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MAXIMALLY KNOTLESS GRAPHS a Thesis Presented to the Faculty Of MAXIMALLY KNOTLESS GRAPHS A Thesis Presented to the Faculty of California State University, Chico In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics Education by Lindsay Eakins Summer 2020 MAXIMALLY KNOTLESS GRAPHS A Thesis by Lindsay Eakins Summer 2020 APPROVED BY THE GRADUATE ADVISORY COMMITTEE: Pamela Morrel Thomas Mattman, Ph.D., Chair Graduate Coordinator Jorgen Berglund, Ph.D., Committee PUBLICATION RIGHTS No portion of this thesis may be reprinted or reproduced in any manner unacceptable to the usual copyright restrictions without the written permission of the author. iii DEDICATION I would like to dedicate this thesis to my mother, Kathi, my brother, Nolan, and my loving husband, Loren, who have always believed in me, even when I did not believe in myself. Without your love and encouragement I would not be where I am today. iv ACKNOWLEDGEMENTS First and foremost, I would like to acknowledge my Thesis Chair, Thomas Mattman, who has been extrememly supportive and encouraging over these past few months while working on this thesis. I could not have accomplished this without your guidance and timely feedback. I truly appreciate your dedication and efforts in helping me complete my work. Second, I would like to acknowledge my Thesis Committee Member, Jorgen Berglund, who has been a guiding light throughout my student career at California State University, Chico beginning in 2009. You have pushed me to think outside the box and to always strive for greatness. My proofs would be less than stellar without your guidance. Lastly, I would very much like to acknowledge my Masters Cohort (2018-2020). I have made many friends who have, in time, become family. Through our shared learning efforts, group study sessions that would run late into the nights, lots of delicious shared dinners with your families (can’t forget about that hiccup cheese!), the laughs, the tears and kayaking trips to relax before tests, I could not have made it this far without each and every one of you: Robert Bond, Yara Fuentes, Kelley Jardin, Sandra Jauregui, Teresa Mercado and Gary Pope. I cherish the memories these last two years have given us and look forward to continuing to watch us all grow as educators. v TABLE OF CONTENTS PAGE Publication Rights ....................................................................................................... iii Dedication ................................................................................................................... iv Acknowledgments....................................................................................................... v List of Tables .............................................................................................................. viii List of Figures ............................................................................................................. ix Abstract ....................................................................................................................... xii CHAPTER I. Introduction ............................................................................................... 1 The Graph 퐸9 ................................................................................ 1 Graph Embeddings and Knots ...................................................... 1 Maximally Knotless Graphs ......................................................... 2 Main Theorems 1-11 ..................................................................... 3 Classification of Maximally Knotless Graphs Through Nine Vertices and Twenty Edges ........................................ 3 Number of Edges in Maximally Knotless Graphs ........................ 4 II. Literature Review...................................................................................... 5 Introduction ................................................................................... 5 Graphs and Degree of a Vertex ..................................................... 5 Minimum Degree of a Graph ........................................................ 5 Complete Graphs .......................................................................... 5 Complement of a Graph ................................................................ 5 Handshake Lemma........................................................................ 6 Planarity ........................................................................................ 6 Isomorphic Graphs ........................................................................ 7 Apex Graphs ................................................................................. 8 Paths and Cycles ........................................................................... 9 Connected Graphs and Components ............................................. 9 Cycles and Linking ....................................................................... 10 Linkless and Intrinsically Linked Graphs ..................................... 10 푌 − Δ Transformations ................................................................. 10 The Petersen Family Graphs ......................................................... 11 Knots, Reidemeister Moves and Equivalent Knots ...................... 12 Knotless, Intrinsically Knotted and Maximally Knotless Graphs 14 vi CHAPTER PAGE II. Literature Review Continued .................................................................... Edge Contraction .......................................................................... 14 Minor of a Graph .......................................................................... 15 The Heawood Family Graphs ....................................................... 15 The Knot Sum Theorem................................................................ 16 III. Methodology ............................................................................................. 17 Introduction ................................................................................... 17 I. Preliminary Theorems ............................................................... 17 Theorem 1 ..................................................................................... 17 Theorem 2 ..................................................................................... 18 Theorem 3 ..................................................................................... 18 II. Classification of Maximally Knotless Graphs Through Nine Vertices and Twenty Edges ........................................ 19 Theorem 4 ..................................................................................... 19 Theorem 5 ..................................................................................... 19 Theorem 6 ..................................................................................... 20 The Complement of 퐺(9,29) ........................................................... 21 Theorem 7 ..................................................................................... 22 Lemma 1 ....................................................................................... 24 The Complement of 퐸9 ................................................................. 27 Theorem 8 ..................................................................................... 28 III. Number of Edges in a Maximally Knotless Graph ................. 29 Theorem 9 ..................................................................................... 29 Theorem 10 ................................................................................... 32 Theorem 11 ................................................................................... 34 IV. Conclusions and Further Research ............................................................ 36 Introduction ................................................................................... 36 References ..................................................................................................................... 37 Appendices A. Exploring Two Consecutive Non-Alternating Crossings ......................... 39 B. Exploring Triples ...................................................................................... 74 vii LIST OF TABLES TABLE PAGE 1. Table 2.1: Isomorphic Graphs ...................................................................... 7 2. Table 3.1: The Exploration of Crossings 1&2 ............................................. 25 3. Table 3.2: The Exploration of Triple {1, 4, 5} ............................................. 26 viii LIST OF FIGURES FIGURE PAGE 1. Figure 2.1: Complement of a Graph ............................................................ 5 2. Figure 2.2: Complete Graphs 퐾1 Through 퐾4 .............................................. 6 3. Figure 2.3: Exploring Crossings in 퐾5 ......................................................... 6 4. Figure 2.4: Exploring Crossings in 퐾3,3 ....................................................... 7 5. Figure 2.5: Apex and 2-Apex Graphs .......................................................... 8 6. Figure 2.6: Path and Cycle in a Graph ......................................................... 9 7. Figure 2.7: Connected and Not Connected Graphs .................................... 9 8. Figure 2.8: Cycles and Linking .................................................................... 10 9. Figure 2.9: 푌 − Δ Transformations .............................................................. 11 10. Figure 2.10: The Petersen Family Graphs ................................................... 12 11. Figure 2.11:
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