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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 4B106 I I I I 75-26,683 WANG, Chin San, 1939- IMBEDDING GRAPHS in the PROJECTIVE PLANE

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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 4B106 I I I I 75-26,683 WANG, Chin San, 1939- IMBEDDING GRAPHS in the PROJECTIVE PLANE INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page{s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. 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Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 4B106 I I I I 75-26,683 WANG, Chin San, 1939- IMBEDDING GRAPHS IN THE PROJECTIVE PLANE. The Ohio State University, Ph.D., 1975 Mathematics Xerox University Microfilms ,Ann Arbor, Michigan 48106 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. EMBEDDING GRAPHS IN THE PROJECTIVE PLANE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School o f The Ohio S ta te U n iv ersity Chin San Wang, B .S ., M.S. * * * # The Ohio State University 1975 Reading Committee: Approved Ry G. Neil Robertson Henry H. Glover Department of Mathematics ACKNOWLEDGEMENTS I wish to thank my advisers Henry Glover and Philip Huneke for giving me this thesis problem; Professor Huneke for his care­ ful detailed supervision of my computations, Professor Glover for his general understanding of this area. I particularly appreciate the time that we spent together checking my work ( including many weekends and holidays. ) i i VITA. December 18, 1939 ••• Bora - Ilan, Taiwan 1958 ......................... Hwa Lien Normal School, Hwa Lien, Taiwan 1963 ............................... Taichung Teachers College, Taichung, Taiwan 1967 - 1968 .................Mathematics Instructor, Brashear High School Brashear, Missouri 1968 ...............................B.S. in Mathematics, Northeast Missouri State University, Kirksville, Missouri 1968 - 1970 .................Teaching Assistant, University of Arkansas Fayetteville, Arkansas 197O ...............................M.S. in Mathematics, University of Arkansas Fayetteville, Arkansas 1970 - 1975 .................Teaching Associate, The Ohio State University Columbus, Ohio 1 9 7 5 ...............................M.S. in Computer and Information Science Ph.D. in Mathematics, The Ohio State University Columbus, Ohio (anticipated) FIELDS OF STUDY Algebra : Erofessor Joseph C. Ferrar Analysis : Erofessor Bogdan Baishanski Topology : Professor Henry H. Glover i i i TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................................................................ i i VITA ..................................................................................................................... H i INTRODUCTION .................................................................................................... 1 SECTION 1 . THE PARTIALLY ORDERED SET l ( M ) ....................................... 5 2 . SOURCES, SINKS AND SYMMETRIES IN l( M ) .............................. 8 3 . THE TOPOLOGY OF THE PROJECTIVE PLANE.............................. 10 k. DISTINCTNESS OF GRAPHS IN l ( P ) ....................................... 15 5* THE MAIN RESULT ......................................................................... 27 6 . SOURCES DO NOT EMBED................................................................. 29 7* SINKS ARE IRREDUCIBLE ............................................................ k2 8 o SOURCES AND SINKS OF A GRAPH ............................................... 67 9« CONJECTURES AND RELATED QUESTIONS................................... 73 APPENDIX............................................................................................................ 75 BIBLIOGRAPHY.................................................................................................... I l l INDEX OF GRAPHS IN l ( P ) .......................................................................... 112 iv INTRODUCTION The four color conjecture is well knowno In 1 8 9 0 p . j . Heawood [ 5 ] fcund the mistake in Kempe's "proof" of this conjecture and proved the corresponding result for five colors* In addition he formulated a generalization of the four color conjecture to general surfaces* He conjectured that the chromatic number n , the number of colors needed to color the closed orientable surface of genus g , M is the integer part C& and for the closed non-orientable surface of genus g t M i s O The Heawood conjecture for non-orient able surfaces was solved by Ring el [ 9 1 in 1952 and by Ringel and Youngs [10] for orientable surfaces in 1968* They used Heawood's "contiguity Lemma” which allowed them to replace th is problem with an embedding problem* Namely they showed that the orientable genus, the smallest genus of any orientable surface containing is the integer hull 1 2 (n - k)(n - 3 ) g(n) 12 and the ncn-orientable genus of is (n - 4Kn - 3) g(n) The idea used in this replacement is an amalgamation technique similar to that in the proof of the five color theorem. Using this technique ve may construct an example of a map on M (*L) with D O exactly n(g) (rT(g)) countries each two of which are contiguous. By considering the dual to the graph of boundaries of this map (the graph of railroads between the capitals of these countries) we then get c Mg (K^ c Mg) and the dual embedding problem (see Ring el [ 8 ]) • Using the solution of the Heawood conjecture as motivation we next ask whether we can find a method to compute the orientable (respectively non-orient able) genus of an arbitrary graph. That is, given arbitrary graph K find the integer g such that M 6 contains K but M ... does not contain K (respectively M & “* o contains K but M_ m, does not contain K). Q An answer to a special case of this question is the theorem 2 of KuratowskL [ 7 ] which says that a graph is planar (K c IR ) if and only if K does not contain or ^ • Equivalently, g(K) - 0 if and only if K does nob contain or ^ • 3 A first generalization of KuratowskL 's theorem would be a similar characterization of graphs that embed in the projective p lan e P • In this paper we prove part of this characterization. In particular we prove that the set of irreducible graphs l(P) for p P (analogous to the set l( R ) consisting of and contains 103 graphs (Theorem 5*1)• Further we strongly conjecture that our list of 103 graphs gives the complete set l(P) • (See section 9 for a more complete discussion of this conjecture.) The technique that we use to construct our lis t of 103 graphs in l(p) is as follows: Given a surface M , Paul Kainen [ 6 ] introduced a partial ordering on l(M) , the set of irreducible graphs for M • Huneke and Glover [ 2 ] have introduced a study of the isotcpy classes of embeddings of K c M which makes it possible to study this partially ordered set. In particular they strongly conjecture that l(P) has exactly five maximal elements. One of these five graphs was found by Hell Robertson. The 103 graphs we have found are obtained by using this partial order to construct graphs below these five. In this paper we do not show that these are all graphs K in l(p) below these five, but we do prove that each of the 103 graphs given are distinct and are in l(p) • The proof that these five graphs are the maximal elements will appear in [ 3 ]• The proof that these 103 graphs are all the graphs in l(P) below these five will appear in [11]. Altogether this will show that l(P) Is precisely the set of these 103 graphs. The plan of the paper follows: In section 1 we give basic definitions and prove that l(P) is a partially ordered set with respect to the order that we introduce (Preposition 1.2). In se ctio n 2 we discuss certain special elements in l(p) which are sinks and sources of a partially ordered subset of l(P) • Many of these sources (minimal minors) were found independently by Robertson. We also discuss symmetries in graphs. In section 3 we give miscellaneous facts about embedding graphs in the projective plane P • In section 4 we prove that all 103 graphs we lis t are distinct (not homeomorphic)* In section
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