A substitution theorem for graceful trees and its applications

A graceful labeling of a graphG = (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.