CHAPTER - 1 INTRODUCTION

Transcription

1 CHAPTER - 1 INTRODUCTION INTRODUCTION This thesis comprises of six chapters and is concerned with the construction of new classes of cordial graphs, even and odd graceful graphs, even and odd mean graphs, prime graphs and strongly multiplicative graphs by using graph operations. It has becomes more and more clear in recent years that the two disciplines of graph theory and computer science have much in common, and that each is capable of assisting significantly in the development of the other. Thus graph theorists are increasingly finding that many of their problems can be solved, or their research furthered by the use of computing techniques. Graph labeling problems are one among them. Numerous variations of labeling have been investigated in the literature. A survey on recent results, conjectures and open problems in labeling graph is presented in the paper by J.A.Gallian [18]. Two important graph labelings called Graceful labeling and Harmonious labeling are defined in graph labeling. Cordial labeling, Even

2 2 graceful labeling, Odd graceful labeling and equitable labeling are variations of Graceful labeling. These labelings are used throughout the thesis. An outline of the results contained in all the chapters is given in the introduction. 1.1 TYPES OF LABELING A labeling (or valuation) of a graph G is an assignment f of labels from a set of positive integers to the vertices of G that induce a label for each edge xy defined by the labels f(x) and f(y). We formally define below some of the graph labelings which are used throughout this thesis. [1,4,8,9] For the other related terminology and notations, we follow Bondy and Murthy [6] and Behzad et al [4]. Definition A simple graph is said to be graceful if there exists an injection such that the induced function defined by, is a bijection. The injection is called a Graceful Labeling of. The values are called graceful labels of the vertex and the edge respectively. Many families of graphs are proved to be graceful and also some families of graphs are proved non graceful (refer Gallian 2008). Recognition of a given graph to be graceful or not is one of the major problems in Graph Theory. [25-28,30]

3 3 An example of graceful graph is given in Figure 1.1. Figure 1.1 Graceful labeling of A cyclic decomposition of is obtained as follows: 1. Choose any graph with graceful labeling. 2. Identify the edges of with suitable edges of, where q = E( G) 3. Each vertex and each edge of is rotated times from the original position. Definition A graceful labeling f of a graph is said to be an α-labeling (interlaced and balanced) if there exists, a smallest value λ such that either or.

4 4 Equivalently, for each edge e = uv of G, one end vertex receives a label from the set while the other end vertex receives a label from the set {λ+1,..., q}. Such a λ is often called the width of graceful labeling. A graph G admitting α-labeling is necessarily bipartite. In this thesis it is verified that Supersubdivision of any non-trivial graph with even or odd graceful graph is α-labeling. Rosa [29] has proved the following significant theorem. Theorem [29] If a graph G with q edges has a graceful labeling, then there exists a cyclic decomposition of the complete graph K 2cq+1 into subgraphs isomorphic to G, where c is arbitrary natural number. Recently, El Zanati and Vanden Eynden [17] have proved that G has q edges and admits α-labeling, then can be decomposed into subgraphs isomorphic to G, for all positive integers m and n. Definition Let f be a function from the vertices of G to {0,1}and for each edge xy assign the label f(x)-f(y). Call f a Cordial Labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differs by at most 1 and the number of edges labeled 0 and the number of edges labeled 1 differs by atmost 1.An example of cordial graph is given in Figure 1.2.

5 5 Figure 1.2 Cordial labeled wheel W 5 In the complete bipartite graph, the part consisting of two vertices is termed as the 2-vertices part of and the part consisting of m vertices is termed as the m-vertices part of. Definition A function f is called an Even-Graceful Labeling of a graph G with q edges, if f is an injection from the vertices of G to the set { 0,1,2,...,2q} such that when each edge uv is assigned the label f ( u ) f ( v ), the resulting edge labels are distinct even numbers, i.e. they range from 2 to 2q. Definition A function f is called an Odd-Graceful Labeling of a graph G with q edges, if f is an injection from the vertices of G to the set {0,1,2,,2q-1} such that when each edge uv is assigned the label f ( u ) f ( v ), the resulting edge labels are distinct odd numbers, i.e. they range from 1 to 2q-1.

