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Embedding of Hypercube Graphs on Orientable Surfaces Prateek Kunwar University of Maine, [email protected]

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Embedding of Hypercube Graphs on Orientable Surfaces Prateek Kunwar University of Maine, Prateek.Kunwar@Maine.Edu The University of Maine DigitalCommons@UMaine Electronic Theses and Dissertations Fogler Library 12-2016 Embedding of hypercube graphs on orientable surfaces Prateek Kunwar University of Maine, [email protected] Follow this and additional works at: http://digitalcommons.library.umaine.edu/etd Part of the Geometry and Topology Commons Recommended Citation Kunwar, Prateek, "Embedding of hypercube graphs on orientable surfaces" (2016). Electronic Theses and Dissertations. 2535. http://digitalcommons.library.umaine.edu/etd/2535 This Open-Access Thesis is brought to you for free and open access by DigitalCommons@UMaine. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of DigitalCommons@UMaine. EMBEDDING OF HYPERCUBE GRAPHS ON ORIENTABLE SURFACES By Prateek Kunwar B.E, Delhi College of Engineering, 2013 A THESIS Submitted in Partial Fulllment of the Requirements for the Degree of Master of Arts (in Mathematics) The Graduate School The University of Maine December 2016 Advisory Committee: Dr. Robert Franzosa, Professor, Advisor Dr. Andrew Knightly, Associate Professor Dr. Benjamin Weiss, Assistant Professor THESIS ACCEPTANCE STATEMENT On behalf of the Graduate Committee for Prateek Kunwar, I arm that this manuscript is the nal and accepted thesis. Signatures of all committee members are on le with the Graduate School at the University of Maine, 42 Stodder Hall, Orono, Maine. Dr. Robert Franzosa, Professor (Date) ii c All Rights Reserved iii LIBRARY RIGHTS STATEMENT In presenting this thesis in partial fulllment of the requirements for an advanced degree at The University of Maine, I agree that the Library shall make it freely available for inspection. I further agree that permission for fair use copying of this thesis for scholarly purposes may be granted by the Librarian. It is understood that any copying or publication of this thesis for nancial gain shall not be allowed without my written permission. Prateek Kunwar (Date) EMBEDDING OF HYPERCUBE GRAPHS ON ORIENTABLE SURFACES By Prateek Kunwar Thesis Advisor: Dr. Robert Franzosa An Abstract of the Thesis Presented in Partial Fulllment of the Requirements for the Degree of Master of Arts (in Mathematics) December 2016 The purpose of this thesis is to study hypercube graphs and their embeddings on orientable surfaces. We use rotation systems to represent these embeddings. We prove some results about the eect of adjacent switches in rotation system and create a rotation system called the ABC rotation system and prove general results about it. Using this rotation system, we give a general theorem about the minimal embedding of Qn. We also look at some interesting types of maximal embedding of Qn, such as the Eulerian walk embedding and the "big-face embedding". We prove a theorem that gives a recursively constructive way of obtaining the latter embedding in Qn ACKNOWLEDGEMENTS I would rstly like to thank the University of Maine and the department of Mathematics and Statistics for giving me an opportunity to pursue my passion in mathematics by oering me admission into the master's program. I am very grateful to my advisor Dr. Robert Franzosa for not only guiding me through the mathematics but also for having great patience in editing and going through my written work and giving me continuous feedback and encouragement. I would also like to thank the members of my committee for their valuable input and support. Lastly, I thank my parents, roommates, friends and the universe for being how they are. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS .......................................................... iv LIST OF FIGURES .................................................................. vii Chapter 1. INTRODUCTION ...............................................................1 1.1 History and Motivation....................................................1 1.2 Graph Theory ..............................................................3 1.3 Topology Background .....................................................8 2. GRAPH EMBEDDING INTO SURFACES.................................... 13 2.1 Graph embedding and Rotation systems ................................. 13 2.2 Boundary Walks ........................................................... 15 2.3 Maximal and Minimal genus of graph embeddings ...................... 21 2.4 Adjacent changes in a Rotation System .................................. 24 3. HYPERCUBE GRAPH EMBEDDINGS ON ORIENTABLE SURFACES.. 33 3.1 Introduction ................................................................ 33 3.2 Common Terminology ..................................................... 