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Western Michigan University ScholarWorks at WMU Dissertations Graduate College 1-2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle C. Kolasinski Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Kolasinski, Kyle C., "Hamiltonicity and Connectivity in Distance-Colored Graphs" (2011). Dissertations. 431. https://scholarworks.wmich.edu/dissertations/431 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. Hamiltonicity and Connectivity in Distance-Colored Graphs by Kyle C. Kolasinski A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment for the requirement for the Degree of Doctor of Philosophy Department of Mathematics Advisor: Ping Zhang, Ph.D. Western Michigan University Kalamazoo, Michigan June 2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle Kolasinski, Ph.D. Western Michigan University, 2011 For a connected graph G and a positive integer fc, the fcth power Gk of G is the graph with V(Gk) — V(G) where uv E E(Gk) if the distance dc{u^v) between u and v is at most fc. The edge coloring of Gk defined by assigning each edge uv of Gk the color da(u,v) produces an edge-colored graph Gk called a distance-colored graph. A distance-colored graph Gk is Hamiltonian-colored if Gk contains a properly colored Hamiltonian cycle. The minimum k for which Gk is Hamiltonian-colored is the Hamiltonian coloring exponent hce(G). It is shown that for each pair fc, d of integers with 4 < k < d, there exists a tree T with hce(T) = k and diam(T) = d. Hamiltonian coloring exponents are determined for several well-known classes of graphs. It is also shown that for each integer k > 2, there exists a tree T/~ with hce(T/c) = k such that every properly colored Hamiltonian cycle in the fcth power of T/c must use all colors 1, 2,..., fc. For a grid G — Pn • Pm with n^m > 2, it is shown that G2 is Hamiltonian-colored if and only if nm = 0 (mod 4) and a complete solution is presented for restricted Hamiltonian-colored cycles in G3. A distance-colored graph is properly p-connected if two distinct vertices u and v in the graph are connected by p internally disjoint properly colored u — v paths. For a connected graph G and an integer fc > 2, the color-connectivity of Gk is the maximum positive integer p for which Gk is properly p-connected. The color- connectivities are determined for some well-known classes of graphs. It is shown that G2 is properly 2-connected for every 2-connected graph that is not complete, a double star is the only tree T for which T2 is properly 2-connected and G3 is properly 2-connected for every connected graph G of diameter at least 3. All pairs fc, n of positive integers for which Pk is properly fc-connected are determined and other related results are presented. Various color-related distance parameters and rainbow concepts in distance- colored graphs are introduced and studied. Results and open problems are also presented in this area of research. UMI Number: 3467827 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Publishing UMI 3467827 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ©2011 Kyle Kolasinski ACKNOWLEDGMENTS First and foremost, I would like to express my gratitude towards Dr. Ping Zhang. Without her constant guidance and inspiration, I would have been lost. I truly appreciate the kind of person that she is, and feel honored to have her as my academic mother. I would next like to thank the other members of my dissertation committee. I always received constructive feedback and support on my thesis and the members of my committee always helped me along the way. I was privileged to work under such brilliant minds who have really given so much to the world of mathematics. I would specifically like to thank Dr. Gary Chartrand. He always went the extra mile for me and I am very grateful for his efforts. I also would like to thank the Mathematics Department at Western Michigan University. Whether I wanted to attend a conference, or needed summer support, they were always there for me. It was nice to know that I had the support and encouragement from many staff and faculty members who wished for my success. Another person who deserves recognition is Dr. Ryan Jones. We went through all of graduate school together and having him as a friend always seemed to make things better. Special thanks go to my family and friends, who had to deal with