International Journal of Engineering, Science and Mathematics Vol. 6Issue 7, November 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Caďell’s DireĐtories of Puďlishing Opportunities, U.S.A 131 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com SOME RESULTS ON CUBIC HARMONIOUS LABELING Mini.S.Thomas 1 Mathew Varkey T.K 2 Abstract A (n,m) graph G =(V,E) is said to be Cubic Harmonious Graph(CHG) if there exists an injective function f:V(G)→{1,2,3,………m 3 +1} such that the induced mapping f *chg: E(G)→ {1 3 ,2 3 ,3 3 ,……….m 3 } defined by f *chg(uv) = (f(u)+f(v)) mod (m 3 +1) is a bijection. Here we will discuss about cubic harmonious labeling techniques of the graph   Θ  1 – and twig graph. Keywords: Comb graph, Cubic harmonious graph, Harmonious graph, Path graph. Twig graph. Author correspondence: Asst.Prof,Department of Mathematics, ILM College of Engineering And Technology Eranakulam, Kerala India 1 1. Introduction By a graph, we mean a finite, undirected graph without loops or multiple edges. A path on n vertices is denoted by Pn. Graph labeling, where the vertices are assigned certain values subject to some conditions, have often motivated by practical problems. In the last five decades, enormous work has been done on this subject [3].Any graph labeling will have the following three common characteristics : (1). A set of numbers from which the vertex labels are chosen. (2). A rule that assigns a value to each edge. (3). A condition that these values must satisfy. Graph labelings is an active area of research in graph theory which is mainly evolved through its rigorous applications in coding theory, communication networks, optional circuits layouts and graph decomposition problems. According to Beineke and Hegde [2] graph labeling serves as a frontier between number theory and structure of graphs. For a dynamic survey of various graph labeling problems along with an extensive bibliography we refer to Gallian [4]. Graham and Sloane[5] introduced harmonious labeling . Square harmonious graphs were introduced in [10]. Cubic graceful graphs were introduced in [6]. Cubic harmonious graphs were defined in [7]. Definition 1.1 The path on n vertices is denoted by Pn. Definition1.2 The Twig graph obtained from the the path Pn by attaching exactly two pendant edges to each internal vertex of the path . Definition1.4 Let PnΘK1 be the Comb which is the graph obtained from a path Pn by attaching pendant edge at each vertex of the path .