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C losed 2 -cell embeddings of 2 -connected graphs in surfaces

Zha, Xiaoya, Ph.D.

The Ohio State University, 1993

UMI 300 N. ZeebRd. 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

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 2 we introduce the vertex version of the face chain method and some new operations in this method. We give a model to describe the attachment(s) of two faces with common boundary points in any surface embedding. In Chapter 3 we prove some topological results, including the classification of the free homotopic classes of simple closed curves in the torus,

Klein bottle and A/ 3 , the sphere with 3 cross-caps. A graph is called jV* embeddable if it can be embedded into the non-orientable surface of Nk (the sphere withk cross-caps).

In Chapter 4 we confirm the closed 2-cell embedding conjecture for N& embeddable graphs (which includes doubly toroidal graphs). This improves the previous result from the sphere and the projective plane to the double torus and N&. In Chapter 5

we study minimal genus surface embeddings. We prove some structural properties

of ( 1 ) minimal surface embeddings, which are either orientable genus embeddings or

non-orientable genus embeddings, and ( 2 ) embeddings with the minimum number of

repeated vertices or edges in facial walks (if the closed 2 -cell embedding conjecture

is true, then this number is zero). An upper bound is also given for the number of

homotopic classes of principal ^-minimal curves for 3-connected graphs embedded in

minimal surface embeddings. This is 3g — 3 on S^, for g > 2 (or 1 if g = 1) and

3[§J - 3 on N k, for k > 4 (or for k = 1,2,3). Examples are given to show that these bounds are the best possible for minimal surface embeddings. These examples also provide a complete list of surfaces where the genus embedding conjecture fails. C H A PTER I

Preliminaries and Survey

1.1 Preliminaries

A graph G is a pair (V, E) where V is the set of vertices of G, E is the set of edges of G, and each edge of G is associated with either one or two vertices. If v G V is associated with e G E then v is said to be incident to e and e is said to be incident to v. The vertices which are incident to an edge e are called the endvertices of e and the set of endvertices of e is denoted by enda(e). If v and w are distinct vertices incident with an edge e then v and w are said to be adjacent. An edge which is associated with only one vertex is called a loop and an edge which is associated with two vertices is called a link. A link e with endvertices v and w is denoted by vw.

If e\ and e 2 are two edges of G with e\ ^ e2 and end(e\) = end(e2) then e\ and e2 are called parallel edges. A graph which contains no loops and parallel edges is called a simple graph. If v G V then the valence of v in G, denoted by val(G, v ) (or val(v), if the circumstance causes no confusion), is the number of links incident to v plus twice the number of loops incident to v.

Let G be a graph. A graph H is called a subgraph of G if V(H) C V(G), E(H) C

E(G) and for all e 6 E(H), endn(e) = endc(e). More generally, a minor of G is a

4 graph K obtained from a subgraph H of G by contracting (sequentially) some edges to

vertices. Let u and v be two not necessarily different vertices of G. By an u-v walk W

we mean an alternating sequence of vertices and edges, say v i,e i,v 2, e2 ,u/, ci,

such that Vi = u, u/+1 = v and e,- = v,ul+i G E(G) for 1 < i < /. When G is a

simple graph we may put W = v\V2...vi+\ since from this form it is clear which edges

are in the sequence. The walk above is a trail if all its edges are distinct and it is

a path or v\-vi path if all its vertices are distinct. A closed trail is a cycle. If / > 1 and v\ = vj+i, but v \,v 2 ,...U| are distinct then we call the walk a circuit. We do not distinguish the walks, paths, cycles and circuits from their associated subgraphs.

A graph is k-connected if there are no k — 1 vertices whose deletion from the graph disconnect the graph. A graph is k-edge-connected if there are no k — 1 edges whose deletion from the graph disconnect the graph. An k-edge-coloring of a graph is an edge coloring with at most k colors such that adjacent edges receive different colors.

A surface is a compact 2-manifold, without boundary. These consist of the ori-

entable surface Sg with orientable genus or handle number g, homeomorphic to the

2 -sphere withg > 0 handles adjoined, and the non-orientable surface withNk non- orientable genus or cross-cap number k, homeomorphic to the 2 -sphere withk > 1 cross-caps adjoined. When S is a topological space, denote by C(S) its set of con nected components. Then a surface embedding is defined to be a triple (E, U, V) where E is a surface, U is a closed subset of E, and V is a finite subset of U such that

C(U — V) is a finite set of homeomorphic copies of the open unit interval (0,1). We denote by V^(^) the vertex set V of \&, by E{^f) the edge set of \k, which is the set of the closures of the elements in C(U — V ), and by F (^ ) the face set C{ E — U) of

The graph G(\&) (we may use G if no confusion is caused) of 'P is the graph with the same vertex set and edge set as 'F, in which each edge is incident with its endvertices.

When all the faces of are homeomorphic to open 2-cells (i.e. open 2-disks) the graph G(\l/) is non-null and connected. Such embeddings are called open 2-cell embeddings and are the most common type of embedding considered in topological graph theory. Open 2-cell embeddings exist for any connected graph; for exam ple any embedding of minimum orientable genus. For open 2-cell embeddings the

Euler characteristic x (^ ) = V — E + F (where we abbreviate the cardinalities of

Vr(’ir),2?(\f,) ,/r(’J!') by V,E,F, respectively) determines the surface S(^) up to ori- entability, and Sg satisfies V — E + F = 2 — 2g and Nk satisfies V — E + F = 2 — k.

Moreover, x W is an integer < 2, with x(^) = 2 only when E(^) is the 2 -sphere.

The boundary df of any / € F(^) in S(^) is a closed subset of U which contains any edge of that it meets at an internal point, and thus determines a subgraph of

G(\l>). Following the circular order of the open disk / this subgraph is traced out by a closed walk in G, unique up to rotations and reversal of direction, called the facial walk of /. Note that each edge of appears exactly twice in the facial walks of 'k and thus has two “sides” along one or two of the faces of and each vertex of $ appears in these facial walks with multiplicity equal to its valence in G(\&).

Let E be a surface. A simple closed curve T (maybe a circuit in an embedded graph) is 2 -sided if it is orientation preserving and 1 -sided if it is orientation revers ing. Define E\r to be the (not necessarily connected) surface with boundary formed 7

by cutting E along T. Either E\r has two connected components and T is called

separating or E\r has one connected component and T is called nonseparating. If T

is separating and one component is homeomorphic to an open 2 -cell then T is null-

homotopic (or trivial). All simple closed curves which are not null-homotopic are said

to be essential. When T is separating, or E is orientable, then T is always 2-sided.

However, in a non-orientable surface a nonseparating simple closed curve T may be

1-sided or 2 -sided, and E\r may be orientable or non-orientable. In this dissertation we do not distinguish between a curve and a homotopy class of curves on a surface when it is not necessary for clarity. Epstein has the following useful result.

L em m a 1 . 1 . 1 ([Eps], Lemma 2.4). Let E be a surface and T i , ^ be two disjoint essential 2-sided simple curves. If T2 is homotopic to Ti, then Ti and T2 bound a cylinder.

Let \& be an embedding of a graph G. When the faces of are open 2-cells bounded by simple circuits in G, the embedding is called a closed 2-cell embedding.

When 'I' is a closed 2 -cell embedding and the subgraph of G(\&) bounding the faces incident with any vertex is a wheel with > 3 spokes and a (possibly) subdivided rim, the embedding is called a wheel-neighborhood embedding.

Let be an embedding of a graph G in a surface E which is not the sphere. Then the representativity of is defined to be

(1.1.1) p(^) = m in {|r fl G| : T is an essential closed curve in E('J')}.

Refinements of this parameter, defined when the corresponding essential curves exist on E. 8

(1.1.2) p.('I') = mm{|r n G| : T is an essential *-sided simple curve in S(^) } for

*' = 1, 2;

(1.2.3) pa(V) = mm{|r fl G| : T is an essential separating simple curve in S(^) }.

A closed curve T is V-minimal if T is simple, essential, TflG C V(G) and |rnG | = p(^). It is easy to see that a ^-minimal curve T always exists in E when p(^) is defined. Clearly, p(^) = mm{/ 9 i(^ ),p 2 (^,),Pj(^')}*

Robertson and Vitray [RV] have the following results:

Proposition 1 . 1 .2 . An embedding is an open 2 -cell embedding if and only if p(&) > 1 and G (^) is connected.

Proposition 1.1.3. An embedding is a closed 2-cell embedding if and only if p('J') > 2 and G(\&) is 2-connected.

Proposition 1.1.4. An embedding \E' is a wbeel-neighborhood embedding if and only if p('i) > 3 and G('P) is simple and 3-connected.

The orientable genus of a graph G is defined to be the smallest number g such that G can be embedded in the orientable surface Sg. The nonorientable genus of a graph G is defined to be the smallest number k such that G can be embedded in the non-orientable surface JV*. The minimum orientable surface of G is Sg if the orientable genus of G is g, and the minimum non-orientable surface of G is Nk if the non-orientable genus of G is k. A minimum surface of G is a surface E that G has an embedding ^ in E such that x(^) is maximum. A minimal surface of G is either the minimum orientable surface or the minimum non-orientable surface of G, or both of 9

them.

Suppose G is a 2-connected graph. A set of vertices {wi,..., w*} C V(G) is called a

k-vertex cutset if subgraphs Gi,G2 of G exist such thatG = G 1 UG2 , |V (C ri)|,\V(Gi)\ >

3 and V(Gi fl G2) = {t>i,..., w*}- Let {vi,V2 ,v3} be a 3-vertex cutset of G. Add a new vertex u and three new edges uv\,uv2, uv3 to G\ and G2 to form new graphs G f and G2 • If the genera of G f and G2 (orientable or non-orientable, depending on the context) do not exceed the genus of G, we say this 3-vertex cutset is good. Note that a 3-vertex cutset caused by a 3-circuit may increase the genus of G* or G2 when G 1 or G2 is a triangle.

Let be an embedding and / be a face of \k. Denote by df the boundary of / and by /(= / U df) the closure of /. Note that if p(^) > 2 and G is 2-connected, then df is a circuit. Let f,g be two faces of VP. If df Ddg ^ 0, we say / and g are attached. If / and g are attached, then dfndg is the union of connected components, called the attachments of / and g. Clearly, if /o(^) > 2 and G is 2 -connected, then each attachment is either a common edge, possibly subdivided, or a vertex. Denote by ||df fl dg\\ the number of connected components of d f fl dg. If ||d f D dg\\ = k, we say the faces / and g have k attachments.