6 6 A graph which admits an Even or Odd-Graceful labeling is said to be Even or Odd-Graceful Graph. An Even or Odd-Graceful labeling f is called an Even or Odd- α- valuation of G. If there exist an even integer λ, so that for each edge uv either f ( u) λ < f ( v) or f ( v) λ < f ( u). It follows that such a λ must be the smallest of the two vertex labels that yield the edge labeled 2 or 1 respectively. bipartite. A graph G admitting an Even or Odd-α- valuation is necessarily Definition Let G be a graph with n vertices and t edges. A graph H is said to be a Supersubdivision of G if H is obtained from G by replacing every edge of G by a complete bipartite graph for some m i (vary for each edge),in such a way that the end of G are merged with the two vertices of the 2-vertices part of after removing the edge from G. The Graph C 4 and its Supersubdivision are shown in the following Figures 1.3 and 1.4.

7 7 Fig 1.3 Cycle Fig 1.4 A Super subdivision of Definition A graph G with p vertices and q edges is called Even Mean Graph if there is an injective function f from the vertices of G to {2,4,6,8,..2q} such that when each edge uv is labeled with [f(u)+f(v)]/2, then the resulting edge labels are distinct. Definition A graph G with p vertices and q edges is called Odd Mean Graph, if there is an injective function f from the vertices of G to {1,3,5,7,..2q-1} such that when each edge uv is labeled with [f(u)+f(v)]/2, then the resulting edge labels are distinct. Definition A graph with vertex set V is said to have a Prime Labeling if its vertices are labeled with distinct integers 1,2,3,, V such that for each edge xy the labels assigned to x and y are relatively prime.

8 8 Definition A graph with p vertices is Strongly Multiplicative if the vertices of G can be labeled with distinct integers 1,2,3, p such that the labels induced on the edges by the product of the end vertices are distinct. Definition A graph of order n is said to be a Strongly * graph if its vertices can be assigned the values 1,2,3,4 n in such a way that, when an edge whose vertices are labeled i and j is labeled with the values i + j + ij, all edges have different labels. 1.2 ORGANIZATION OF THESIS The chapters are organized based on the following classifications: 1. Supersubdivision of any non-trivial Graph 2. One vertex union of Grid Graph and Structured Web Graph 3. Silo graph and Chaplet Graph 4. Even Graceful Graph and Odd Graceful Graphs 5. Even Mean Graph, Odd Mean Graph, Prime Graph, Strongly Multiplicative Graph and Strongly * Graph. In Chapter 2, an algorithm is given to construct certain Super subdivision of any non trivial connected graph. It is shown that such Supersubdivision graph of any non trivial connected graph admits a graceful

9 9 labeling.it follows from Rosa s theorem [29] that some complete graphs can be cyclically decomposed into subgraphs isomorphic to such Supersubdivision graph of any non trivial connected graphs and by theorems of El Zanati and Vanden Eynden [15]. It follows that some complete bipartite graphs can be decomposed into sub graphs isomorphic to such certain Supersubdivision of any non-trivial connected graph. In Chapter 3, an algorithm is given for cordial labeling of one vertex union of any non trivial grid graph and diagonal grid graph.. An algorithm is also given for cordial labeling of structured web graph. Motivated by the work of Sethuraman et al [18] in this theorem, we propose an algorithm to get Cordial Labeling for one vertex union of t copies of grid graph P P, mn>, 2, i.e.. ( P m have also proved the following graphs are cordial: n m ) t n P, mn>, 2, t > 1. We One vertex union of Diagonal Grid Graph d( p p ) m, n mn, 1 and One vertex union of Structured Web Graph W n S ( P C ) m, n 1 = n m n Further it is proved that Fence Graph P m d P mn, is Total Product n Cordial Graph. In Chapter 4, another family of graphs with Cordial Labeling of Silo graph and Cordial Labeling of Chaplet Graph is considered. I. Cahit [7] and

10 10 Sethuraman et al [31,32,33] have proved that that every tree is cordial and one edge union of Shell Graphs and one vertex union of complete bipartite graphs are cordial. Here cordial labeling for the following graphs are given as: 1. Silo graph C4r tpn rt, 1 and n > 1 t 2. Chaplet graph C C, p, qand t p q In Chapter 5, Even graceful labeling for the following graphs are studied. Grid, Comb Θ and Multiple fans, n mean graphs. In Chapter 6, the following graphs are prove to be even and odd Comb Θ K 1, n, Path, n, Cycle with odd length,

11 11 Star K 1,n, n. The following graphs are proved to satisfy the prime labelling t C n ( Isomorphic) n, t C n ( Non-Isomorphic) n, C n. K 1, n, W n When is odd, Strongly Multiplicative and Strongly * Labeling results for Prism Graph is given.