Hamilton decompositions of 6-regular abelian Cayley graphs

122 Pages·2016·1.27 MB·English

by  Erik E. Westlund

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Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's
Reports - Open Reports
2010
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Erik E. Westlund
Michigan Technological University
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Copyright 2010 Erik E. Westlund
RReeccoommmmeennddeedd CCiittaattiioonn
Westlund, Erik E., "Hamilton decompositions of 6-regular abelian Cayley graphs", Dissertation, Michigan
Technological University, 2010.
https://doi.org/10.37099/mtu.dc.etds/206
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Part of the Mathematics Commons
HAMILTON DECOMPOSITIONS OF 6-REGULAR ABELIAN CAYLEY GRAPHS
By
ERIK E. WESTLUND
A DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
(Mathematical Sciences)
MICHIGAN TECHNOLOGICAL UNIVERSITY
2010
(cid:13)c 2010 Erik E. Westlund

This dissertation, “Hamilton Decompositions of 6-Regular Abelian Cayley Graphs”, is hereby ap-
proved in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in
the field of Mathematical Sciences.
DEPARTMENT:
Mathematical Sciences
Signatures:
Dissertation Advisor
Donald L. Kreher, Ph.D.
Department Chair
Mark S.Gockenbach, Ph.D.
Date

To my family.

Contents
List of Figures xii
List of Tables xiii
Acknowledgments xv
Abstract xvii
1 Cayley Graphs and Hamilton Cycles 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Hamilton Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Hamilton decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Lovász Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Alspach Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Pseudo-Cartesian Products 11
2.1 The Pseudo-Cartesian Product of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Edge Color-Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Hamilton Decompositions for Graphs of Odd Order 19
3.1 Lifting to the 6-Regular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Decompositions for Odd Order Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 22
vii
4 A Decomposition for Non-Minimal Connection Sets 27
4.1 Using a Subgroup of Index 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Hamilton Decompositions Using Quotient Graphs 37
5.1 Preliminaries and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Decomposing Layered Pseudo-Cartesian Products . . . . . . . . . . . . . . . . . . . . 39
5.3 Decompositions for Low-Order Quotient Graphs . . . . . . . . . . . . . . . . . . . . 49
5.3.1 Odd-order quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3.2 Even-order quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Conclusions and Further Research Problems 65
6.1 Open cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.1 Quotient connection sets with involutions . . . . . . . . . . . . . . . . . . . . 66
6.1.2 General connection sets with involutions . . . . . . . . . . . . . . . . . . . . . 66
6.2 Fundamental Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A Data 69
1.1 Abelian Groups of Order 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1.2 Abelian Groups of Order 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1.3 Abelian Groups of Order 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.4 Abelian Groups of Order 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B Source code 81
2.1 MAGMA code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.2 C code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.2.1 Constructing the Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.2.2 Hamilton cycles via a randomized greedy algorithm . . . . . . . . . . . . . . 85
2.2.3 Obtaining Hamilton decompositions . . . . . . . . . . . . . . . . . . . . . . . 87
2.2.4 Outputting to LATEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
viii
2.3 Shell Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.4 Mathematica Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography 99
Index 103
ix