For a given embedding of a graph the clockwise order of all edges incident to v

(“rotation” of edges at vertex v) at each vertex v, and its local disk neighborhood is fixed. HefFter [H], and independently Edmonds [Ej realized that the converse is also true; i.e., for every vertex v, if all edges incident to v are assigned a rotation order, then this will also define a graph embedding in some surface. They have developed a 10

useful technique, called the rotation scheme method of embedding graphs in surfaces.

One of the descriptions of a rotation scheme, called a rotation projection, is demon

strated as follows: Let G be a connected graph. Let O (G) be a plane drawing of G

on the plane such that edges are allowed to cross each other in finite many times. At

each vertex v, all edge incidences form a clockwise edge rotation order. An edge is

called twisted if it is marked with an 'x '. Each edge has two sides and these two sides

are called edge-sides. If an edge is twisted then its two sides are also twisted. A facial

walk is a closed edge-sides sequence obtained by tracing edge-sides from one edge- side to another according to the edge rotation at each vertex. By closed edge-sides sequence we mean the tracing of edge-sides terminate when same pair of edge-sides repeats. By pasting an open disk on each facial walk and gluing all these open disks together, the plane drawing ©(G) induce an embedding of G in a surface E. Such an embedding is an open 2-cell embedding and the surface E can be determined by cal culating its Euler characteristic and checking the orientation. More detail of rotation schemes (rotation projections) can be found in Gross and Tucker [GT].

1.2 Survey

1 .2 . 1 . Conjectures.

As mentioned in the Introduction, the closed 2-cell embedding conjecture comes from the folklore of the subject. It is one of a series of conjectures. We will state four of them, including the closed 2 -cell embedding conjecture, which is central to much of this dissertation. Of these four conjectures, each is stronger than the previous one. A cycle double cover of a graph G is a family of cycles C = (C i,...,C n) such that every edge e G E(G) is contained in exactly two of the C,’s. An edge strong

embedding is defined to be an embedding such that every edge is on the boundary of two distinct faces; i.e., in the facial walk of each face, there are no repeated edges

(repeated vertices are allowed).

(i) The cycle double cover conjecture: Every 2-edge-connected graph has a cycle double cover.

(ii) The edge strong embedding conjecture: Every 2-edge-connected graph has an edge strong embedding in some surface.

(iii) The closed 2-cell embedding conjecture: Every 2-connected graph has a closed 2-cell embedding in some surface.

(iv) The orientable closed 2 -cell embedding conjecture: Every 2-connected graph has a closed 2-cell embedding in some orientable surface.

Next is a false conjecture. Since we will study some related topics in this disser tation, we put this conjecture here for our reference.

(v) The genus embedding conjecture (see [Sey], Conjecture 3.4): Every trim- lent graph has a closed 2-cell embedding in a surface of its own genus.

1.2.2. Cycle double covers.

A cycle double cover of a graph is not defined in terms of a surface embedding.

However, it is closely related to a surface embedding. Clearly, a closed 2 -cell embed ding of a graph will give a cycle double cover of that graph, simply by using all the facial walks of the embedding as cycles. On the other hand, in some cases a cycle 12

double cover of a graph can induce an embedding of that graph in the following way.

Let G be a 2-connected graph and S = {ci, ...,c/} be a cycle double cover of G.

Suppose each c, E S is a circuit in G. Then c,- contains at most one pair of edges incident to any fixed vertex v. Let E v be the set of edges which are incident with v E V(G). For each v E V(G), define a graph Gv with vertex set E v and such that two vertices e and e' of Gv are adjacent if and only if e and e' are contained as edges in some circuit c, E S. Note that Gv is a 2-regular graph and when connected it is a circuit. Since whenever Gv is a circuit if and only if the edge rotation at v guarantees a disk neighborhood of v in the 2 -complex formed by pasting disjoint open disks onto the circuits Cj,..., c/, the following proposition is obvious:

Proposition 1.2. 2 .1 . A 2-connected graph G has a closed 2-cell embedding in a surface if and only if G has a cycle double cover S {ci,...,c/} = such that c, is a circuit in G for 1 < i < I, and the associated graph Gv is connected (i.e., is a circuit) for all v E V(G).

Clearly, if G is a trivalent graph, then Gv is a 3-circuit, which is connected, for all v E V(G). Therefore, as a consequence, we have the next proposition.

Proposition 1.2.2.2. Let G be a 2-connected trivalent graph. Then the existence of a cycle double cover of G is equivalent to the existence of a closed 2-cell embedding o f G.

Since confirmation of the cycle double cover conjecture can be reduced to trivalent graphs ([Fll], [Jae]), we can thus approach the cycle double cover conjecture by constructing surface embeddings of trivalent graphs. 13

Similarly to the closed 2 -cell embedding conjecture, the cycle double cover con jecture is also from the folklore of the subject. The question is raised in particular

in [Sey] and [Sze]. Since then cycle double covers of graphs and more general cycle

covers of graphs have been very active topics in graph theory.

The most important recent result on the existence of cycle double covers is due to

Alspach and Zhang [AZ]. They have shown that a 2-connected graph G has a cycle double cover if G contains no Petersen graph as a minor. Recently, Hoffman, Locke and Meyerowitz [HLM] have shown that all Cayley graphs have cycle double covers.

A minimum counterexample to the cycle double cover conjecture is a 2-edge- connected graph with no cycle double cover which has a minimum number of edges among graphs with these properties. It follows that such a minimum counterexample

G must be a trivalent graph which is cyclically 4-connected [Jae]. A trivalent inter nally 4-connected graph which does not have an edge-coloring with 3 colors is called a snark (see [Gar]). Since a 3-edge coloring of a trivalent graph implies a cycle double cover of that graph (an edge set of one color is a 1 -factor of the graph and an edge set of two colors is a disjoint union of circuits which spans the graph. We have three such two-color sets and they form a cycle double cover of the graph). Therefore the following is true.

Proposition 1.2.2.3 ([Jae], Proposition 4). A minimum counterexample to the cycle double cover conjecture must be a snark.

In [Cel] various families of snarks are shown to satisfy the cycle double cover con jecture. One may hope that the study of minimum counterexamples will be developed until eventually they are shown not to exist. Goddyn [Go] has proved that a minimum

counterexample to the cycle double cover conjecture has girth at least 11. This is

quite interesting since no snark of girth at least 7 is known. In fact, it is conjectured

in [JS] that such snarks don’t exist.

There are many topics which are related to that of cycle double cover. These topics

include the cycle decompositions of Eulerian graphs, integer flows, etc. More can be

found in [F12], [FI3], [Jae], [JS]. We conclude this sub-section with the following quote

of Tutte (see [F12], page V.5): “I too have been puzzled to find an original reference.

I think the conjecture is one that was well established in mathematical conversation

long before anyone thought of publishing it”.

1.2.3. Edge strong embeddings.

Let 'k be an embedding. A face coloring of is said to be proper if two sides of each edge receive different colors. A k-coloring of 'P is a proper face coloring of

with at most k colors.

The following are equivalent:

(i) G has an edge strong embedding in some surface;

(ii) G has an embedding such that the faces can be properly colored;

(iii) G has an embedding such that there are no loops in the dual embedding.

A Tati coloring of a trivalent graph is a 3-coloring of the edges of that graph such that the three edges incident with a vertex receive the three distinct colors.

Tait [Tai] proved in 1880 that one can four-color the faces of a trivalent graph embedded on the plane if and only if one can Tait-color the edges. 15

Tutte examined the relationship between Tait colorings of trivalent graphs and

the face 4-colorings of embeddings of these graphs in surfaces other than the sphere.

He proved the following result:

Theorem 1.2.3.1 ([Tut]). Let G be a cubic graph. The following are equivalent:

(i) G is Tait colorable;

(ii) G embeds in an orientable surface such that the faces can be 4-colored;

(iii) G embeds in some surface such that the faces can be 4 colored;

(iv) G embeds in some surface such that the faces can be three colored.

Tutte also proved that a trivalent graph embeds in an orientable surface such that

the faces can be 3-colored if and only if it is bipartite.

Let G be a graph and II be a subgraph of G. The subgraph H evenly spans G if

(i) each vertex of G is of even, nonzero degree in //; and

(ii) in each component of II there are an even number of vertices which are of odd

degree in G.

A 3-splitting of G is any trivalent graph that can be converted to G by a sequence of edge contractions and subdivisions. A proper ^-coloring of the faces is packed if for each vertex v of G the set of faces incident with v receives at least three distinct colors. Note that every proper face-coloring of a trivalent graph is necessarily packed.

Archdeacon generalized Tutte’s result to non-trivalent graphs:

Theorem 1.2.3.2 [Arc]. Let G be a graph with minimum degree at least 3. The following are equivalent:

(i) G has a 3-splitting which is a Tait-colorable trivalent graph; 16

(ii) G embeds in some orientable surface with a packed 4-coloring of the faces;

(iii) G embeds in some surface with a packed 4-coloring of the faces;

(iv) G embeds in some surface with a packed 3-coloring of the faces; and

(v) G is evenly spanned by a subgraph H.

Since the following graphs are evenly spanned [Arc], they have edge strong em beddings in some surfaces:

(i) Hamiltonian graphs;

(ii) Eulerian graphs (connected graphs in which all vertices have even valency);

(iii) super-Eulerian graphs (graphs containing spanning Eulerian subgraphs);

(iv) graphs containing two edge-disjoint spanning trees; or

(v) 4-edge-connected graphs.

Note that even though almost all graphs are 4-edge-connected [BH], and therefore evenly spanned, there are still infinitely many graphs which are not evenly spanned

[Arc],

Recently, Huneke, Richter and Younger [HRY] proved that any 2-edge connected graph which can be embedded in 7V3 has an edge strong embedding in some surface.

1.2.4. Closed 2-cell embeddings.

If G is a simple graph, then clearly any embedding with face size at most 5 will be a closed 2-cell embedding. In particular, the triangulations and quadrangulations of a surface are closed 2-cell embeddings for some graphs. If G is a simple bipartite graph, then any embedding of G with face size bounded by seven is a closed 2-cell em bedding. 17

In the 1960’s, in their remarkable work on chromatic numbers of surfaces which are not spheres, Ringel and Youngs actually proved that the complete graph K n, and the complete bipartite graph K m,n have closed 2-cell embeddings. They constructed orientable genus embeddings of these graphs, which happen to be closed 2-cell em beddings. Part of these results were obtained by Gustin [Gus], Helfter [Hef] and

Jungerman [Jun].

