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Ann Arbor, MI 48106 C lo sed 2 -cell E m b e d d in g s of 2 -c o n n e c t e d G r a ph s in S urfaces DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Xiaoya Zha, B.S., M.S., The Ohio State University 1993 Dissertation Committee: Approved by Thomas Dowling l Philip Huneke P Adviser Randolph Moses Department of Mathematics Dijen Ray-Chaudhuri Neil Robertson to my wife Qiong Shi, and my parents Binmo Zha and Zhilan Zhou for all they have done through the years ii A cknowledgements I am deeply indebted to my advisor, Dr. Neil Robertson, for his guidance, en­ couragement, many insights and tireless assistance through the preparation of this dissertation. I would like to express my thanks to Dr. Ruth Charney and Dr. Henry Glover, for their valuable help in topology, which has been specially important to this disser­ tation. I am grateful to my committee members Dr. Thomas Dowling, Dr. Philip Huneke, Dr. Randolph Moses and Dr. Dijen K. Ray-Chaudhuri, for their suggestions and the time they have spent reading this dissertation. I would like to express my special appreciation to Qiong Shi, my wife, for her encouragement, faith and many sacrifices, without which this work would have been impossible. Finally, I would like to thank my friend Yining Xia, for many stimulating discus­ sions, help and friendship. V ita August 14, 1959 Born in Tunxi, China. 1982 B.S., Anhui University, Hefei, China. 1984 ..................................M.S., Department of Mathematics, Huazhong University of Science and Technology, Wuhan, China. 1990 ..................................M.S., Department of Mathematics, The Ohio State University, Columbus, Ohio. F ie l d s o f S t u d y Major Field: Mathematics iv T a b l e o f C o n t e n t s ACKNOWLEDGEMENTS ........................................................................................ iii VITA ............................................................................................................................... iv LIST OF FIGURES ..................................................................................................... vii LIST OF TABLES ........................................................................................................ ix INTRODUCTION ........................................................................................................ 1 CHAPTER PAGE I Preliminaries and Survey ..................................................................................... 4 1.1 Preliminaries ............................................................................................... 4 1.2 Survey ............................................................................................................ 10 II The Face Chain M ethod..................................................................................... 20 2.1 Circuit chains and an edge version of face chains .............................. 20 2.2 Vertex version of the face chain m eth o d .............................................. 21 2.3 Two new operations .................................................................................. 26 2.4 Dual circuits and the face attachment model ........................................ 28 2.5 Good face c h a in s ......................................................................................... 30 III Some Topological Results ................................................................................. 41 3.1 Classification of nonseparating simple curves in the Klein bottle . 41 3.2 Homology and homotopy intersections of T and T ~ ........................... 44 3.3 Classification and properties of simple curves in A 3 ........................... 50 IV Closed 2-Cell Embeddings of Lower Surface Embeddable Graphs ............ 57 4.1 R e d u c tio n s ............................................................................................ 57 4.2 Closed 2-cell embeddings of some planar graphs with new edges . 59 4.3 Face chains in the projective plane and the Klein b o ttle .......... 65 4.4 Face chains in ........................................................................................ 70 4.5 Closed 2-cell embeddings of some projective graphs with new edges 76 4.6 Closed 2-cell embeddings of lower surface embeddable graphs . 81 V Minimal Surface Embeddings ........................................................................... 89 5.1 Notation and some technical results .................................................. 90 5.2 A structure theorem ........................................................................... 93 5.3 A pp lication ............................................................................................ 99 5.4 E x a m p le s ............................................................................................... 107 BIBLIOGRAPHY ......................................................................................................... 114 vi L is t o f F ig u r e s 1 24 2 25 3 26 4 27 5 28 6 29 7 30 8 32 9 35 10 37 11 38 12 39 13 42 14 43 15 49 16 51 vii 17 54 18 55 19 56 20 61 21 64 22 66 23 71 24 74 25 75 26 78 27 80 28 94 29 95 30 98 31 99 32 108 33 111 34 112 viii Introduction The closed 2-cell embedding conjecture (called the strong embedding conjecture in [Jae], [LR] and circular embedding conjecture in [RSS]) states that every 2-connected graph G has a closed 2-cell embedding in some surface; i.e., an embedding in a compact closed 2-manifold in which the boundary of each face is a circuit in the graph. This conjecture from the folklore of topological graph theory has been open for at least 30 years. The first result may be due to H. Whitney [Whi] in 1932, who proved that a plane embedding of any 2-connected planar graph is a closed 2-cell embedding. Since then, not much progress has been made. During the last 30 years, topological graph theory has become an active research area which combines graph theory and lower dimensional topology. As a topic of topological graph theory, closed 2-cell embedding has attracted more and more attention. A closed 2-cell embedding can be viewed as a “good” embedding, since embeddings (even genus embeddings) in which the facial walks have repeated vertices and edges may be viewed as “bad” embeddings. Many problems in topological graph theory require “good” embeddings. For instance, it is hard to work with the (surface) dual of an embedded graph if a facial walk has repeated edges because loops will be created. Therefore, if the closed 2-cell embedding conjecture is true, then it will provide a very general class of embeddings of graphs satisfying this useful condition. As an existence 2 problem, it has its own interest as a topic in topological graph theory. On the other hand, for the past 15 years, the topic of cycle double covers of 2- edge-connected graphs has been very active. The cycle double cover conjecture states that every 2 -edge-connected graph has a cycle double cover, i.e. a list of cycles in the graph with each edge in exactly two of these cycles. Part of the reason that this topic has been active is it is related to the well-known 4-color theorem of planar graphs, which was proved in 1974-76 by computer techniques, and to flow theory, including the famous 4-flow and 5-flow conjectures of Tutte. Since the existence of a closed 2- cell embedding of a graph implies the existence of a cycle double cover of that graph, simply by choosing all face boundaries as cycles, research on closed 2 -cell embeddings can also be viewed as a topological approach to the cycle double cover problem. In this dissertation, we try to develop some techniques to approach the closed 2 -cell embedding conjecture and to confirm this conjecture for some classes of graphs. Chapter 1 contains the preliminaries and a survey. In Chapter