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Colorings of Hamming-Distance Graphs University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2017 Colorings of Hamming-Distance Graphs Isaiah H. Harney University of Kentucky, [email protected] Digital Object Identifier: https://doi.org/10.13023/ETD.2017.194 Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Harney, Isaiah H., "Colorings of Hamming-Distance Graphs" (2017). Theses and Dissertations-- Mathematics. 49. https://uknowledge.uky.edu/math_etds/49 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Isaiah H. Harney, Student Dr. Heide Gluesing-Luerssen, Major Professor Dr. Peter Hislop, Director of Graduate Studies Colorings of Hamming-Distance Graphs DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Isaiah Harney Lexington, Kentucky Director: Dr. Heide Gluesing-Luerssen, Professor of Mathematics Lexington, Kentucky 2017 Copyright c Isaiah Harney 2017 ABSTRACT OF DISSERTATION Colorings of Hamming-Distance Graphs Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain param- eters. Additionally, explorations are made into subgraph structures whose identification may be useful in determining the chromatic number. KEYWORDS: Hamming distance, graphs, coloring, q-ary block codes Author's signature: Isaiah Harney Date: April 28, 2017 Colorings of Hamming-Distance Graphs By Isaiah Harney Director of Dissertation: Heide Gluesing-Luerssen Director of Graduate Studies: Peter Hislop Date: April 28, 2017 ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Dr. Heide Gluesing-Luerssen, for her invaluable guidance and enduring patience with my wandering mind. You have continually gone above and beyond the call of duty as an advisor and I cannot express my appreciation strongly enough. I would also like to thank all the members of my committee for sacrificing their time and energy in reviewing this thesis, and for the valuable feedback they provided. I especially want to thank Dr. Ben Braun for serving as my master's advisor and being a source of wise advice and feedback throughout my graduate studies. Special thanks is due to Sheri Rhine, without whom I would still be lost in a mire of academic bureaucracy. Your kindness and unwavering willingness to help continue to make the math department a brighter place. Thank you to my parents without whose encouragement, love, and support I could not be where or who I am today. Thank you for always believing in me. Above all, thank you to my wonderful wife Anne. It is hard to imagine having made it through this process without you. Thank you for all the sacrifices you have made to provide me with this opportunity. As in every area of my life, your love and support have made all the difference. iii To all the nights I didn't sleep. And all the dreams that still came true. TABLE OF CONTENTS Acknowledgments . iii Table of Contents . iv List of Figures . vi List of Tables . vii Chapter 1 Introduction . 1 1.1 Error-Correcting Codes . 1 1.2 Hamming-Distance Graphs . 1 1.3 Outline . 2 Chapter 2 Basic Properties of Hamming Graphs . 4 2.1 Notation . 4 2.2 Descriptions of Hamming-distance Graphs. 4 2.3 Girth and Diameter . 6 2.4 Hamiltonicity of Hq(n; d)............................ 10 Chapter 3 On the Chromatic Number of the Hamming Graphs . 17 3.1 Lower Bounds . 17 3.2 An Upper Bound from Coordinate Colorings . 19 3.3 Known Values of χ(Hq(n; d)) . 20 3.4 Other relationships . 21 Chapter 4 Colorings of H (n; n 1) . 23 2 − 4.1 Minimal Colorings of H2(n; n 1) . 23 4.2 Maximally Robust 4-colorings− of H (n; n 1) . 34 2 − Chapter 5 Fractional Chromatic Number of the Hamming Graphs . 41 5.1 Definitions and Known Values . 41 5.2 Gap between χf (H2(n; d)) and χ(H2(n; d)) . 43 5.3 Expressing Aq(n; d) in terms of fractional colorings. 46 Chapter 6 Implicit Edges and Entanglements . 48 6.1 Implicit Edges . 49 6.2 Entanglements . 57 6.3 Results and Applications . 62 Chapter 7 Open Problems . 67 7.1 Concerning Hamming Graphs . 67 iv 7.2 Concerning Implicit Edges and Entanglements . 68 Chapter A The Mycielski Construction . 69 Bibliography . 71 Vita . 73 v LIST OF FIGURES 2.1 H2(3; 2) ....................................... 5 4.1 H2(3; 2) ....................................... 24 4.2 (2; 3)-coordinate coloring with transition edges . 25 4.3 Coloring K6 with transition edges . 25 4.4 H2(4; 3) ....................................... 26 4.5 (1; 2)-coordinate coloring of H2(4; 3) with transition edges . 27 6.1 Implicit edge K4 .................................. 50 6.2 Cani(K)....................................... 51 i 6.3 Expk(Can (K))................................... 51 6.4 K3-free implicit edge of order 3 . 54 6.5 Proper 3-coloring of Figure 6.4 . 54 6.6 Failed 3-coloring of Figure 6.4 . 54 A.1 Mycielski Construction . 69 A.2 Mycielski graph of orders 2, 3, and 4 . 69 vi LIST OF TABLES 2.1 (2,3)-Gray code . 14 4.1 All 4-colorings of H2(3; 2) ............................. 25 4.2 Noncoordinate Colorings of H2(4; 3) . 26 vii Chapter 1 Introduction 1.1 Error-Correcting Codes The study of error-correcting codes is concerned with the transmission of information via \noisy" channels in which the original message may be corrupted as it travels from sender to receiver. It is often infeasible or inefficient to simply resend the message several times. For instance, consider the cases of digital broadcasting or cell phone communication. By encoding messages before transmission, it is possible to recover the intended message even if \noise" on the channel results in an erroneous transmission. Finding the most efficient methods of transmitting data while simultaneously allowing for the correction of errors plays a central role in information theory. A code is simply a collection of known messages, called codewords. If we are wise about how weC choose our code, it is possible to recover the original message if the size of the error in the transmission is not too large. A common technique is to choose a code such that all the codewords are very dissimilar from each other. The \dissimilarity" of the codewords is measured by a metric called the Hamming distance which simply counts the number of coordinates in which two codewords disagree. If all the codewords in a code have a large Hamming distance from one another, it is usually possible to determine what the intended message was as there will be a unique codeword that is closest to the received message. A code generally has four important parameters, n; d; q; and M. An (n; M; d)q-code has M codewords of length n over an alphabet of size q such that any two codewords have Hamming distance at least d. The rate of transmission increases with the size M of the code so coding theorist are concerned with finding large codes of given parameters (n; d; q). However, finding the maximal size of a code with given parameters is prohibitively difficult and energy has instead been focused of finding good bounds. 1.2 Hamming-Distance Graphs The problem of finding large error-correcting codes can be given a graphical interpretation n via the Hamming-distance graphs. Hq(n; d) is defined as the graph with vertex set Zq where two vertices are adjacent if the Hamming distance between them is at least d.
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