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Graceful Trees: Statistics and Algorithms Michael Horton

i

Graceful Trees: Statistics and

Algorithms

By Michael Horton, BComp

A dissertation submitted to the

School of Computing

in partial fulfilment of the requirements for the degree of

Bachelor of Computing with Honours

School of Computing

University of Tasmania

November, 2003

Graceful Trees: Statistics and Algorithms Michael Horton

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Statement

I, Michael Horton do hereby declare that this thesis contains no material that has

been accepted for the award of any other degree or diploma in any tertiary institution.

To the best of my knowledge and belief it contains no material previously published

by another person, except where due reference is made in the text of the thesis.

Signed:................................................

Graceful Trees: Statistics and Algorithms Michael Horton

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Abstract

The Graceful Tree Conjecture is a problem in graph theory that dates back to 1967.

It suggests that every tree on n nodes can be labelled with the integers [1..n] such that

the edges, when labelled with the difference between their endpoint node labels, are

uniquely labelled with the integers [1..n-1]. To date, no proof or disproof of the

conjecture has been found, but all trees with up to 28 vertices have been shown to be

graceful. The conjecture also leads to a problem in algorithm design efficiently

finding graceful labellings for trees. In this thesis, a new graceful labelling algorithm

is described and used to show that all trees on 29 vertices are graceful. A study is

also made of statistical trends in the proportion of tree labellings that are graceful.

These trends offer strong additional evidence that every tree is graceful.

Graceful Trees: Statistics and Algorithms Michael Horton

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Contents

GRACEFUL TREES: STATISTICS AND ALGORITHMS.............................................................I STATEMENT ...................................................................................................................................... II ABSTRACT ....................................................................................................................................... III CONTENTS ........................................................................................................................................IV LIST OF TABLES..............................................................................................................................VI LIST OF FIGURES.......................................................................................................................... VII ACKNOWLEDGEMENTS ...............................................................................................................IX DEFINITIONS..................................................................................................................................... X 1 INTRODUCTION....................................................................................................................... 1 2 LITERATURE REVIEW........................................................................................................... 2

2.1 THE GRACEFUL TREE CONJECTURE ..................................................................................... 2 2.2 APPROACH 1: CLASSES OF GRACEFUL TREE ......................................................................... 2

2.2.1 Chains............................................................................................................................. 3 2.2.2 Caterpillars..................................................................................................................... 3 2.2.3 m-stars ............................................................................................................................ 3 2.2.4 Trees with diameter five.................................................................................................. 4 2.2.5 Olive trees....................................................................................................................... 4 2.2.6 Banana trees ................................................................................................................... 4 2.2.7 Tp-Trees.......................................................................................................................... 5 2.2.8 Product trees................................................................................................................... 5

2.3 APPROACH 2: EXHAUSTIVE LABELLING ............................................................................... 6 2.3.1 Graceful labelling algorithms......................................................................................... 7

2.3.1.1 Exhaustive labelling algorithms...........................................................................................7 2.3.1.2 Forward-thinking labelling algorithms ................................................................................7 2.3.1.3 Approximation labelling algorithms....................................................................................7

2.3.2 Tree construction ............................................................................................................ 8 2.3.2.1 Constructing all trees ...........................................................................................................8 2.3.2.2 Constructing random trees...................................................................................................8

2.4 RELATED PROBLEMS ............................................................................................................ 9 2.4.1 Ringels Conjecture ........................................................................................................ 9 2.4.2 Strong graceful labelling .............................................................................................. 10 2.4.3 Graceful graphs ............................................................................................................ 10 2.4.4 Harmonious graphs and trees....................................................................................... 11 2.4.5 Cordial graphs and trees .............................................................................................. 12

2.5 SUMMARY .......................................................................................................................... 13 3 METHODS ................................................................................................................................ 14

3.1 INTRODUCTION................................................................................................................... 14 3.2 DRAWING EVERY SIZE N TREE ............................................................................................ 14

3.2.1 Encoding the trees ........................................................................................................ 14 3.2.2 Generating the trees ..................................................................................................... 14

3.3 DRAWING RANDOM SIZE N TREES ....................................................................................... 15 3.3.1 Simple random tree construction.................................................................................. 16 3.3.2 Evenly distributed random tree construction................................................................ 16

3.3.2.1 Random rooted unlabelled trees ........................................................................................16 3.3.2.2 Random rootless unlabelled trees ......................................................................................16

4 THE EDGE SEARCH ALGORITHM.................................................................................... 18 4.1 INTRODUCTION................................................................................................................... 18 4.2 THE BASIC ALGORITHM (EDGESEARCHBASIC) ................................................................... 18 4.3 EXAMPLE ........................................................................................................................... 20

4.3.1 Correctness ................................................................................................................... 23 4.3.2 Termination .................................................................................................................. 23 4.3.3 Run-time analysis.......................................................................................................... 23

Graceful Trees: Statistics and Algorithms Michael Horton

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4.3.3.1 Theory ...............................................................................................................................23 4.3.3.2 Results ...............................................................................................................................25

4.4 EXTENSIONS....................................................................................................................... 28 4.4.1 Restarting after excess time .......................................................................................... 28 4.4.2 Restarting after excess failures (EdgeSearchRestart)................................................... 29 4.4.3 Identifying mirrored nodes (EdgeSearchRestartMirrors) ............................................ 32

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