Graceful and harmonious labeling of the disjoint union of certain cycles and paths /

Abstract

A vertex labeling of a graph G is an assignment f of labels to the vertices of G. Each edge xy is assigned a label depending on the vertex labels f (x) and f (y). One of the best known type of labeling methods is called graceful labeling. A function f is called a graceful labeling of a graph G with m edges if f is an injection from the vertices of G to the set {a, 1, ... ,m} such that, when each edge xy is assigned the label If(x) - f(y)l, the resulting edge labels are distinct. Over the past few decades, several different graphs have been gracefully labeled. Another popular labeling method is harmonious labeling. A function f is called a harmonious labeling of a graph G with m edges if it is an injection from the vertices of G to the group of integers modulo m such that when each edge xy is assigned the label (J(x) + f(y)) (mod m), the resulting edge labels are distinct. If the graph G is a tree or has components that are trees, then exactly one label may be used on two vertices. In this case f would not be an injection. In this thesis, we present a technique for labeling the disjoint union of the path and some cycles. We use both labeling methods, graceful and harmonious.