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Perfect Matchings: Modified Aztec Diamonds, Covering Graphs Andn-Matchings

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Perfect Matchings: Modified Aztec Diamonds, Covering Graphs Andn-Matchings University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Fall 1997 Perfect matchings: Modified Aztec diamonds, covering graphs andn-matchings Adriana Badauta Cransac University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Cransac, Adriana Badauta, "Perfect matchings: Modified Aztec diamonds, covering graphs andn- matchings" (1997). Doctoral Dissertations. 1969. https://scholars.unh.edu/dissertation/1969 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. 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PERFECT MATCHINGS: MODIFIED AZTEC DIAMONDS, COVERING GRAPHS AND n-MATCHINGS BY Adriana Badauta Cransac B.S., University of Bucharest (1987) M.S., University of Bucharest (1988) DISSERTATION Submitted to the University of New Hampshire in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics September 1997 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9807555 Copyright 1997 by Cransac, Adriana Badauta All rights reserved. UMI Microform 9807555 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ALL RIGHTS RESERVED ©1997 Adriana Badauta Cransac ^ ____________ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation has been examined and approved. Director, David V. Feldman Associate Professor of Mathematics Donovan H. Van Osdol Professor of Mathematics Kenneth I. Appel y J Professor of Mathematics Eric A. Nordgren Professor of Mathematics Edward K. Hinson Associate Professor of Mathematics J-frn > , m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Please Note Page(s) missing in number only; text follows. Filmed as received. iv UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DEDICATION To my parents and my grandmother, to Didier, to aH my mathematics teachers. v I __ .. _____ ___________ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Please Note Page(s) missing in number only; text follows. Filmed as received. vi UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I want to show my appreciation to all the people that offered their help, moral support and friendship during my studies at UNH. I would like to express my warm gratitude to Professor David Feldman, for being an excellent advisor and teacher, for his constructive help during this research and also for being available, patient and friendly. Many thanks to the Committee members for proofreading my thesis. I wish to thank Professor Ken Appel for his concern, advice and encouragement during all my stay at UNH I am obliged to Professor Edward Hinson, not only as a teacher and member of the Committee, but also as the person who solved the ’’administrative” problems caused by my absence. Thanks to Professor Don van Osdol for his most instructive and enjoyable reading classes. Many, many thanks to Ioana and Bogdan Mihaila, for being ready to help at any time and for all the good advice (including the choice of University). To all my family in Philadelphia, thank you from my heart. My love goes to my parents who, through many years of care prepared me for this work, and kept supporting me even from fax away. Finally, to my husband Didier, for his patience, permanent availability and love. May we “discover new worlds... ” together. v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS DEDICATION ................................................................................................................ v ACKNOWLEDGMENTS............................................................................................ vii LIST OF TABLES ...................................................................................................... x LIST OF FIGURES ...................................................................................................... xiii ABSTRACT ................................................................................................................... xiv CHAPTER PAGE INTRODUCTION ......................................................................................................... I 0.1 Graph Theory ........................................................................................... 2 0.2 Topology and Topological Graph Theory ............................................ 7 I. STATISTICS FOR RANDOM MATCHINGS OF MODIFIED AZTEC DIAMONDS ......................................................................................................... 9 1.1 The Initial Problem ................................................................................. 9 1.2 The Dual Problem .................................................................................... 16 1.3 Modifications of the Problem ................................................................ 21 1.4 Cylindrical Aztec Diamonds .................................................................... 28 1.5 Mobius Aztec Diamonds .......................................................................... 31 1.6 Klein Aztec Diamonds .............................................................................. 36 1.7 Projective Aztec Diamonds .................................................................... 41 1.8 Torus Aztec Diamonds .............................................................................. 45 viii f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.9 Conclusions .................................................................................................. 48 II. MATCHINGS AND COVERING GRAPHS ................................................... 51 2.1 Perfect Matchings, 2-Matchings and n-M atchings ............................. 51 2.2 Structure and Decomposition of Non-Ramified Perfect n-Matchings 55 2.3 Covering Graphs and Perfect n-Matchings ........................................... 58 2.3.1 Branched Covering G r a p h s ........................................................ 59 2.3.2 Permutation Derived G raphs ..................................................... 61 2.4 Perfect 2-Matchings ................................................................................. 66 2.4.1 Matchings of G2 ............................................................................ 67 2.4.2 A Mean Value Theorem for Perfect Matchings of 2-Covering Graphs ......................................................................... 68 2.5 Conclusions and C o m m en ts ..................................................................... 73 APPENDIX A ................................................................................................................ 81 APPENDIX B ................................................................................................................... 100 LIST OF REFERENCES .............................................................................................. 112 ix I _ ______ __ ______ Reproduced with permission
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