An analysis of labeling algorithms for trees

Abstract

The second part of this thesis deals with graceful labelings. There is a famous unproven conjecture in computer science that all trees have a graceful labeling. This claim has been proven for many classes of trees, but there is currently no proof for general trees. This thesis expands upon some of these proofs and provides numerical support for the graceful tree conjecture for certain other classes of trees. The classes that are focused on are symmetric trees and spider trees. Regarding symmetric trees, fast recursive and iterative algorithms for gracefully labeling symmetric trees are given. Work on spider trees includes a constructive proof of existence of alpha-labelings, a subset of graceful labelings, for a certain class of spider trees, and a conjecture that all spider trees have a graceful labeling with a special property.

Alexander Rosa conjectures that every tree has a rho-labeling. The first part of this thesis sets out to experimentally examine whether certain efficient labeling algorithms will yield a rho-labeling for all trees up to a certain size.