [] F.Buckley and F.Harary, Distance in Graphs, Addison-Wesley, Reading, 0. [] R.Balakrishnan, A.Selvam and V.Yegnanarayanan, On felicitous labelings of graphs, Graph Theory and its Applications, (6), 4 6. [] J.A.Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, (0), # DS6. [4] K.Manickam and M.Marudai, Odd mean labelings of graphs, Bulletin of Pure and Applied Sciences, E() (006), 4. [] R.Ponraj, J.Vijaya Xavier Parthipan and R.Kala, Some results on pair sum labeling of graphs, International Journal of Mathematical Combinatorics, 4 (00), 6. [6] S.Somasundaram and R.Ponraj, Mean labelings of graphs, National Academy Science Letter, 6 (00), 0. [] Selvam Avadayappan and R. Vasuki, Some results on mean graphs, Ultra Scientist of Physical Sciences, ()M (00), 84.