International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 04 Issue: 07 | July -2017 www.irjet.net p-ISSN: 2395-0072 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2303 Cordial labelings in the context of triplication S. Gurupriya 1 , S.Bala 2 1 B.E(Final year), Department of Computer Science Sri Sairam Engineering College, West Tambaram, Chennai, Tamilnadu, India 2 Assistant professor, Department of Mathematics S.I.V.E.T.College, Gowrivakkam, Chennai, Tamilnadu, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - In this paper, we introduce the extended triplicate graph of a ladder and investigate the existence of cordial labeling, total cordial labeling, product cordial labeling, total product cordial labeling and prime cordial labeling for the extended triplicate graph of a ladder graph by presenting algorithms. Key Words: Ladder graph, Triplicate graph, Graph labelings. 1.INTRODUCTION Graph theory has various applications in the field of computer programming and networking, marketing and communications, business administration and so on. Some major research topics in graph theory are Graph coloring, Spanning trees, Planar graphs, Networks and Graph labeling. Graph labeling has been observed and identified for its usage towards communication networks. That is, the concept of graph labeling can be applied to network security, network addressing, channel assignment process and social networks [3]. In 1967, Rosa introduced the concept of graph labeling [4]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of mapping is the set of vertices (edges) then the labeling is called a vertex(an edge) labeling. In 1987, Cahit introduced the notion of cordial labeling [2]. A graph G is said to admit a cordial labeling if there exists a function f : V → {0, 1} such that the induced function f* : E → {0, 1} defined as f*(vivj) = | f(vi) - (f(vj) | or (f(vi) + f(vj)) (mod 2) satisfies the property that the number of vertices labeled ǮͲǯ and the number of vertices labeled Ǯͳǯ differ by atmost one and the number of edges labeled ǮͲǯ and the number of edges labeled Ǯͳǯ differ by atmost one. A graph G is said to admit a total cordial labeling if there exists a function f : V → {0, 1} such that the induced function f* : E → {0, 1} defined as f*(vivj) = | f(vi) - (f(vj) | or (f(vi) + f(vj)) (mod 2) satisfies the property that the number of vertices and edges labeled with ǮͲǯ and the number of vertices and edges labeled with Ǯͳǯ differ by atmost one. In 2004, Sundaram, Ponraj and Somasundaram have introduced the concept of product cordial labeling [ 5,6 ]. A graph G is said to admit product cordial labeling if there exists a function f : V → {Ͳ, ͳ} such that the induced function f* : E → {Ͳ, ͳ} defined as f*(vivj) = {(f(vi) × f(vj) | vivj ∈ E} satisfies the property that the number of vertices labeled ǮͲǯ and the number of vertices labeled Ǯͳǯ differ by atmost ͳ and number of edges labeled ǮͲǯ and the number of edges labeled Ǯͳǯ differ by atmost ͳ. A graph that admits product cordial labeling is called product cordial graph. A graph is called total product cordial graph if there exists a function f : V → {Ͳ, ͳ} such that the induced function f* : E → {Ͳ, ͳ} defined as f*(vivj) ={(f(vi) × f(vj) | vivj ∈ E} satisfies the property that the number of Ͳǯs on the vertices and edges taken together differ by atmost one with the number of ͳǯs on the vertices and edges taken together. In 2011, Bala and Thirusangu introduced the concept of the extended triplicate graph of a path Pn ((ETG(Pn)) and proved many results on this newly defined concept [1]. Let V = { v1, v2,…,vn+1} and E = { e1, e2 , …. , en} be the vertex and Edge set of a path Pn. For every vi ∈ V, construct an ordered triple {vi , vi Ԣ , vi ԣ } where ͳ≤ i ≤ n+1 and for every edge vivj ∈ E, construct four edges vivj Ԣ , vj Ԣ vi ԣ , vjvi Ԣ and vi Ԣ vj ԣ where j = i +1, then the graph with this vertex set and edge set is called a Triplicate Graph of a path Pn. It is dentoted by TG(Pn). Clearly the Triplicate graph TG(Pn) is disconnected. Let V1 = {v1, v2 …,v3n+1} and E1 = { e1, e2,…., e4n}be the vertex and edge set of TG(Pn). If n is odd, include a new edge (vn+1 , v1) and if n is even, include a new edge (vn ,v1) in the edge set of TG(Pn). This graph is called the Extended Triplicate of the path Pn and it is denoted by ETG(Pn). In 2014 , Thirusangu et.al proved some results on Duplicate Graph of Ladder Graph [7]. A ladder graph Ln is a planar undirected graph with 2n vertices and 3n– 2 edges. It is obtained as the cartesian product of two path graphs, one of which has only one edge: Ln,1 = Pn × P1, where n is the number of rungs in the ladder. Motivated by the study, the present work is aimed to provide label for the extended triplicate graph of a ladder graph and prove the existence of cordial labeling, total cordial labeling, product cordial labeling