the constant rambling of my dissertation. Specifically, I would like to thank my parents, Steve and Laurie, my brothers, Steve and Jake, and my sisters, Karissa and Jessica for ii Acknowledgments-Continued their continuous moral support. I could not have done this without your encour­ agement. I know it was a long and boring ride from Howell, but I appreciate how many times you drove it just to see me. Lastly, I would like to thank my fiance, Jessica. She has been my rock through­ out my entire college experience, and I really could not have done this without her. She was always the one who told me to keep going, and knowing that she believed in me made all the difference in the world. She really is a special person. Kyle Kolasinski in TABLE OF CONTENTS ACKNOWLEDGMENTS ii LIST OF FIGURES vi CHAPTER 1. Introduction 1 1.1 History and Motivation 1 1.2 Definitions and Notation 8 2. Color-Hamiltonicity 12 2.1 Distance-Colored Graphs 12 2.2 Hamiltonian Coloring Exponents of Graphs 21 2.3 Graphs with Hamiltonian Coloring Exponent 2 26 2.4 Hamiltonian Coloring Exponents of Trees 29 2.5 Some Well-Known Classes of Trees 32 3. Colored Bridge Problems 49 3.1 The Konigsberg Bridge Problem 49 3.2 A Colored Bridges Problem 52 3.3 The Colored Bridges Problem for Grids 55 3.4 Restricted Colored Bridge Problems 59 iv Table of Contents-Continued CHAPTER 4. Color Distance and Connectivity 82 4.1 Color-Connection Exponent and Color-Distance 82 4.2 Color-Eccentricity, Color-Radius and Color-Diameter 88 4.3 Color-Center and Color-Periphery 97 4.4 Color Connectivities of Graphs 103 4.5 On //-Colored and //-Chromatic Graphs 123 5. Rainbow Connected Graphs 128 5.1 Introduction 128 5.2 Rainbow Pancyclic Graphs 135 5.3 Rainbow Connectivity 142 6. Topics for Further Study 150 6.1 Questions on Color-Hamiltonicity 150 6.2 Questions on Color-Distance 152 6.3 Questions on Rainbow Subgraphs 153 6.4 Questions on Rainbow Connectivity 154 BIBLIOGRAPHY 156 v LIST OF FIGURES 1 1 A connected graph whose square is not Hamiltonian 2 1 A triangle in K7 2 2 A 3-edge coloring of K7 2 3 A properly colored Hamiltonian cycle m K7 2 4 Properly colored Hamiltonian cycles in C| and C% 2 5 A properly colored Hamiltonian cycle in P2 2 6 A connected grapg G of order 6 and A(G) = 4 such that G3 contains a properly colored Hamiltonian cycle 2 7 Properly colored Hamiltonian cycles in P^ for 4 < n < 6 2 8 Properly colored Hamiltonian cycles in P3 for 7 < r? < 16 2 9 A step in the proof of Theorem 2 13 3 2 10 Hamiltonian r2 — rn paths in P for 3 < r? < 8 2 11 Illustrating the graphs in the proof of Theorem 2 2 2 for k=A and k=5 2 12 Labeling the vertices of G = Spq 3 2 13 A colored Hamiltonian cycle in S(K\ 4) 2 14 Illustrating the proof of Proposition 2 5 3 2 15 Properly colored Hamiltonian cycles in T$ for 4 < d < 11 2 16 A step in the proof of Theorem 2 5 4 2 17 A step in the proof of Theorem 2 5 4 2 18 The graphs Gk for k E {4,5, 6} in the proof of Proposition 2 VI List of Figures-Continued 2.19 Illustrating the properly colored Hamiltonian cycles of Tk(4 < k < 7) in the proof of Theorem 2.5.8 46 3.1 A map of Konigsberg 50 3.2 a multigraph representing Konigsberg 51 3.3 A portion of a town 53 3.4 The graph representation of the town in Figure 3.3 55 3.5 A properly colored Hamiltonian cycle in the graph Gs of Figure 3.4(c) 55 3.6 Properly colored Hamiltonian cycle in the cubes of PnDPm when n, m E {2, 3, 5, 6} 57 3.7 Properly colored Hamiltonian cycles in the distance-colored graph P3 for n E {4, 7,8, 9,10} 58 3.8 A properly colored Hamiltonian cycle in 3 the distance-colored graph (P7DP6) 59 3.9 A step in the proof of Theorem 3.4.1 61 3.10 Properly colored Hamiltonian cycles in the squares of PnDP4 and PnOP$ for n E {3,9} 65 3.11 Properly colored Hamiltonian paths in the squares of PnDP4 for n E {4,6, 8,10} 66 3.12 Properly colored Hamiltonian cycles in the squares of PnDP6 for n E {4,10} 67 3.13 The graph P2DP7 68 3.14 The situation where C contains the edge x\x± 68 vn List of Figures-Continued 3.15 The situation where C contains the edges #52/7, zij/3,2/12/3 and yby7 69 3.16 The graph P2DP5 70 3 3.17 Properly colored Hamiltonian cycles in (P2DPm) for m E {4, 5, 6,11} containing v^mv2,m, vi,m-2Vi,m-i and ?;2,m_2?;2,m_i 70 3 3.18 Constructing a properly colored Hamiltonian cycle in (P2DPm+4) dotfill 71 3.19 The vertices

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