A substitution theorem for graceful trees and its applications

A graceful labeling   of a graph G=(V,E)G=(V,E) assigns |V||V| distinct integers from the set {0,…,|E|}{0,…,|E|} to the vertices of GG so that the absolute values of their differences on the |E||E| edges of GG constitute the set {1,…,|E|}{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.