33 3.2.1 The "ABC" rotation system ...................................... 35 3.2.2 The embedding corresponding to the ABC rotation system .... 36 v 3.3 The Minimal Embedding of Qn ........................................... 39 3.3.1 Denition ........................................................... 39 3.3.2 Examples of minimal embedding ................................. 39 3.3.3 Minimal Embedding Theorem .................................... 42 3.4 Maximal Embedding....................................................... 43 3.4.1 Various maximal embeddings of Q3 .............................. 45 3.4.2 Maximal Eulerian Circuit Embedding............................ 48 3.4.3 Big-Face Maximal Embedding .................................... 51 3.4.4 The Big-Face Maximal Embedding Theorem ................... 53 4. FURTHER THOUGHTS ........................................................ 61 4.1 The spectrum of embeddings from minimal to maximal ................ 61 4.2 Some open questions....................................................... 63 REFERENCES ....................................................................... 65 BIOGRAPHY OF THE AUTHOR ................................................. 66 vi LIST OF FIGURES Figure 1.1 The Konigsberg Bridges ...............................................1 Figure 1.2 The utility problem....................................................2 Figure 1.3 The bi - partite graph K3;3 ...........................................2 Figure 1.4 The utilities layout on a torus ........................................3 Figure 1.5 The complete graph K5 with labeled edges and vertex .............4 Figure 1.6 A Graph showing the degree of it's vertices .........................4 Figure 1.7 Examples of a regular graph and a non-regular graph ..............5 Figure 1.8 Example of a simple graph............................................5 Figure 1.9 A graph that is not simple ............................................5 Figure 1.10 Examples of Paths and a Cycle.......................................6 Figure 1.11 Example of an Eulerian Path .........................................6 Figure 1.12 The Konigsburg Bridges Problem modeled by a graph .............7 Figure 1.13 Hypercube graphs for n = 1; 2; 3......................................8 Figure 1.14 Example of an embedding of the cube on a sphere..................9 Figure 1.15 A two hole torus .......................................................9 Figure 1.16 Folding a square into a torus ......................................... 10 Figure 1.17 Folding an octagon into a two hole torus ............................ 10 Figure 1.18 Example for orientation preserving gluing ........................... 11 Figure 1.19 A convex icosahedron ................................................. 12 vii Figure 2.1 Embedding of K5 on a Torus ......................................... 13 Figure 2.2 Q3 on a sphere : Front and Rear view ............................... 14 Figure 2.3 Boundary Walk ........................................................ 17 Figure 2.4 The boundary walks ................................................... 18 Figure 2.5 Top and bottom view of embedding of Q3 on a torus............... 18 Figure 2.6 Mapping of regions on Q3 ............................................. 19 Figure 2.7 20-gon corresponding to the walk from the modied rotation system .................................................................. 22 Figure 2.8 Top and bottom view of Q3 in a 2-hole torus after 1 switch ....... 23 Figure 2.9 The conguration prior to the switch of edges and polygons ....... 26 Figure 2.10 The polygons P1;P2 and ;P3 and their boundary walks ............ 27 Figure 2.11 Parts of the polygon(s) after switching 01 and 02................... 28 Figure 2.12 3 faces have combined into 1.......................................... 28 Figure 2.13 Case 2: Initial conguration .......................................... 29 Figure 2.14 Case 2: The boundary walk after switching edges 01 and 02....... 29 Figure 2.15 Case 2: The boundary walk before switching edges 01 and 02 ..... 30 Figure 2.16 Case 2: The boundary walk after switching edges 01 and 02....... 31 Figure 3.1 Vertices connected to 7 in Q3 ......................................... 34 Figure 3.2 Polygons corresponding to the ABC rotation system in Q3 ........ 36 Figure 3.3 Top and bottom view of the embedding of Q3 with the ABC rotation system ........................................................ 37 viii Figure 3.4 Top and bottom views of one of the faces in the embedding of Q4 with the ABC rotation system ................................. 38 Figure 3.5 ABC Proof sketch ..................................................... 39 Figure 3.6 Faces in the minimal embedding of Q3 ............................... 40 Figure 3.7 Front and rear view of the minimal embedding of Q3 on a sphere . 40 Figure 3.8 Lattice
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