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Some pages may have indistinct La qualite d'impression de print especially if the original certaines pages peut laisser a pages were typed with a poor desirer, surtout si les pages rypewriter ribbon or if the originales ont €96 university sent us an inferior dactylographiees a I'aide d'un photocopy. ruban 2s; ou si I'universite nous a fait parvenir une photocopie de ... ~ qualite inferieure. Reprodlrction in full or in part of La reproduction, m6me partielle, this microform is governed by de ce??emicroforme es? soumise the Canadian Copyright Act, & la Loi canadienne sur le droit R.S.C. 1970, c. C-30, and d'a~teur,SRC 1970, c. C-30, et subsequent amendments. ses amendements subsequents. The Block-Intersection Graph of Pairwise Balanced Designs Donovan Ross Hare B. Sr. (Hr,nours), University of Victoria, 1986 31. Sr... University of Alberta; 1987 a. THESIS SUBMITTED IX PARTIAL FULFILLMEYT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of llathernatics and Statistics @Donowin Ross Hare i991 SIXON FRASER UNIVERSITY July 139i -&I1 rights reserved. ?'his work may not be rcyrociuced in whole or in part, by photocopy or other Incam, wit hc *I;: permission of the author. The author has granted an L'auteur a accorde une licence irrevocable non-exclusive licence irrevocable et non exclusive allowing the National Library of permettant a la Bibliothkque Canada to reproduce, loan, nationale du Canada de distribute or sell copies of reproduire, priiter, distribuer ou his/her thesis by any means and vendre des copies de sa these in any form or format, making de quelque maniere et sous this thesis available to interested quelque forme que ce soit pour persons. mettre des exernplaires de cette these a la disposition des personnes interessees. The author retains ownership of L'auteur conserve la proprikti. du the copyright in his/her thesis. droit d'auteur qui protege sa Neither the thesis nor substantial these. Ni la these ni des extraits extracts from it may be printed or substantiels de celie-ci ne otherwise reproduced without doivent gtre irnprimes ou his/her permission. autrement reproduits sans son autorisation. ISBN a-315-79174-2 APPROVAL Name: Donovan Ross Hare Degree: Doctor of Philosophy Title of Thesis: The Block-Intersection Graph of Pairwise Balanced Designs Examining Committee: Chairman: Dr. Alistair Lachlan -- - -, - fl .- ~r.~Brian Alspac h, Professor Senior Supervisor Dr. Tom Brown, Professor - -7- LZ/,Y -, Dr. Alan Mekler, ?rofes&r Dr. Heinz Ang, $@essy Technical University, Berlin Dr. Chris Rodger, ~rbfessor Auburn ~nive&ty External ~xarniner . Date Approved: July 29. 1991 PARTIAL CC)fJ'r'lZIGIIT C ICI.NSI: . I hereby grant- to Sitriotl FI-asc.1- ilnivcr-s i t y tfji, r i\:jlii to /crlcf my thesis, project or extended essay (the til la of which is shown boiow) to users of tho Simon Frasor Univcrsi ty Library, and to makc pal-1 ijI or single copies only for such users or in response lo a roqucst from tile l i brary of any other un ivcrs i ty , or at her educat iona I i ns-f i .t u t- i ort, on 'iSs own behalf or for om of its users. I furthcr agroc tlmf permission for multiple copying of this work for scholarly purposes may bc granted by me or the Dean of Graduate Studies. It is undcrst-ood that copy i INJ or publication of this work for financial gain shall not bc allowcd without my written permission. - Title of Thesis/Project/Extended Essay . Author: / (s ignaturef Abstract. The focws of this thesis is to investigate certain graph theoretic proper- tir-s of a class of graphs \shicL arise from combinatorial design theory. The Itlock-i~~tc.rscc.tiongraph of a pairwise balanced design has as its vertices tlw Mocks of tllc (hasign ;irici has as its edges precisely those pairs of blocks wliicll llrtvr. nori-cm~ptyilitc~rsc~c'ticm. These graphs have a particular local strllrt~xrc~which is ~sploit(~1i:i tlw proofs of the results. In Chapter 1 an ovc-rvicw is given. Defi~iitioxls,ill1 esarnple. and some motivational material (a 1,ric.f history, conrlc~.tior~sto otlicr work) are included here. Cycles of ttwse graphs are investigittctl in Chapter 2. In particular, it is shown that thiw graphs are harrli1toni;m. 111 C'llal>ter 3 the connectivity of the block- interscdon RS~LI)~is d~tel-mili(d for balanced incomplete block designs and for 'large' pairwise I>al:trtct.tf designs. Chapter 4 contains the proofs of a nri~xh~rof restilts wllich pt3rtairl to coloring the block-intersection graph. hforc specifically, it is shown that the ~~eighborhoodof a vertex of the