Relation Between Different Types of Graceful Graphs

Abstract

A graceful labeling (or β-valuation) of a graph G = (V, E) with m =ǀVǀ vertices and n =ǀEǀ edges is a one-to-one mapping f of the vertex set V(G) into the set {0,1,2, …, n} with the following property: If we define, for any edge e = {u,v}∈E(G), the value f*(e) = |f (u) −f (v)| then f* is a one-to-one mapping of the set E(G) onto the set {1,2,…,n}. In this paper we present some relations between graceful labeling and other graceful labeling such as odd-even graceful labeling and even graceful labeling. We also provide all even graceful graphs are graceful if they have no odd vertex.