It is well known that a plane embedding of any 2-connected planar graph is a closed 2-cell embedding (see [Ore], Section 1.2). Negami ([Neg], Lemma 2.1), and independently Robertson and Vitray([RV], page 303) showed that every 2-connected projective planar graph has a closed 2-cell embedding either in the sphere or in the projective plane. Richter, Seymour and Siran [RSS] proved that every 3-connected planar graph has a closed 2-cell embedding in some surface other than the sphere; also they characterized those planar graphs which have this property. Zhang [Zhan] has shown that every 2-connected graph without as a minor has a closed 2-cell embedding in some surface.

A graph G is Nk embeddable if G can be embedded in the non-orientable surface

Nk. We will prove in this dissertation that 2-connected graphs which are Ns em beddable (including double toroidal graphs) have closed 2-cell embeddings in some surfaces. This improves the previous results from the sphere and the projective plane to the double torus and N 5. As a corollary, such graphs have cycle double covers.

1.2.5. Orientable closed 2 -cell embeddings.

There are not many general results about the existence of orientable closed 2- 18

cell embeddings. However, as mentioned earlier, the orientable genus embeddings of

K n and K m>n are closed 2-cell embeddings. Archdeacon [Arc] had a probably the strongest conjecture which is related to the orientation. The conjecture states that every 2-connected graph has an edge strong embedding for which the faces can be

5-colored.

1.2.6. Genus embeddings.

Since the orientable genus embeddings of any 2-connected graph must be open

2-cell embeddings, one may expect that some of the orientable genus embeddings are also closed 2-cell embeddings. It was a folklore conjecture and was first stated explicitly in [Sey] (conjecture 3.4) that every cubic 2-connected graph has a closed

2-cell embedding in its minimal orientable surface. Unfortunately, this is not true.

Nguyen Huy Xuong [Xuo] presented a 2-connected trivalent graph of orientable genus

1 which has no closed 2-cell embedding in the torus. Another counterexample can be found in [HRY] which also has the torus as its minimum orientable surface.

Since any plane embedding of a 2-connected graph is a closed 2-cell embedding and a projective planar graph has a closed 2-cell embedding either in the sphere or in the projective plane, the genus embedding conjecture is true when the surface is the sphere or the projective plane.

In this dissertation, we will prove some structural properties of (1) minimal surface embeddings, which are either orientable genus embeddings or non-orientable genus embeddings, and (2) embeddings with the minimum number of repeated vertices or edges in facial walks (if the closed 2-cell embedding conjecture is true, then this number is zero). Also an upper bound for the number of principal ^-minimal curves

(defined in Chapter V), which is 3g — 3 on Sg for g > 2 (or 1 if g = 1) and 3[fj — 3 on Nk , for k > 4 (or for k = 1,2,3), is given for each surface which is not a sphere or a projective plane. Examples are given to show that these bounds are the best possible for minimal surface embeddings. These examples show that the genus embedding conjecture fails for all surfaces but the sphere and the projective plane. C H A P T E R II

The Face Chain Method

In this chapter, we discuss the face chain method, by which we will derive new closed

2-cell embeddings from old ones. In Section 2, we generalize the edge version of face chains to the vertex version. In Section 3, we introduce two new face chain operations.

In Section 4, we give a model of face attachments, which is very important if the surface is not a sphere. In Section 5, we discuss so-called “good face” chains, which enable us to construct new closed 2-cell embeddings from old ones.

2.1 Circuit chains and an edge version of face chains

In previous approaches to the cycle double cover conjecture, the circuit chain method is widely used (see [AGZ], [AZ], [Sey]). Given two non-adjacent vertices x and y in a graph G, a sequence of circuits G = (G i,...,C n) is called a circuit chain joining x and y if x € C \,y € C„, and G, and C,+i have common vertices, for i = l,...,n — 1.

Suppose G is a graph which has a cycle double cover. The circuit chain method shows that if we can find a certain type of circuit chain joining two non-adjacent vertices x, y in a graph G, then we may obtain a cycle double cover of the graph G U {xy} from the original cycle double cover of G, where xy is a new edge added to G.

There is a surface embedding version of this method which can be found in [HRY],

20 21

[LR]. The surface embedding version of the circuit chain method involves finding a face chain joining two non-adjacent vertices x, y and using this to construct an embedding o f G U {xt/}. Suppose 'Pis a closed 2-cell embedding in some surface. Since the sequence of face boundaries in a face chain forms a circuit chain, a face chain is a special kind of circuit chain. But on the other hand, a face chain involves the surface embedding and hence one can hope to apply some topological argument.

The face chains used in [HRY] and [LR] require that any two consecutive faces must have at least one edge in common. Therefore in the resulting embedding one may avoid repeated edges, but repeated vertices are harder to avoid since non-adjacent faces in the sequence may have common vertices on their boundaries. Hence embed dings formed by using such face chains are edge-strong embeddings.

2.2 Vertex version of the face chain method

In order to derive new closed 2-cell embeddings from old ones, we will improve the face chain method used in [HRY], [LR] and elsewhere to its vertex version. We allow here consecutive faces in a face chain to have common vertices but not necessarily common edges.

A n edge e is said to be a monofacial edge of an embedding 'P if it appears in a facial walk twice. A monofacial edge e is consistent if e is traversed twice in the same direction in the facial walk, otherwise, it is inconsistent (all the monofacial edges in an orientable embedding are inconsistent). Similarly, a vertex v is said to be a multiple vertex of a face / if it appears more than once in the facial walk of /.

As usual, the two appearances of the endvertex of a facial walk are counted as one. 22

If a multiple vertex appears in a facial walk only twice, we call it a double vertex.

Define consistent and inconsistent double vertices in a similar way. Consistent and

inconsistent double vertices are well defined because each vertex has a small closed

disk neighborhood which meets the incident edges in initial segments only. These

segments are given a circular order by their intersections with independent of the

choice of the disk neighborhood boundary. Each time a facial walk traverses a vertex

it uses consecutive segments in the order. Two traversals cannot cross in the order

and consequently will be either consistent or inconsistent. Call the monofacial edges

and double vertices of a face / the double-attachments of / .

Let / and g be two attached faces of ty. Assign local orientations to the facial

walks of / and g. The directions of the two sides of each attachment of / and g will

be either the same or opposite. For a choice of local orientations, call an attachment

consistent if the directions are the same and inconsistent if the directions are opposite.

Let a and b be two attachments of / and g. We say these two attachments are of the

same type if they are both consistent or inconsistent under any local orientation, and

otherwise they are of different types.

Let x and y be two vertices. A face chain C = (/i, / 2,..., /„) which joins x and y is a sequence of faces of 'I' such that x 6 d f \,y € d f n and dfi fl £?/,+1 ^ 0, for i = 1,2, ...,n — 1. A face chain is simple if dfi D d fj ^ 0 im plies — j | =1. Clearly any face chain joining x and y contains a simple sub-face chain which also joins x and y. Therefore we may always assume that face chains are simple.

We now introduce the vertex versions of three operations which generalize their 23

edge versions. The edge version of Operation 2.2.1 can be found in [Hag]. The edge

versions of Operations 2.2.2 and 2.2.3 can be found in [HRY] and [LR].

Operation 2.2.1. Let ^ be an open 2-cell embedding of G in £ and e = xy be

a consistent monofacial edge with facial walk xeyP\xeyP2. By viewing the local

embedding as a rotation projection and putting an 'x' on e, we obtain an embedding

of G U {xy} in a surface £'. In Figure 1, the 'x' on an edge means this edge is

twisted; that is, the facial walk will cross the edge from one side to the other side. If an

edge has an 'x' on it originally, after adding another 'x', it becomes an ordinary edge.

The facial walk xeyP \xeyP 2 in is divided into two facial walks xeyPi and xeyP 2 in

VI//, with all other facial walks unchanged. The edge e is no longer a monofacial edge

in VP'. Similarly, if u is a consistent double vertex of face / (so that v appears in this

facial walk twice), and ab...cd...a is the rotation of all edges at v with a, 6, c, d G d f,

we change the rotation projection as shown in Figure 1(b). Again the original facial

walk of / is broken into two facial walks with all other facial walks unchanged. We

obtain a new embedding of G with fewer double vertices.

Remark 1: Suppose a face / of has two or more consistent double-attachments.

If we apply Operation 2.2.1 on one of these consistent double-attachments, all other

consistent double-attachments are either no longer double-attachments or remain con

sistent double-attachments of the new embedding. Thus if an open 2-cell embedding

has no inconsistent double-attachments and no other multiple vertices, then this em

bedding can be transformed, by repeatedly applying Operation 2.2.1, into a closed

2-cell embedding. 24

P P2

\ / / \

Figure 1:

Remark 2: For the vertex version of Operation 2.2.1, the condition that v is a

double vertex is not necessary. In this chapter, it is convenient for us to assume that

v is a double vertex. The case when v is a general multiple vertex will be studied in

Chapter V.

The next operation is similar to the one above except that the edge or vertex is

not a monofacial edge or a double vertex, respectively.

O p eratio n 2 . 2 .2 . Let 4* be an open 2-cell embedding of a graph G and / i , / 2 be

two faces of $ such that d fi fl d f 2 0. Choose an attachment from d fi fl dfi. If this

attachment is an edge, put an 'x' on this edge. If this attachment is a vertex u, and

ab...cd...a is the rotation of all edges at v with a,i 6 d fi and c,d £ dfa, change the

rotation projection as indicated in Figure 1(b). Call such an attachment a passing

attachment from fi to fa. The rotation schemes are similar to those in Figure 1.

Operation 2.2.3. Let 'I' be a closed 2-cell embedding of G in S and x,y be two non-adjacent vertices of G. Let C =(/i, / 2,..., /„) be a simple face chain joining x 25

■•r

y y (b)

Figure 2:

and y. For i = l,...,n — 1, choose a passing attachment from /, to /,+i, and apply

Operation 2.2.2 on these attachments. Change the rotation projection by adding a new edge xy as shown in Figure 2(a) to obtain an embedding 'P/ of G+ = GU {x y } in

S'. The faces / x, ...,/„ of $ are replaced by two faces g\ and § 2 of #', while the other faces of are all unchanged. The 'o' on the edge xy means one 'x' if the number of edges with 'x'’s on the facial walk of g\ (or of <72> whose number of 'x '’s has the same parity as for <71) is odd, and zero 'x' otherwise.

