International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 04 Issue: 07 | July -2017
p-ISSN: 2395-0072
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Cordial labelings in the context of triplication S. Gurupriya1, S.Bala2 1B.E(Final
year), Department of Computer Science Sri Sairam Engineering College, West Tambaram, Chennai, Tamilnadu, India 2Assistant 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.
exists a function f : V → {0, 1} such that the induced function f* : E → {0, 1} defined as f*(vivj) = {(f(vi) × f(vj) | vivj ∈ E} satisfies the property that the number of vertices labeled ‘0’ and the number of vertices labeled ‘1’ differ by atmost 1 and number of edges labeled ‘0’ and the number of edges labeled ‘1’ differ by atmost 1. A graph that admits product cordial labeling is called product cordial graph.
Key Words: Ladder graph, Triplicate graph, Graph labelings.
A graph is called total product cordial graph 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) | vivj ∈ E} satisfies the property that the number of 0’s on the vertices and edges taken together differ by atmost one with the number of 1’s on the vertices and edges taken together.
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 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′ , vi ″} where 1≤ i ≤ n+1 and for every edge vivj ∈ E, construct four edges vivj′, vj′ vi″ , vjvi′ and vi′ vj″ 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 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 ‘0’ and the number of vertices labeled ‘1’ differ by atmost one and the number of edges labeled ‘0’ and the number of edges labeled ‘1’ 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 ‘0’ and the number of vertices and edges labeled with ‘1’ differ by atmost one.
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
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
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