On the labelings of graphs and their applications

Abstract

Graph labeling deals with the assignment of integers to the vertices or edges of the graph. In this thesis, linear arrangement and graceful labeling which are well-known graph labeling problems are investigated. Also several new graph labeling problems are proposed and investigated. The problems are motivated from various problems of practical interest and approached from a viewpoint of graph labeling. First, we introduce a new graph labeling problem called d-edge labeling. The constraint in this labeling is placed on the allowable edge label which is the difference between the labels of endvertices of an edge. Each edge label should be a power of d. We show that every full quaternary tree admits 2-edge labeling schemes to them. The labelings on the trees are applied to their embeddings into hypercubes and recursive circulants. Second, we show that min-sum and min-cut linear arrangement problems on trees can be represented as path labeling problem on trees. In a path labeling of a tree, we successively find paths in the tree and assign labels to the edges of each path. The labels assigned to edges are used to find a linear arrangement satisfying some conditions. As an application of the path labeled tree, two-row arrangement problem is considered. The problem is to minimize the crossing number of the edges when the vertices of a tree are placed in two rows and the edges are connected by straight lines. A formula on the crossing number of two-row arrangement obtained from a path labeled tree is given in terms of the cost of the path labeling. Also the relation between the min-sum path labeling of a tree and the optimal two-row arrangement of the tree is investigated. Third, we propose transitive labeling on comparability graphs. A new expression on the cost of a linear arrangement is derived. And we show that a transitive labeling on the comparability graph minimizes a term in the cost expression. The term is concerned with the complement graph of a comparabilit...