A substitution theorem for graceful trees and its applications

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. ve 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.