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Photographs included in the original manuscript have been reproduced xerographicaliy in this copy. Higher quality 6" x 9* black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SOME TOPICS IN COMBINATORIAL DESIGN THEORY AND ALGEBRAIC GRAPH THEORY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Nick C. Fiala, M . S. ***** The Ohio State University 2002 Dissertation Committee: Approved by Dr. Akos Seress, Advisor < 4 * £ * « , Dr. Dijen Ray-Chaudhuri Advisor Dr. Surinder Sehgal Department of Mathematics Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 3049023 UMI’ UMI Microform 3049023 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor. Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT This dissertation consists of three chapters, each one based upon papers by the author. At the risk of being redundant, we have made each chapter self-contained for ease of reading. Chapter 1 is based upon two papers by the author and deals with a conjecture in combinatorial design theory and extremal set theory known as the A-design con­ jecture. A A-design on v points is a set of v subsets (blocks) of a u-element set (points) such that any two distinct blocks meet in exactly A points and not all of the blocks have the same size. Ryser’s and Woodall’s A-design conjecture states that all A-designs can be obtained from symmetric designs by a certain complementation pro­ cedure. In Chapter 1, we prove that the A-design conjecture is true when v = 6p + 1, where p is any prime number, and when v = 8q -F 1, where q = 1 or 7 (mod 8) is a prime. Chapter 2 deals with the determination of all strongly regular graphs with chro­ matic number equal to 5. A strongly regular graph is a finite simple regular graph, not complete or edgeless, such that the number of common neighbors of any two distinct vertices depends only upon whether or not they are adjacent. The chromatic number of a graph is the fewest number of colors that are required to color its ver­ tices such that adjacent vertices always receive different colors. Haemers determined ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all strongly regular graphs with chromatic number 3 or 4. In Chapter 2, we use eigenvalue techniques and computer enumerations to begin the determination of all strongly regular graphs with chromatic number 5. We show that there are at most 43 possible sets of parameters for such a graph. We deal completely with 32 of these sets and obtain a partial result for one additional set. Chapter 3 deals with a generalization of strongly regular graphs that the author calls strongly regular vertices and partially strongly regular graphs. A strongly regular vertex in a finite simple graph is such that the number of common neighbors it has with any other vertex depends only upon whether or not they are adjacent. A partially strongly regular graph is a regular graph with at least one strongly regular vertex. In Chapter 3, we prove some basic properties of strongly regular vertices and partially strongly regular graphs, provide several constructions of partially strongly regular graphs, determine all partially strongly regular graphs on at most 10 vertices, and make several conjectures regarding partially strongly regular graphs. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Akos Seress, for giving me the freedom to work on whatever I liked. I would like to thank Dr. Ronald Solomon for serving on my advisory committee, and Dr. Dijen Ray-Chaudhuri and Dr. Surinder Sehgal for serving on both my advisory and dissertation committees. I would like to thank Adam Wolfe for computer programming help, and Ted Spence for supplying me with combinatorial data. I would also like to thank my family and God. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA October 11, 1973 ...................................... Born in Oakes, North Dakota 1996 ......................................................... B. S. in Mathematics and Physics, minor in Economics, Rose-Hulman Institute of Technology, Indiana 2000 ......................................................... M. S. in Mathematics, The Ohio State University 1996 - Present ........................................ Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major field: Mathematics Specialization: Combinatorics Studies in Algebraic Graph Theory Dr. Akos Seress Design Theory and Coding Theory Dr. Dijen Ray-Chaudhuri Group Theory Dr. Surinder Sehgal Group Theory Dr. Ronald Solomon Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS A b s trac t ....................................................................................................................... ii Acknowledgments........................................................................................................ iv V i t a ............................................................................................................................. v List of Tables.............................................................................................................. viii CHAPTER PAGE 1 A-designs on 6p + 1 and 8p + 1 points ......................................................... 1 Introduction .............................................................................................. 1 Preliminary results ..................................................................................... 4 The lonin-Shrikhande method.................................................................. 5 A-designs with g = 6 ................................................................................. 10 A-designs with g = 8 ................................................................................. 23 Conjectures ................................................................................................. 36 2 5-chromatic strongly regular graphs ............................................................ 38 Introduction .............................................................................................. 38 Basic properties of strongly regular graphs ............................................ 40 Some families of strongly regular g ra p h s ............................................... 41 The Paley graphs ........................................................................................ 41 The triangular graphs .............................................................................. 41 The lattice graphs ........................................................................................ 42 Strongly regular graphs from quasi-symmetric designs ........................ 42 Strongly regular graphs from latin squares........................................... 43 Strongly regular graphs from systems of linked symmetric designs . 43 Strongly regular graphs from partial geometries.................................. 44 Strongly regular graphs from rank 3 permutation groups.................. 44 Matrix-theoretic to o ls .............................................................................. 45 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenvalues , cocliques, and chromatic number of strongly regular graphs 48 Parameters of 5-chromatic strongly regular graphs ............................ 50 The parameter sets 1, 2, 4, 5, 6 , 7, 8, 9, 10, 11, 14, 16, 19, 20, 21, 25, 28, 29, and 4 3 ........................................................................ 54