CUBO, A Mathematical Journal Vol.22, N◦03, (299–314). December 2020 http://dx.doi.org/10.4067/S0719-06462020000300299 Received: 28 December, 2019 | Accepted: 01 September, 2020 Odd Harmonious Labeling of Some Classes of Graphs P. Jeyanthi 1 and S. Philo 2 1Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur - 628 215, Tamil Nadu, India. 2Department of Mathematics, St.Xavier’s College, Palayamkottai, Tirunelveli -627002, Tamilnadu, India. jeyajeyanthi@rediffmail.com, lavernejudia@gmail.com 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 Tp- tree, T ◦̂Pm, T ◦̂2Pm, regular bamboo tree, Cn◦̂Pm, Cn◦̂2Pm and subdivided grid graphs are odd harmonious. RESUMEN Un grafo G(p, q) se dice impar armonioso si existe una inyección f : V (G) → {0, 1, 2, · · · , 2q − 1} tal que la función inducida f∗ : E(G) → {1, 3, · · · , 2q − 1} definida por f∗(uv) = f(u) + f(v) es una biyección. En este art́ıculo probamos que los grafos Tp-árboles, T ◦̂Pm, T ◦̂2Pm, árboles bambú regulares, Cn◦̂Pm, Cn◦̂2Pm y cuadŕıculas subdivididas son impar armoniosos. Keywords and Phrases: harmonious labeling, odd harmonious labeling, transformed tree, sub- divided grid graph, regular bamboo tree. 2020 AMS Mathematics Subject Classification: 05C78. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000300299 300 P. Jeyanthi and S. Philo CUBO 22, 3 (2020) 1 Introduction Throughout this paper by a graph is implied as a finite, simple and undirected. For standard terminology and notation we follow Harary [3]. 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. An extensive survey of various graph labeling problems is available in [1]. Graham and Sloane [2] 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 an odd harmonious labeling was due to Liang and Bai [14]. A graph G 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. If f : V (G) → {0, 1, 2, · · · , q} then f is called as strongly odd harmonious labeling and G is called a strongly odd harmonious graph. The odd harmoniousness of a graph is useful for the solution of undetermined equations. The following results have been proved in [14]: 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 ≤ p ≤ (2q − 1). 3. If G is an odd harmonious Eulerian graph with q edges, then q ≡ 0, 2(mod 4). Followed by this, Vaidya and Shah [18], [19] showed that shadow and splitting graphs are odd harmonious. Selvaraju et al. [17] established that some path related graphs are odd harmonious. Jeyanthi et al. proved that the following graphs are odd harmonious: double quadrilateral snake and banana tree [5], cycle related graphs [6], plus graphs [7], super subdivision graphs [8], subdi- vided shell graphs [9], spider and necklace graphs [10], m-shadow, m-splitting and m-mirror graphs [11] and [12], grid graphs [13]. We use the following definitions in the subsequent section. Definition 1.1. Let G = (V, E) be a graph. G is called a path Pn if V = {v1, v2, · · · , vn} such that 1 ≤ i ≤ n, (vi, vi+1) ∈ E. Definition 1.2. The Cartesian product of graphs G and H denoted as G�H, is the graph with vertex set V (G) × V (H) = {(u, v)|u ∈ V (G) and v ∈ V (H)} and (u, v) is adjacent to (ú, v́) if and only if either u = ú and (v, v́) ∈ E(H) or v = v́ and (u, ú) ∈ E(G). The Cartesian product of two paths Pm and Pn denoted by Pm × Pn is known as a grid graph on mn vertices and 2mn − (m + n) edges. CUBO 22, 3 (2020) Odd Harmonious Labeling of Some Classes of Graphs 301 Definition 1.3. Let G be a graph with p vertices and H be any graph and x be a vertex of H. A graph G◦̂H is obtained from G and p copies of H by identifying vertex x of ith copy of H with ith vertex of G. Definition 1.4. [4] Let T be a tree and u0 and v0 be the two adjacent vertices in T . Let u and v be the two pendant vertices of T such that the length of the path u0 − u is equal to the length of the path v0 − v. If the edge u0v0 is deleted from T and u and v are joined by an edge uv, then such a transformation of T is called an elementary parallel transformation (or an ept) and the edge u0v0 is called transformable edge. If by some sequence of ept’s, T can be reduced to a path, then T is called a Tp- tree (transformed tree) and such sequence regarded as a composition of mappings (ept’s) denoted by P is called a parallel transformation of T . The path, the image of T under P is denoted as P(T ). A Tp- tree and the sequence of two ept’s reducing it to a path are illustrated in Figure 1. a) A Tp- tree T b) An ept P1(T ) c) Second ept P2(T ) r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r u0 v0 x0 y0 u v x0 u v x y y0 Figure 1: Transformed tree Definition 1.5. [15] Let T be a Tp-tree with n vertices v1, v2, · · · , vn. The graph T ◦̂Pm is obtained from T and n copies of Pm by identifying a pendant vertex of i th copy of Pm with vertex vi of T . Definition 1.6. [16] Consider k copies of paths Pn of length n − 1 and stars Sm with m pendant vertices. Identify one of the two pendant vertices of the jth path with the centre of the jth star. Identify the other pendant vertex of each path with a single vertex u0 (u0 is not in any of the star and path). The graph obtained is a regular bamboo tree. 302 P. Jeyanthi and S. Philo CUBO 22, 3 (2020) 2 Main Results In this section, we prove that Tp- tree, T ◦̂Pm, T ◦̂2Pm, regular bamboo tree, Cn◦̂Pm, Cn◦̂2Pm and subdivided grid graphs are odd harmonious. Theorem 2.1. Every Tp- tree is strongly odd harmonious. Proof. Let T be a Tp-tree with n vertices. By definition, there exists a parallel transformation P of T , we have V (P(T )) = V (T ) and E(P(T )) = (E(T ) − Ed) ∪ Ea, where Ed is the set of deleted edges and Ea is the set of newly added edges through the sequence P = (P1, P2, · · · , Pl) of the ept’s used to obtain P(T ). Hence Ed and Ea have the same number of edges. Let u1, u2, · · · , un be the vertices of P(T ) successively, from one pendant vertex of P(T ) right up to the other. This Tp-tree has n vertices and n − 1 edges. We define a labeling f : V (G) → {0, 1, 2, · · · , q = n − 1} as follows: f(ui) = i − 1, 1 ≤ i ≤ n. Let (uiuj) be an edge of T , 1 ≤ i