Western Michigan University ScholarWorks at WMU Dissertations Graduate College 6-2000 Rainbow Ramsey Numbers Linda Eroh Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Statistics and Probability Commons Recommended Citation Eroh, Linda, "Rainbow Ramsey Numbers" (2000). Dissertations. 1448. https://scholarworks.wmich.edu/dissertations/1448 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]. RAINBOW RAMSEY NUMBERS by Linda Eroh A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of requirements for the Degree of Doctor of Philosophy Department of Mathematics and Statistics Western Michigan University Kalamazoo, Michigan June 2000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RAINBOW RAMSEY NUMBERS Linda Eroh, Ph.D. Western Michigan University, 2000 We investigate a new generalization of the generalized ramsey number for graphs. Recall that the generalized ramsey number for graphs Gi,G 2,--- , Gc is the minimum positive integer N such that any coloring of the edges of the complete graph K n with c colors must contain a subgraph isomorphic to G, in color i for some i. Bialostocki and Voxman defined RM (G) for a graph G to be the minimum N such that any edge-coloring of K n with any number of colors must contain a subgraph isomorphic to G in which either every edge is the same color (a monochromatic G) or every edge is a different color (a rainbow G). This number exists if and only if G is acyclic. Expanding on this definition, we define the rainbow ramsey number RR(G j, Gi) of graphs Gi and G'2 to be the minimum N such that any edge-coloring of Kn with any number of colors contains either a monochromatic G 1 or a rainbow G 2. This number exists if and only if G x is a star or G2 is acyclic. We present upper and lower bounds for RR(KUn, Km), RR(Kn,T m), RR(Kn, RR(Ki,n,mK2) and R R (n K 2, where Tm is an arbitrary tree of order m. We also define the edge-chromatic ramsey number C R (G i,G 2 ) to be the minim um N such that any edge-coloring of K n must contain either a monochro­ matic Gi or a properly edge-colored G2. When both are defined, C/?(Gi,G2) < RR{G u G2). We consider bounds for GR(G„,Pm), CR{KUn,Pm), CR{Pn,P m), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the corresponding rainbow ramsey numbers. These two new ramsey numbers can be further generalized as the JF-free ramsey number. For a set of graphs T, an ^-free coloring of a graph G is a coloring so that G does not contain any monochromatic subgraph isomorphic to any graph in J-. The T-free ramsey number of graphs Gi and G i, denoted Rj:{G\.Gi). is the minimum N such that every edge-coloring of Kw contains either a monochromatic copy of Gi or an JF-free copy of GV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. 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Further reproduction prohibited without permission. 0 2000 Linda Eroh Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would particularly like to thank my advisor, Professor Allen Schwenk, for the guidance, advice, and encouragement he has given me, not only as my advisor the last two years, but during my entire time as a student at Western Michigan University. I would also especially like to thank Professor Gary Chartrand, who helped kindle my love for mathematics and continues to show an interest in my mathematical development. Professor Ortrud Oellermann has been remarkably kind and helpful as an outside reader, even when I sent copies of my dissertation by regular mail to Canada and they took two weeks to arrive, leaving her little time to review them. In addition, I would like to acknowledge the other members of my dissertation committee, Professor Art White, Professor Yousef Alavi, and especially Professor Ping Zhang, my second reader. As both an undergraduate and graduate student, I received a great deal of encouragement and support from Professor John Petro. And I have received valuable guidance in my teaching and career advice from Professor Nil Mackey. Fi­ nally, I would certainly not have met deadlines or Graduate College requirements without the help of Margo Chapman, Administrative Secretary for Graduate Pro­ grams, and Maryann Bovo. Thank you also to my husband, Brian, and my parents, Bob and Lois Hansen, who have not always understood why anyone would want a Ph.D. in mathematics but have never failed to be proud and supportive. Linda Eroh iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGEMENTS ....................................................................................... iii LIST OF FIGURES ...................................................................................................... vi INTRODUCTION ......................................................................................................... 1 1.1 Background and Basic Definitions .................................................... 1 1.2 Some Traditional R esults ..................................................................... 5 1.3 Where the Rainbow Begins .................................................................. 8 RAINBOW RAMSEY NUM BER ............................................................................. 16 2.1 Existence of Rainbow Ramsey N u m b e rs ......................................... 16 2 .2 Lower Bounds .......................................................................................... 24 2.3 S t a r s ........................................................................................................... 26 2.4 Rainbow Ramsey Numbers and Matchings .................................. 27 2.5 Matchings and Stars ............................................................................... 45 GENERALIZATIONS OF THE RAINBOW RAMSEY NUMBER 53 3.1 Edge-Chromatic Ramsey N um ber ................................................... 53 3.2 Bounds for Cycles and P aths .............................................................. 53 3.3 Bounds for Stars and Paths ................................................................. 58 3.4 Bounds for Paths and P a th s .............................................................. 63 3.5 The .F-free Ramsey N um ber .............................................................. 67 3.6 The 2 /v 2-free Ramsey N um ber .......................................................... 70 3.7 Bounds for Stars and C ycles .............................................................. 74 DISCONNECTED GRAPHS .................................................................................... 79 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents-Continued SYMMETRY IN F-FREE RAMSEY NUMBERS ............................................. 82 DIGRAPH RAINBOW AND EDGE-CHROMATIC RAMSEY NUMBERS 84 POSSIBLE DIRECTIONS FOR FURTHER STUDY ....................................... 92 REFERENCES ............................................................................................................... 94 v Reproduced with permission of the copyright