Remark: Note that Operation 2.2.3 may create double-attachments in the new facial walks as shown in Figure 2(b). In this example /, and /;+x have three attach m ents a, b and c. If we use a as the passing attachment, then b is part of the common boundary of two new faces, but c becomes an inconsistent double-attachment of the dotted face. In general, if ^ is a closed 2-cell embedding and Operation 2.2.3 does not create inconsistent double-attachments, then either is a closed 2-cell embedding already, or it can be transformed using Operation 2.2.1 into a closed 2-cell embed- 26

v u u V

Figure 3:

ding. Moreover, if x and y are not on the boundary of the same face in then the

embedding obtained by contracting the edge xy in is also a closed 2-cell embedding.

2.3 Two new operations

We introduce two new operations in this section which can be used to construct new

closed 2-cell embeddings based on given closed 2-cell embeddings.

Operation 2.3.1. Let ^ be a closed 2-cell embedding with x ,y ,u ,v € V(G) and

x y , uv £ E(G). Let G+ = G U {xy, uv}, where xy and uv are two new edges. Suppose

C\ is a simple face chain joining x and u, Ci is a simple face chain joining y and v,

and no face of C\ is attached to a face of Ci. For every two consecutive faces in C\

and C i, choose a passing attachment and apply Operation 2.2.2. Change the rotation

projection by adding two edges xy and uv as shown in Figure 3. The 'o' on the edge

xy is defined by the same rule as in Operation 2.2.3. Similar remarks to those for

Operation 2.2.3 hold here.

Operation 2.3.2. Let 'P be a closed 2-cell embedding of G in a surface E and Figure 4:

x , y ,z € V(G). Without loss of generality, we may assume no two of x ,y ,z are on

the boundary of the same face. Let C\ be a simple face chain joining x and y, and

C 2 be a simple face chain joining z and a vertex on a face boundary of C\ and such

that only the last face of C2 is attached to some face of C\ (if z is on the boundary

of a face in C\ then C2 is not needed). Let u be a new vertex, u i, uy, uz be three

new edges, and G+ = G U {u, u i, uy, uz}. Let /i,/2 be faces of C i,C 2 , respectively,

such that d fi fl 8 / 2 ^ 0, and let e be an attachment edge or vertex between f\ and

f 2. For every two consecutive faces in C\ and C 2 , choose a passing attachment and apply Operation 2.2.2. Change the rotation projection by adding the new vertex u and three new edges ux,uy,u2, as shown in Figure 4. The 'o' on the edge ux means one 'x' if the number of the edges with 'x'’s in the segment of the new facial walk between x and / is odd, and zero 'x' otherwise. The V ’s on the edges uy and uz are defined similarly. 28

/ \ \ /

Figure 5:

2.4 Dual circuits and the face attachment model

In this section we discuss various cases of how two faces are attached in an embedding.

By the remark after Operation 2.2.3, one can see that the way that two faces in a

face chain are attached (actually, in a non-orientable surface) is very important. Two faces in a non-orientable surface can be attached in a very complicated way. Figure 5 gives an example showing that two faces of an embedding in the Klein bottle have four attachments.

In order to discuss the attachment model for two attached faces, we first introduce the concept of “dual circuit”. Let ^ be an embedding of G in the surface E, and /, <7 be two faces of <-?('&). Let a, b € d f fl dg, where a, 6 may be vertices or edges. Put two vertices /* and g * in the faces / and g, and draw lines la and h through a and

6, respectively, joining /* and gm. The circuit ra{, formed by la and /*, is called the dual circuit of / and g through the attachments a and 6. Note that if both a and b 29

' - ' - s H ' t / \ \ r 2

X - M J L J U

~ O C “ (a) (b)

Figure 6:

are edges, then the dual circuit Tab is formed by two parallel edges in the ordinary

surface dual of G.

Let a, b be two attachments of two faces / and g. For any local orientation of the

facial walks of / and g , the following is true.

Lemma 2.4.1. The dual circuit Fab is 2-sided if and only if the attachments a and

b are of the same type (i.e., both consistent or inconsistent).

Proof: Figure 6(a) and Figure 6(b) give the rotation schemes of the two cases. If

a and b are both consistent or inconsistent, r a6 is homotopic to the curve Tj, which

is 2-sided. If a, b are of different types of orientations, Ta6 is homotopic to T2, which

is 1-sided. This proves the lemma. □

Remark: Clearly the above lemma is independent of the assignment of local ori entations.

Now, by using the facial walk to represent the face, we will give a model to describe the arrangement of all attachments between two faces. 30

(a) (b)

Figure 7:

Let fi and / 2 be two faces with n attachments. Draw two circuits representing

the facial walks of f x and / 2, and mark n points on each circuit to represent the n attachments (again, these attachments may be either edges or vertices). Choose an attachment as the first attachment, and give a local orientation to the facial walks of fi and / 2 so that this first attachment is inconsistent (in general, it is not necessary that a first attachment be inconsistent). For each attachment draw a segment if it is inconsistent or a segment with an x if it is consistent to connect the corresponding points on these two circuits. Figure 7 gives an example following these instructions.

Figure 7(a) is a local embedding of two attached faces given by rotation projection, while Figure 7(b) is a model to show in a more clear way how these two faces are attached .

2.5 Good face chains

As mentioned in the remark after Operation 2.2.3, applying that operation may create inconsistent double-attachments in the resulting embedding. Sometimes one may change the choice of the passing attachment to avoid this. However, in some cases, 31 one cannot find suitable attachments in a face chain to avoid double-attachments in the resulting embedding. Our purpose in this section is to find so called “good” face chains and apply some of the operations in Sections 2.3 and 2.4 to derive new closed

2-cell embeddings.

Let E be a surface and ^ be a closed 2-cell embedding of G in E. Let C =

( / i i •••> fn) be a simple face chain joining x and y , w ith x, y e V(G) and x y 0 E(G).

The face chain C is good if the resulting embedding obtained by applying Operation

2.2.3 on C has no inconsistent double-attachments; otherwise it is bad. Let /,• and fi-fi be two attached faces in C. If there is an attachment e of /,• and /l+i such that using e as a passing attachment will not create inconsistent double-attachments, then we say C is good at this part of C. Let C be a face chain joining x and y. In general the attachments of and /,• and the attachments of /,• and /<+i may be in no particular order on the face boundary of If dfi is a disjoint union of two paths

P i and p i+i such that dfi-1 f id / , C p, and d/. n d / .+ i C p1+1, then we say dfi-\ D dfi and dfi fl d/,+i are separated on dfi, otherwise they are alternated on dfi. A closed

2-cell embedding is called a 2 -attachment closed 2-cell embedding if any two faces have at most two attachments.

Let C be a simple face chain in a closed 2-cell embedding. Suppose there exists three consecutive faces and /,+1 in C such that ||d/i_ind/,j| = ||d/,nd/i+i|| =

2 , dfi-1 Cidfi and dfi nd/,+i are alternated on dfi, with the attachments of /,_2 and fi+2 placed as in Figure 8, and such that two attachments of dfi-1 H dfi (as well as dfi fl d fi+i) are of different types. Then there does not exist a suitable choice 32

fi+2

f i-2

Figure 8:

of passing attachments from dfi-\ H dfi and dfi fl dfi +1 for us to apply Operation

2.2.3 and obtain an embedding without inconsistent double-attachments. Hence such

a face chain is bad.

In a given face chain, two consecutive faces can be attached in a very complicated

way which often creates inconsistent double-attachments when applying Operation

2.2.3. However, the attachments of two faces in an orientable face chain is not so

important. We have the following theorem.

Theorem 2.5.1. Let be a closed 2-cell embedding and C be a simple face chain

in 'P. Denote by U(C) the union of the closure of all faces in C. If U(C) C 2 does not contain an orientation reversing simple curve, then C is good.

Proof: After applying Operation 2.2.3, all the faces in C, together with the new edge xy, have turned into two faces

faces remain unchanged. Our only concern is whether the face boundaries of g\ and

<72 are simple circuits or not. If not, suppose g\ has inconsistent double-attachments.

Since the face chain is simple, these double-attachments must be components of the common boundaries of two consecutive faces, say /, and /,+i. By the construction, the part of the facial walk of gi from /, to /,-+1 contains only one 'x'. Since U(C) is orientable, all the attachments of /, and /,+i are inconsistent. Therefore double attachments in <71 is consistent, and by applying Operation 2.2.1 we separate the facial walk g\ into two facial walks. Note that if there are other consistent double attachments in the facial walk of g\, then after we apply Operation 2.2.1, these double attachments are no more double-attachments or they remain consistent. Hence C is good. □

The next two corollaries follow immediately.

Corollary 2.5.2. Let ^ be a closed 2-cell embedding in an orientable surface. Then any simple face chain C in *P joining non-adjacent vertices x and y is good.

Corollary 2.5.3. Let G be a 2-connected graph with x,y £ V(G) and xy E(G).

Let G + = G U { iy } be a new graph obtained by adding a new edge xy to G. If there exists a closed 2-cell embedding *P of G in some orientable surface, then there exists a closed 2 -cell embedding VP' of G+ in some non-orientable surface.

Remark: If x and y are not on the boundary of the same face in *P, then the embedding obtained by contracting the edge xy in 'P/ is also a closed 2-cell embedding.

The behavior of a simple face chain in a non-orientable embedding is much dif ferent. Generally, it is hard for a simple face chain in a non-orientable embedding to 34 be good. However, under certain circumstances, we are still able to find good simple face chains. We use attachment instruction to discuss how to implement Operation

2.2.3.

Let C = (/i, ■•.,/„) be a simple face chain of an embedding ^ which joins two non-adjacent vertices x and y. Let /,_i,/,+i be three consecutive faces in the face chain C. Assume we have already chosen all passing attachment from fj to fj+\ for j = 1 ,2 ,..., i — 2. For the face pair of /,•_i and /,-, call the passing attachment between fi—2 and fi—i the entry, and the passing attachment between /,- and / l+i the exit. The question is how to find a suitable passing attachment between /,•_i and /,■ to avoid inconsistent double-attachments in new facial walks. We state the following facts for better understanding.

