A substitution theorem for graceful trees and its applications

Abstract

A graceful labeling of a graph G = (V, E) assigns | V | distinct integers from the set {0, ..., | E |} to the vertices of G so that the absolute values of their differences on the | E | edges of G constitute the set {1, ..., | E |}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful. We prove a Substitution Theorem for graceful trees, which enables the construction of a larger graceful tree through combining smaller and not necessarily identical graceful trees. We present applications of the Substitution Theorem, which generalize earlier constructions combining smaller trees. © 2008 Elsevier B.V. All rights reserved.