block- intc~rsc~ctio~igraph of .large' balanced incomplete block designs can always I>ccoiortd in an optinial \say. I wish to thark my supervisor Professor Brian Alspach for his guidance and support. He and Professor Katherine Heinrich helped expose me to the gw>graphicaland sociologicd aspccts of the world of combinatorial math- cntatics (not to mention the theoretical aspects as well). I also wish to thank Dr. William McCuaig for the discussions that led to the joint work of Chapter 3. For the extra work that was created by being in other departments dur- ing the researcll of the thesis I wish to thank the Graduate Secretary, Mrs. Sylvia Holmes. I dso wish to tha~lkthe Department of Mathematics and Statistics, University of Victoria, for their hospitality during the prepara- tion of the thesis. In particular, my appreciation is extended to Dr. Chris Bose for letting me use his KeXT computer to draw the figures. Finally, I wish to thank the B. C. Science Council, the Faculty of Grad- uate Studies, the Department of Xiathe~naticsand Statist.ics, and Professor Brian Alspach for their financial support. Contents 1 Introduction 1.1 Definitions ............................ 1.1.1 Block Designs ...................... 1.1.2 Hypergraphs ...................... 1.1.3 Graphs ......................... 1.2 Basic Properties of t lrc: Block-111t.c.rsc.c.tio11 Graph ...... 1.3 History and Results of the Tfwsis ............... 1.3.1 Cycles .......................... 1.3.2 Connectivity ...................... 1.3.3 Coloring ......................... 1.4 Other Needed Results ..................... 2 Cycles 2.1 Blocks of Cardinality Two Only ................ 2.2 Blocks of Differing Cardinalities ................ 2.3 Conclusion ........................... 3 Connectivity 41 3.1 Balanced Incorriplcte Block Designs ............. 41 3.2 Pairwise Balanced Designs ................... 46 4 Coloring 53 4 .I Erdos-Faher-LovAsz Conjecture ................ 53 4.1.1 The Dual of ;I Liricar Hypergraph .......... 55 4.1.2 The Proof of the Equivalence ............. 55 4.1.3 Pairwise Balancd Designs ............... 58 4.2 Balanced Incomplete B!ock Designs .............. 58 4.2.1 IC3-factors ........................ 59 4.2.2 I<[-factors ........................ 64 4.2.3 Coloring thc Ncighhorllood of a Block ........ 66 References 68 List of Figures 1.1 The block-intersection gr;tpf~of ;L (9.3. 1).tliGgn ....... G 1.2 The structurc of a ~lc.i~l~Lorhir,c1 ill B(L3) ........... 8 2.1 Proposition 2.2 gu;ir;mtc~~stliv c~sistcwwof a p-piitll ill 11 . 21 2.2 Case3whenr>l........................ 28 2.3 C~3(bj............................. 29 2.4 Case 3(c)............................. 30 2.5 Case 3(d)............................. 31 2.6 Case 3(e)............................. 33 4.1 The structure of the graph H = (2.C) ............ 61 4.2 Counter-example for t = 5 ................... 64 ... Vlll Chapter 1 Introduction This tl~esisinvestigates a ccr-tail1 class of graphs which arise from combina- torial design theory. We prescwt in this chapter the background material for the problems being studied. The first see tion gives t hc necessary definitions as well as an example of a block-i~ltersectiongraph. Some of the basic prop- erties of the block-intersection graph are discussed in the following section. A brief history is given in the third section of the three particular areas of the block-intersection graph investigated in the subsequent chapters. The contributions the thesis makes to these areas are stated in this section as well. The final section presents a couple of results about matchings that will be used in later the remaining chapters. 1.I Definitions In this section we give definitions of the less familiar combinatorid objects and properties. For the basic graph tlleoretic terminology the reader is refe: .ed to 151. We will start wit11 designs, followed by liypc3rgr;tphs ZUNI then Dove to some non-standard graph theory dcfini tions. Some dcfirii t ions will be given in the other chapters, however, they have been invc~itcdto facilitate the reading of the proofs and so are not included h8re. 1.1.1 Block Designs Let K be a finite set of positive inregers, and let X and '1) be positivc integers such that v > max I< (here niax I< is the maxi~numclemcnt in I<; similarily for minh'). A pairwzse balanced design, denoted PBD(w,K, A), is a pair (V,D) where I/' is a finite set clenlents are called points, B is a collection of subsets of V,called block^, such that IVI = v, the blocks have their cardinalities from Ir' and any pair of distinct points is contained in exactly X blocks.