(2.5.1) We know that Operation 2.2.2 will add an 'x ; in the facial walk. Under the attachment description, if the passing attachment is inconsistent, after Operation

2.2.2, it becomes a segment with an 'x', and therefore the new facial walks should cross the passing attachment. If the passing attachment is consistent, namely, with an 'x' on it, then after Operation 2.2.2, two 'x's will cancel out, and so the passing attachment is a segment without an 'x', and therefore the new facial walks will not cross the attachment.

(2.5.2) We may choose two passing attachments between two consecutive faces. Sim ply by observation, these two passing attachments must be of the same type; both are segments with 'x'’s or segments without 'x ;’s. The four possible two passing attachments are shown in Figure 9. If we use two passing attachments, the resulting 35

i-...... Il'& 'jO (C-~ " " " 7 - j ) : 1 : ; w* • • i • i *.\S ......

Hi"

Figure 9:

embedding will have more than two faces. Only two of these facial walks use the

piece of arc from each facial walk in the face chain, with x and y on their facial walks.

All the other facial walks are closed locally (by the pieces of arc from two consecu tive facial walks). The dashed and two dotted curves represent new facial walks (see

Figure 9).

(2.5.3) The resulting facial walks may be not simple; i.e., they may have double attachments. If they are consistent, we can reduce them easily by Operation 2.2.1.

If they are inconsistent, then the double-attachments are caused by a consistent at tachment if the passing attachment is inconsistent, or by an inconsistent attachment if the passing attachment is consistent.

Therefore, we immediately have the following three lemmas.

Lemma 2.5.4. Let C = (/i,/,+ i,..., /„ ) be a simple face chain in tf. If all the attachments of fi and /,+ 1 are of same type, then C is good at this part.

Lemma 2.5.5. Let C = ( / i , ...,/ , _ i , /,-, / l+i , ...,/„ ) be a simple face chain in ty. If

Ci = (f i, ..., /;_i) and C2 = ( /l+i , ...,/„) are two good sub-face chains, and all the 36

attachments of /,_ i and fi are of same type, and all the attachments of fi and /,•+\

are of same type, then C is a good simple chain.

Lemma 2.5.6. Let C = ( f \ , ..., f i , ..., f n) be a simple face chain in 'If. If for some i

the sub-face chain C\ = (/i, ...,/i) and C 2 = (fi, ...,fn) are good, and dfi - 1 D d fi and

dfi fl d fi+ 1 are separated on dfi, then C is a good simple face chain.

If the attachments of two consecutive faces in a face chain are of both type, the

case is much harder. We don’t have general results about this. The next lemma will

discuss the cases that two faces have small number of attachments.

Lemma 2.5.7. Let be a closed 2-cell embedding of G in some surface. Let x,y

be two non-adjacent vertices and C = (f \ , . ..,/„ ) be a simple face chain joining x

and y. For a given i, if ||dfi fl d /,+ i || < 4, then with the exception of one case, by

choosing suitable passing attachments, Operation 2.2.3 will not create inconsistent

double-attachments at this part in the resulting embedding.

Proof: Assume we have located the entry and all the points representing these

attachments of /,• and / I+j. Each attachment segment / has one end in the facial walk

of fi and one end in the facial walk of /,+i. Call these the bottom end and top end,

respectively. By our convention, we assume that the first attachment segment is the one without an 'x'.

Case 1: The exit w is on the right of the top end Ui of the first segment /j.

Use lx as the passing attachment. It is clear that the resulting facial walks are simple at this part. The local rotation projection is shown in Figure 10. Two new facial walks are represented by the dotted curve and the dashed curve. 37

Case 2: The exit w is on the left of the top end uj of the first segment /j.

If we use /i as the passing attachment and the resulting facial walk has an inconsis

tent double-attachments, then by the remarks in (2.5.3), there must be an attachment

segm ent l2 with an 'x' on it whose top end u2 is on the left of w and the bottom

end v2 is also on the left of Uj. Now we use l2 as the passing attachment. If the

resulting facial walks have an inconsistent double-attachment, there must be an or

dinary attachment segment /3 in one of the three cases in Figure 11. We have now

three attachment segments, and one of them is an attachment segment with an 'x'.

In the cases of Figure 11(a) and 11(b), use two ordinary attachment segments as

two passing attachments; and in the case of Figure 11(c) use the third segment /3

as the passing attachment. By the same argument, if the result fails to be a closed

2-cell embedding, then there must be an attachment segment with an 'x' appearing

in one of the thirteen sub-cases shown in Figure 12. Among these thirteen sub-cases,

the first four are illustrated in Figure 11(a), the second four in Figure 11(b) and the last five in Figure 11(c). The choices of the passing attachments in some cases are 38

■ V_J V _ 1 w* 3) ( C IW ) C 12 b 11 12 13 11 ll 12 ’v C ) c D c )

(a) (b) (c)

Figure 11:

not unique, however, except for the case of Figure 12(m), we can find a single (or a

pair of) attachment segment(s) with an 'x' (or 'x'’s) as the passing attachment(s) to

apply Operations 2.2.2 and 2.2.3. In Figure 12, we put a ' □ ' to identify the passing

attachment. It is easy to check that the resulting embedding is simple at this portion.

Thus the lemma is true. □

As mentioned earlier in this section, even though any two attached faces have

no more than 4 attachments and without the case of Figure 12(m), it is sometimes

still impossible for us to apply Lemma 2.5.7 successively in the face chain to obtain

a new closed 2-cell embedding. The problem arises from the mixed attachments of

d fi -1 C\dfi and dfDdfi+i on df, as shown in Figure refFigure2.5.1. However, if the

attachments dfi -1 C\dfi and d fn df+ i are separated on dfi for i = 2 — 1, then

the following lemma is obvious simply by applying Lemma 2.5.7 successively.

Lemma 2.5.8. Let be a closed 2-cell embedding of G in some surface, and x and y be two non-adjacent vertices. Let C = ( / i ,.be a simple face chain joining x and y with ||dfi fl 0 /,> i || < 4 and not satisfying the case of Figure 12(m). If for each 39

C :> c C :>

C xu j s i uu is (a) (b) (c) (d ) C :> c :> K E E £ z r C J c (e) ( f ) (g) (h ) C . i-7—U C u n 11 — C i. _ Z p) C c i . U ) C... i .

ciSi _mJ c x i :> cn . (O ( j ) (k) (1) (m)

Figure 12:

i, d fi-\ fl dfi and dfi fl dfi+\ are separated on dfi, then C is a good simple face chain joining x and y.

The significance of Lemma 2.5.8 is that for an embedding of a 3-connected graph, if the surface has small orientable or non-orientable genus, then two faces cannot have a large number of attachments.

If an embedding is a 2-attachment closed 2-cell embedding, then the next lemma will show that the case shown in Figure 8 is the only reason to make a face chain to be bad.

Lemma 2.5.9. Let C = (/i,.••,/«) he a face chain such that any two consecutive faces have no more than two attachments. If there are no three consecutive faces 40

and /,+ 1 with the attachments as shown in Figure 8, then C is a good face chain.

Proof: If for each i, d fi -1 fl dfi and dfi fl c?/,+i are separated on d fi , then by

Lemma 2.5.8, C will be a good face chain. Therefore, if C is bad, there exists some i s such that d fi -1 fl dfi and dfif)dfi +1 are alternated on dfi, with both dfi -1 fl dfi and dfi fl <9/,+i having two attachments.

Let dfi -1 fl dfi = {a, 6} and dfi fl dft+i = {c,d}. Suppose we choose a as the passing attachment. If a and b are of same type attachments, then when applying

Operation 2.2.3, either b is not a double-attachment, or b is a consistent double- attachment, which will lead to a closed 2-cell embedding by applying Operation 2.2.1.

Lemma 2.5.9 is true. □ CHAPTER III

Some Topological Results

In this chapter we derive some topological results for use later. We prove some

interesting properties of essential simple curves on the torus and the torus with a

disk removed. We will also classify all non-homotopic nonseparating simple curves

in the Klein bottle and all non-homotopic 2-sided nonseparating simple curves in

./V3, as well as the homotopic intersection number between any two of them. These

properties are developed not only in preparation for later chapters, but also for their

own interest from the topological point of view.

3.1 Classification of nonseparating simple curves in the Klein bottle

In this section we will prove a topological theorem which will classify all non-homotopic

nonseparating simple curves in (the Klein bottle).

Theorem 3.1.1. Let the surface be N 2 , then

(3. 1.1) there is only one homotopic class of2 -sided nonseparating simple curves, and

any of them passes through each cross-cap exactly once;

(3.1.2) there are only two homotopic classes ofl-sided (and therefore nonseparating) simple curves, and each of them passes through only one cross-cap and passes through

Proof: The Klein bottle can be viewed as the sphere with two cross-caps. A non-null-homotopic simple curve T must pass these two cross-caps a couple of times.

Since in the Klein bottle any two consecutive passes over one cross-cap is homotopic to a simple curve without these two passes (not necessarily true if the surface has more than two cross-caps), the passes over these two cross-caps must alternate. For a 2-sided simple curve, the number of passings over the cross-cap must be even and for a 1-sided simple curve, the number of passings must be odd.

Clearly, the simple curve C\ in Figure 13(a) which passes each cross-cap once is a 2-sided nonseparating simple curve, and the simple curves C2 and C3 which pass through the cross-cap A and B. respectively, are two non-homotopic 1-sided simple curves. We will show that these are the only three non-homotopic nonseparating simple closed curves in N2 .

Let f be a 2-sided simple curve in the Klein bottle. Suppose the total passing 43

Figure 14:

number (over two cross-caps) is greater than two, and the first pass is over the cross

cap A (see Figure 13(b)). After T passes the cross-cap B and comes back to A, if

view the whole surface as a sphere (i.e., replace two cross-caps by two disks), then

T separates the sphere into two areas, shaded and unshaded. Assume T starts at

unshaded area. After the second pass over A , in order to keep T simple, all passes

over B must be from unshaded area to shaded area, and passes over A must be from shaded area to unshaded area. Therefore, after the total even number of passes, T always falls into shaded area, and we are not able to close the simple curve T. Thus

T only passes each cross-cap once.

