A Substitution Theorem for Graceful Trees and its Applications¤

A graceful labeling of a graphG = (V;E) assigns jV j distinct integers from the set f0; : : : ; jEjg to the vertices of G so that the absolute values of their di®erences on the jEj edges of G constitute the set f1; : : : ; jEjg. 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 com-bining smaller trees