Since T passes through A and B alternately and the total passing number over A and B is odd, any 1-sided simple curve can only passes through one cross-cap and the passing number is one. □

Lemma 3.1.2. Let Ti and be two 1 -sided simple curves in the Klein bottle. If

Ti intersects T2 only once, then Tt and T2 pass through the same cross-cap as shown 44

in Figure 14. Furthermore, T) U P2 separates the surface into two parts, with one

part homeomorphic to a disk and the other part homeomorphic to a disk containing

a cross-cap in it.

Proof: By Theorem 3.1.1, we know that there are only two homotopic classes of

1-sided simple curves and each of them passes through only one cross-cap. If Tj and

r2 pass through different cross-caps, then Tj and T2 are disjoint up to homotopic.

Therefore they must pass through same cross-cap as shown in Figure 14. Clearly,

P] U r 2 separates the surface into two parts, with one part homeomorphic to a disk

and the other part homeomorphic to a disk containing a cross-cap in it. □

3.2 Homology and homotopy intersections of T and T~

The torus T, as the “first” non-sphere orientable surface, has some interesting proper

ties. First of all, it’s first homology group and fundamental group are same. Secondly,

we will prove in this section that the algebraic intersection and geometric intersection

on the torus are same. Moreover, all these hold if the surface is T ~ , the torus with a

disk removed.

Let 7Ti(S), H i(E) and [S'1, S] denote the fundamental group, first homology group

with Z coefficients and the set of free homotopy classes of closed simple curves in surface S, respectively, we have:

(3.2.1) tt \{T) — H\(T) = Z ® Z.

(3.2.2) (n, b) £ Z 0 Z can be represented by a simple curve if and only if a and b are

co-prime ([ZVC], page 93). 45

(3.2.3) [S'1,T] = Z@Z (this is because [5'1,T] is the set of conjugacy classes of K[(T)

and m(T)= Z 0 Z is abelian).

In topological graph theory, essential circuits in an embedded graph are viewed as elements in [-S^S] up to orientation. When we say two circuits are homotopic we mean they are freely homotopic. Because of the above facts, we are able to use homology theory to solve some of the free homotopic problems in the torus and some related surfaces.

In homology theory, the intersection number of two simple curves is the number of the transverse crossing between these two simple curves up to the cancellation of positive and negative directions induced by any orientation. We call this type of intersection a homology intersection ([ZVC], page 87). In some books, it is called an algebraic intersection. Let Cj, Cj € H \(T ) = Z © Z be any two simple curves represented by ( a,b) and (c, d), denote Cj • Cj as the homology intersection number, we know ([ZVC], page 89)

(3.2.4) Ci • Cj —\det (® * ) |.

The intersection of any two (free) non-homotopic simple curves is defined to be the number of transverse crossings up to eliminating any trivial intersections (the arcs between two intersections of two curves bound a disk). We call this intersection the homotopic intersection (in some books, called the geometric intersection). Generally, the number of homology intersection and the number of homotopy intersection of two simple closed curves are different. But the next lemma will show that they are the same when the surface is the torus or the punctured torus (homeomorphic to the 46

torus with a disk removed).

Lemma 3.2.1. Let C \,C i be two essential simple curves in T and T~ (torus with a

disk removed). Then the homology intersection number and homotopy intersection

number of C\ and C 2 are same.

Proof: Clearly, any homology intersection is also a homotopy intersection. There

fore, it suffices to show that for any given C,, Cj with minimal homotopy intersections,

all intersections have same induced orientation (therefore there is no cancellation of

positive and negative directions when counting the number of homology intersections).

Suppose there are some intersections having different orientation. Fix the direction

of Ci, we can find two consecutive intersections x and y on C 2 such that C\ and C 2

have different orientation on these two intersections. Since C\ is 2-sided, say left side

and right side, if the cross of C2 over C\ at x is from left to right, then the cross of

C2 over C\ at y must be from right to left.

First we assume the surface is the torus T. Cut the surface along C\, we get a

cylinder. Now it is easy to see that the intersections x and y are trivial and we delete

them. Thus the lemma is true for the case of T.

Now we suppose the surface is T~. Since two disjoint homotopic closed simple curves in any surface bound a cylinder (Lemma 1.1.1), the homotopic intersection

number are invariant among the homotopic. class. Observe that two homotopic essen tial simple curves C and C' are homotopic in T~ if and only if they are homotopic in

T. This is because if we view C and C' as homotopic simple curves in T , C and C' bound two cylinders and the boundary of T~ only sits in one of these two cylinders. 47

Now the lemma follows immediately for the case of T~. □

Lemma 3.2.2. Let T~ be the torus with a disk removed, then there is an one-to-one

correspondence between essential simple curves in T and nonseparating simple curves

(which must be essential) in T~

Proof: Clearly, any essential simple curve in T is an essential simple curve in T~,

and any two homotopic essential simple curves in T are also homotopic in T~. On

the other hand, any nonseparating simple curve in T~ is also a simple curve in T.

It must be essential because any null-homotopic simple curve in T is either a null-

homotopic simple curve in T~ or an essential separating simple curve in T~ (separates

the boundary and the rest part of T~). Therefore, we have one-to-one correspondence

between essential simple curves in T and non-separating simple curves in T~. □

Now we need to find an upper bound of the number of non-homotopic nonsepa

rating 2-sided simple curves, which intersect each other at most once. We first give

the same bound for those simple curves in the torus.

Let H\(T)=Z © Z be the first homology group of the torus. Any simple curve in

Hi (T) can be represented by a pair (a, 6) with a and b co-prime. Let ( a,b),(c,d ) 6

Z © Z, and (a, b) • (c, d) denote the number of homology intersection of (a, b) and

(c, d). We know that (a, b) • (c, d) = |det |. We have the followings:

Lemma 3.2.3. Let C\ , ..., Cn be different elements in H \(T)= Z © Z with

(3.2.5) Ci's are unimodular, i.e., C, = (a,-, £>,■) with a,- and bi co-prime, i = 1,2, ...,n;

(3.2.6) C,»Cj < 1, 48 then n < 3.

Proof: It is well known that C E Z © Z is unimodular if and only if 3 g € GL2(Z) such thatC • g =(1,0). Moreover, for any g € GL2(Z)

(Ci • g) • (Cj • g) = |det ( ^ | = |def ( = | det \\det(g)\ = C, • Cj

i.e., the homology intersection number is invariant under the right multiplication of elements in GL2(Z).

Now suppose C i,..., Cn € Z ® Z satisfy (3.2.5) and (3.2.6).

If C, • Cj = 0, then C, and Cj are multiples of each other and they cannot both be unimodular. Hence we must have C< • Cj — 1 for any i and j. Therefore if g = e GL2(Z) then Ci • 7<- 1,...,C „ • g _1 also satisfy (3.2.5) and (3.2.6). But

C,-<7_1 = (1,0) and Cj-g~x = (0,1). Hence, without loss of generality, we may assume

Ci =(1,0) and C2 = ( 0 ,l). For each C,- = (ai,6,) we have |det ^ 6 ) ^ = ^ = * an<*

|det ^ ^ | = |a,-1 = 1. Thus the only possibilities for C3, ..., Cn are (1,+ 1) (since we identify C,- and — C,, we can always take the first coordinate to be positive). As

(1,—!)•(!,1 ) = 2, we can’t have both of them. Therefore n < 3. □

The next lemma will show that the union of two nonseparating simple curves in

T or T~ sometimes separates the surface, which allows us to discuss problems in a lower surface.

Lemma 3.2.4. Let S be T or T~, and Ci,Cj be two 2-sided non-separating simple curves in E such that Ci • Cj = 2. Then Ci U Cj separates the surface into two con nected components as shown in Figure 15. For the case ofT~, one of the component 49

/ V / \

(a) T (b)T-

Figure 15:

contains the boundary.

Proof: Since H i(T) = H\{T~) = Z®Z, if C ,= (1,0) and Cj=( 1,2), then the lemma

is true. It suffices to show that for any (7, , Cj G Z © Z such that C,- • Cj=2, then

3g € GL2(Z ) so that C, • g = (1,0) and Cj ■ g = (1,2).

Suppose Ci = (a, b) and Cj = (c, d). Form a matrix ) = The ele

mentary transformation of adding the multiple of one column to another is in GL 2 {Z)

and this transformation will not change the co-primeness of the entries of the same

row. Since a and b are co-prime, by a series of above elementary transformations,

we can make the first row as (1,0) and the entries of the second row are still rela

tive prime. Since |det ^ | = 2 and the determinant of the above elementary

transformations is 2, we know that the second entry of the second row must be 2.

Therefore the first entry of the second row must be odd. Adding a multiple of the second column to the first column gives □ 50

3.3 Classification and properties of simple curves in N%

In this section we classify all non-homotopic 2-sided nonseparating simple curves in iV3, as well as the homotopic intersection number between any two of them. Our argument is very much dependent on the structure of the torus T.

In this section, we always view N3 as T#P, the connected sum of a torus and a projective plane. The first theorem shows an interesting property of the nonseparating sim ple curves in N3. The analogue is not true for non-orientable surfaces with more cross-caps.

Theorem 3.3.1. Write N3 as T#P, then

(3.3.1) If C is a 2 -sided nonseparating simple curve (homology non-null), then C does

not pass through the cross-cap.

(.3.3.2) If C is a 1-sided simple curve, then C passes through the cross-cap exactly

once.

Proof: As shown in Figure 16, we use a circle L to represent the cross-cap in

Ar3 with antipodal points identified. Let C be a 2-sided nonseparating simple curve in N 3 . Suppose C passes through the cross-cap a number of times. In order to keep C 2-sided, C must pass through the cross-cap even many times. To avoid the trivial passing through the cross-cap, C must travel back and forth between the torus and the cross-cap. Let S={a’i..., x„, y \,..., yn) be all the end points of C on L, and

C i,C 2, ...Cn be all sections of C in the torus part such that x,-,?/, be end points of C,-

011 L ,i = 1,2, ...,7i. Every two points in S which are antipodal on L are identified. 51

Figure 16:

The question now is how to pair these end points and join them by C,. For the end points {a:,-, r/,}, draw a straight line in the removed disk to join a:,- and ?/,- and close C, to form a closed simple curve C f in T. A ll the C f ’s are homology non-null elements in T . C simple implies C f and C f have at most one nontrivial intersection, and whenever C f and C f have an intersection, it must occur in this removed disk. Since in a torus any two essential simple curves do not intersect if and only if they are homotopic, it implies that if C f and C f are two homotopic elements, then Xj and yj must be in the same part of L separated by X{ and ?/<. Therefore, if Cf,...yCf are homotopic, all end points of Cf. on L must be in the following order on L.

(3.3.3) x ilXiJ...xi,y i....yi:iyi1

Case 1: There is only one homotopic class among all C.^’s.

The order of all points in S on L is XiX2 -..xnyn...y 2 yi. Since C passes through

the cross-cap even number times, n is even. The pair and {ari+i, y^+i} will separate all other end points on L. Moreover, is and y^+i, as well as x±+i and j/i,

are antipodal points. Therefore we obtain a simple curve which is the union of Co. 52 and Cn.+\.

If n > 2, this is a contradiction since Ca U C n+) C C and C is a circuit. 2 2 T

If n = 2, since C f is homotopic to C f in T, they bound a cylinder in T. This im plies that C is a separating simple curve in N3 , which also contradicts our assumption.

Thus (3.3.1) is true in this case.

Case 2: There are more than one homotopic classes among all Cf's.

First we look at Cf and all C.^’s which are homotopic to Cf. Without loss of generality, say C,+, C f,..., Cf are homotopic to each other with their end points x i, x 2, ..., x,-, j/i, j/2»--M Vi being arranged on L in the order as in (3.3.3). Let Cf+X be the first curve which is not homotopic to C f. Since the homotopy intersection number between different homotopic classes is invariant, Cf+l and C f have only one intersection, and so x,+i can not be on the section of L between any xa and x»+i, for s = 1, ...,i — 1 (otherwise we can find j such that Cf and Cf are homotopic but

Cf has no intersection with Cf+i). Therefore x,-+i must be on the section between x\ and j/i, or x, and t/,-. Suppose xt+j is on the section between X\ and y\. This implies that 7/,+i is on the section between x, and y,-. All the C f’s which are homotopic to

Cft must have x* and ytt on the same side of xl+i and y,+i, with x* on the section between Xi and t/i, yk on the section between x,- and y,-. Namely, all x’s and y’s on

L are separated in groups in terms of homotopic classes. For any i, xt- and y,- will be separated by all x ’s and y’s of different homotopic classes, and all x ’s (as well as y’s) in the same homotopic class will be on L one after another in the order as in (3.3.3).

Now it is easy to see that if we identify antipodal pairs of x’s and y’s, we get 53

more than one simple curve, which contradicts the assumption. Thus C can not pass

through the cross-cap. This proves (3.3.1).

Note that any 1-sided simple curve in T#P must pass through the cross-cap an

odd number times, and so by a similar argument, we can prove (3.3.2). □

Since H\(T~) = H\(T ) = Z © Z, all simple curves in T~ are 2-sided and they are

nonseparating if and only if they are non-null in By Lemmas 3.2.1, 3.2.2

and 3.2.3 and Theorem 3.3.1, we have

Theorem 3.3.2. Let C \ ,..., Cn be a set of 2 -sided nonseparating and non-homotopic

simple curves in N,3. If the homotopic intersection number for any pair is at most

one, then n< 3.

For any 1-sided simple curve C in N3, we define C* in T as the following: By

Theorem 3.3.1, C passes through the cross-cap exactly once. Let the antipodal points

be x and y. By viewing the cross-cap as a disk and drawing a straight line through

the disk to join x and y, we obtain a simple curve C" in a torus. Hence we can

establish an one-to-one correspondence between the 1-sided simple curve C in N 3

and the simple curve C * in T.

Lemma 3.3.3. Let C be an 1 -sided simple curve in N 3 = T#P. If C* is null-

homotopic in the torus, then the resulting surface obtained by cutting N 3 along C is

T~.

Proof: Let N 3 be the surface obtained by cutting N 3 along C. The curve C being 1-sided implies that the boundary of is a simple curve. By counting the 54

Figure 17:

Euler characteristic, N3 is either T~ or N 2 with a disk removed. Suppose N3 is non-

orientable, then there must exist an 1-sided simple curve C\ in N'z which is disjoint

from the boundary of N'z. Any 1-sided simple curve C\ in N '3 is also an 1-sided simple

curve in ;V3. C\ being disjoint from the boundary of implies that C\ is disjoint

from C in JV3. Since C * is null homotopic, any 1-sided simple curve in N3 must

intersect C, which is a contradiction. Therefore N3 must be orientable; i.e., = T~.

□

The next four lemmas will show that the union of two nonseparating simple curves

in N 3 sometimes separates the surface, which allows us to discuss problems in a lower surface.

Lemma 3.3.4. Let Ci,Cj be two 2-sided non-separating simple curves in N 3 such

that Ci • Cj = 2. Then C, U Cj separates the surface into two connected components, with one of the component containing a cross-cap, as shown in Figure 17.

Proof: By Theorem 3.3.1, all 2-sided simple curves do not pass through the cross- cap. Therefore we can treat iV3 as T~ by viewing the cross-cap as a hole. By Lemma 55

r, ///////////////// //////////// // / // / // > . ✓✓✓✓//✓✓/✓/ >//✓////////. //////////// V / / / / / / / / ’/ / / / / / / / , y / y y / y / ^ '////////, ... 777 . '///////////

V //////// 1 '/////////^///////

(a) (b)

Figure 18:

3.2.4, this lemma is true. □

Lem m a 3.3.5. Let Ci and C2 be two 1 -sided simple curves in N 3 with C\*Ci = 1. If both Cf and Cf are essential in T, then either CJ1 and Cf arehomotopic or C f• C f = 2.

Proof: Ci • C2 = 1 implies Cf • Cf < 2. If Cf • Cf = 1, by removing a disk centered at Cf fl Cf and replacing this disk with a cross-cap, we get Ci and C2 in iV3 with Cf • Cf = 0, which is impossible. Therefore Cf • Cf = 0 or 2. □

Lemma 3.3.6. Let Ci and C2 be two 1-sided simple curves in jV3 with Ci • C2 =

1. If both C f and C f are essential, then C\ U C2 separates N 3 into two connected components. Either both components are open disks each containing a cross-cap, or one component is a open disk and the other component is an open disk containing two cross-caps.

Proof: By Lemma 3.3.5, we know that either Cf and Cf are homotopic or Cf»Cf =

2. If Cf and Cf are homotopic, since Hi(T) = Z®Z, we may assume Cf is (1,0). It is easy to see that Cf and Cf have two trivial intersections and C1UC2 separates N 3 into 56

//sr.

*7 7 7 7 7 7 7 7 7 ?

Figure 19:

two connected components, one is an open disk and the other is an open disk with

two cross-caps (see Figure 18(a)). If Cf »Cf = 2 then we may assume that Cf = (1,0)

and Cf = (1,2) (same argument as we did in Lemma 3.2.4). Remove a disk centered

at one of the intersection of Cf and Cf, and identify its antipodal points (i.e., to

cap off it with a cross-cap), we see that C\ U C2 separates N3 into two connected

components, both are open disks each containing a cross-cap (see Figure 18(b)). □

L e m m a 3 .3 .7 . Let C \, C2 and C3 be three essential 2 -sided simple curves in N3 with

Cj • Cj = 1,1 < i < j < 3, then C\ U C2 U C3 separates N3 into three connected

components, two o{ them are open disks and the third one is an open disk containing

a cross-cap (see Figure 19).

Proof: By the similar argument to those in Lemma 3.2.3 and 3.2.4, we know that

C i, C i and C3 can be represented as (1,0), (0,1) and (1,1) in Z ® Z. Thus the lemma is true. □ CH APTER IV

Closed 2-Cell Embeddings of Lower Surface Embeddable Graphs

By a lower surface we mean a surface with a small number of handles or cross-caps,

or equivalently, a surface with higher Euler characteristic. The lower surfaces we will

discuss in this chapter are T (the torus), S? (the double torus), P (the projective

plane), N 2 (the Klein bottle) and Nk, for 3 < k < 5. A graph is called toroidal or

double toroidal if its minimum orientablc surface is the torus or the double torus.

Similarly, a graph is called Nk embeddable if its minimum non-orientable surface is

Nk- In this chapter, we prove that any Ar5 embeddable graph has a closed 2-cell

embedding in some surface. The main result is in Section 4.6.

4.1 Reductions

In this section we make some reductions on the connectivity of the graphs. These re ductions apply to any 2-connected graphs. However, since in this chapter we are only able to discuss the graphs with high Euler characteristic, when we make reductions we have to be careful not to increase the genera of the minimum orientable surface or non-orientable surface of the resulting graphs.

It is obvious that a necessary condition of a graph G having a closed 2-cell em-

57 58

bedding is that G does not contain any cut vertex. In particular, the graph does not

contain any loop. Furthermore, we may assume that G is simple, because if we have

a closed 2-cell embedding of the underlying simple graph for G , we can subdivide the

faces by the multiple edges and get a closed 2-cell embedding of G.

N ext, if G has a non-trivial 2-vertex separation, say G = Gi U G 2 , G\ f) G? =

{x, y } C V(G), we may separate G at x and y and add a virtual edge e to G\ and G2

to form G\ and G2. This will not increase the genera of the resulting graphs. If we

have closed 2-cell embeddings of G\ and G2 in the surfaces Ei and E2, respectively,

vve may remove the edge e from Ei (E2) by deleting the interior of small disks around

c and then identifying the two boundaries (the disk-sum of and E2 at e). The

resulting embedding is also a closed 2-cell embedding. Therefore we may assume that

graph G is topologically 3-connected (it may contain some divalent vertices, which

have no elTect on finding the closed 2-cell em bedding).

Now suppose G has a non-trivial 3-vertex separation, say G = G\ U (?2 , G\ flG^ =

{x,y,z} C V{G), where G\ and G2 each contain circuits. Separate G at x,y,z and

add a new vertex u and three new edges ux,uy,uz to G\ and G'2 to form G\ and G2,

respectively. This reduction may increase the genera of the resulting graphs. We call

such a 3-vertex separation good if the genera of the resulting graphs are not greater

than the genus of the original graph, otherwise we call that 3-separation bad. We perform the above mentioned reduction for every good 3-vertex separation and leave the bad 3-vertex separations alone. If we have closed 2-cell embeddings for both G[ and G 2 in surfaces Ei and Ej, respectively, we can obtain a closed 2-cell embedding 59

of G by removing the interior of small closed disks around the new vertex and edges

in Ei, E2, and taking the disk-sum of E and E2 over these disks. When taking the

disk-sum, the order of x ,y and z on the boundary of the disk in Ei must match the

order of x,y,z on the boundary of the disk in Ej. If the orders do not match, we

can easily make them same by taking the 'm irror1 embedding in E2. The resulting

embedding is also a closed 2-cell embedding of G.

In general, if we drop the requirement of not increasing the genera of resulting

graphs, we know that the discussion of the existence of closed 2-cell embedding of

a 2-connecleu graph can be reduced to the case of cyclically 4-connected graphs. A

cyclically 4-connected graph is a graph without such a 3 vertex-cut for which each

component (separated by this 3 vertex-cut) contains a circuit.

4.2 Closed 2-cell embeddings of some planar graphs with new edges

In this section, we show the existence of closed 2-cell embeddings of some planar

graphs with new edges.

Lemma 4.2.1. Let G be a topologically 3-connected planar graph with possible

divalent vertices x,y,z € V(G). Let G+ be the graph obtained by adding a new

vertex u and three new edges ux, uy and uz to G. Then there is a closed 2 -cell

embedding of G + in some surface.

Remarks: We may also obtain closed 2-cell embeddings by performing any of the following: (1) contracting one of the edge ux, uy or uz; (2) contracting the edges ux 60

and uy if x and y are not on the boundary of the same face of ty; (3) contracting the edges ux , uy and uz if x,y and z are not on the boundary of the same face of 'P.

Proof of Lemma 4.2.1: If we ignore the divalent vertices, G is 3-connected. There fore G has a unique plane embedding. Let be the unique plane embedding of G, which is a closed 2-cell embedding.

If x, y and z are on the boundary of the same face, place the vertex u in that face and draw the edges ux, uy and uz in that face.

If only x and y are on the boundary of a face, add in the path xuy in that face and add uz by applying Operation 2.2.3.

If no two of x,y,z are on the boundary of the same face, choose a simple face chain C\ = (/i, •••,/n) that joins x and y. If 2 is on the boundary of a face in chain

C\, apply the degenerate form of Operation 2.3.2 (the degenerate form of Operation

2.3.2 means that 2 is on the boundary of the face chain joining x and y, or 2 is in the middle of an attachment edge). If 2 is not on the boundary of any face in C\, then since the embedding 'I' is unique and has a wheel neighborhood, any two faces have at most one attachment. Therefore U /, is simply connected and A = E — U/,- is connected, and we can find a simple face chain Ci = ( y i ,... ,gm) in A joining 2 and

C\. By minimizing the sum of the lengths of C\ and C2> gm will meet at most two faces in C\, say /, and /,+j. If gm m eets C\ only at /,, we apply Operation 2.3.2 and obtain a closed 2-cell embedding. If gm m eets C\ at both /,• and /,+1, applying

Operation 2.3.2 may create an inconsistent monofacial edge (or a double vertex). In order to prevent this, we must carefully choose a suitable attachment between gm and 61

z Ii+1 fi+1

x X

z z fi+1

y (a)

* u z m z

i+1

y y (b ) Figure 20:

Ci. Consider the order of these attachments appearing in the facial walk of gm. There are four basic types as shown in Figure 20(a). The choice of the passing attachment

(without loss of generality, we may assume is is an edge) is an edge with an 'x'. If fi, fi+i and gm all meet at a single vertex v, the rotation scheme is slightly different.

See Figure 20(b). □

Lemma 4.2.2. Let G be 2-connected planar graph and x,y,u,v € V(G). Let G + be the graph obtained by adding two new edges xy and uv. Then there is a closed

2-cell embedding of G + in some surface.

Remark: If x and y(u and v) are not on the boundary of the same face of then 62

the embedding obtained by contracting xy (uu) is still a closed 2-cell embedding.

Proof of Lemma 4.2.2: The basic idea is to find two distinct (may have attach

ments) simple face chains joining x and y, as well as u and t>, and then apply Oper

ation 2.2.3. Without loss of generality, we assume x and y (u and u) are not on the

boundary of the same face.

Case 1: G is 3-connected.

Let '!> be the unique embedding of G in the sphere Sq. Choose a simple face chain

C\ = (/i,...,/n) joining x and y. If neither u nor v is a divalent internal vertex of

a subdivided edge which is an attachment of C\, we can choose a simple face chain

C2 from A = E — U /; joining u and r>. This is because any two faces in have at

most one attachment, the simple face chain C\ is contractible, and A is connected. By

applying Operation 2.2.3 on C\ and Cz separately, we obtain a closed 2-cell embedding

of G + . If both u and v are divalent vertices and are internal vertices of attachment

edges of C\, we choose a sub-face chain from C\ that joins u and u, and a simple face

chain from the remaining part of E that joins x and y, and then apply Operation

2.2.3. If only u is a divalent vertex that is an internal vertex of an attachment edge e

of Ci with e = dfi fl d/,+i, we choose a simple face chain in A, which together with

fi forms a simple face chain Cz joining u and v. We also choose another simple face

chain C3 joining x and y, and then apply Operation 2.2.3 to C2 and C3.

Case 2: G has nontrivial 2-vertex cuts.

Case 2.1: Suppose x, y and u are in the same 3-component and v is in a different

3-component. 63

Choose a plane embedding such that the two cut-vertices are on the boundary of

the unbounded face. Form a simple face chain C\ joining x and y within the outer

most circuit of the 3-component. If u is not a divalent vertex that is an internal

vertex of an attachment edge of C \y we can easily form a simple face chain by using

the unbounded face to join u and v, and then apply Operation 2.2.3. Now suppose u

is an internal vertex of an attachment edge. We form a simple face chain C 2 w ithout

using the unbounded face to join u and v. If x and y are not internal divalent

vertices of attachment edges of C 2 , we can form a simple face chain, possibly with

the unbounded face, to join x and y, and then apply Operation 2.2.3. At most one

of x and y can be an internal divalent vertex of an attachment edge of C 2 . In this

instance, we extend C 2 to y to form a simple face chain C3 joining y and v, w hile x

and u are internal divalent vertices of two attachment edges, with the order of the

appearance in C3 being y, u, x and v. We now apply Operation 2.3.2 to obtain a closed 2-cell embedding.

Case 2.2: Suppose x,u are in the same 3-component and y,t> are in another 3- com ponent.

We can find two simple face chains to join x ,y and u,v, one by using the un bounded face and one without using the unbounded face.

There are three more sub-cases. These correspond to the following instances:

Case 2.3, x, y, u and v are all in different 3-components;

Case 2.4, x,u are in same 3-component and y,v are in other two different 3- components; 64

u u

y y

Figure 21:

Case 2.5, x, y are in same 3-component and u, v are in other two different 3-

components. These cases may be dealt with analogously. □

Lemma 4.2.3. Let G be a 2 -connected planar graph and x,y,u € V(G). Let G + he

the graph obtained by adding two new edges ux and uy. Then there is a closed 2-cell

embedding of G+ in some nonorientable surface.

Proof: This proof is quite similar to that of the previous lemma. Hence we shall only show the parts of this proof that were not covered by the proof of that lemma.

Let $ be a plane embedding of G. Once again we attempt to find two simple face chains that join u,x and u, y, respectively. We shall first find a simple face chain C\ that joins u and x. If y is not an attachment vertex of C\, or not an internal divalent vertex of an attachment edge of C\, we can find another simple face chain C2 that joins u and y. By discussing the 3-connected and 2-connected cases, we can prove this as we did previously. If y is an attachment vertex or is an internal vertex of an attachment edge, we can form three new simple facial walks from C\ and the edges ux and uy. The rotation scheme is shown in Figure 21. The 'oV on the edges ux 65

and uy represent a cross or not, depending on the parity of the cross number. For

an attachment edge, which is subdivided after drawing the edge uy, the side of the

subdivided edge that will have an ' x' on it is also determined by the parity of the

cross number of the edge uy. □

4.3 Face chains in the projective plane and the Klein bottle

In this section we will show that if the surface is a projective plane or a Klein bottle

and the embedding has representativity of at least 2, then under some circumstances

there exists a good face chain to join two non-adjacent vertices.

As was defined in Section 2.5, a 2-attachment closed 2-cell embedding is a closed

2-cell embedding such that any two attached faces have at most two attachments.

The next two lemmas are about the existence of face chains in projective plane.

Lemma 4.3.1. Let ^ be a 2-attachment closed 2-cell embedding of a 2 -connected graph G in a projective plane. Then there exists a good simple face chain joining any

two non-adjacent vertices.

Proof: Let x and y be any two non-adjacent vertices. By Lemma 2.5.9, it suffices to show that we can find a simple face chain C = (C\,..., Cn) joining x and y such that the case as in Figure 8 will not occur. Suppose Ofi-iDdfi = {a, b) and d/,T)/l+1 = {c, d}, such that a and b (as well as c and d) are of different types, and a, b are separated by c, d on dfi. Since both F(1t and Fcj are 1-sided and they intersect once in the face we have the case shown in Figure 22. Clearly A = P — j = Ai U A

i+1 AS-;x ■.'.'.tv *Xlt ///////# A .

entry

Figure 22:

) is good. If both x and y are in A\ (or A?), by Theorem 2.5.1, there exists a good simple face chain to join x and y. If x is in A\ and y is in A 2 , we find a good simple face chain C\ = ( / u , ..., / i , n, ) in A\ to join x and and a good simple face chain

C2 “ (_/*2i ,..., f 2 ,712) in A.2 to join y and d/,_j. Note that all the attachments (at most two) of / i in, and d/,_i are of same type, as well as the attachments of / 2,nj and d/,_j.

By Lemma 2.5.5, C\ U /,•_ 1 U C2 is a good simple face chain joining x and y. Thus the lemma is true. □

Lemma 4.3.2. Let be a closed 2-cell embedding of G in a projective plane P, x ,y G V(G) and xy ^ E(G). Let the minimum non-orientable surface of G U {x y } be N3. Suppose G U {x y } is 3 -connected and has no good 3-vertex separation, then there is a good simple face chain in 9 joining x and y.

Proof: By Lemma 4.3.1, it suffice to show that is a 2-attachment closed 2- 67

cell embedding. Let / and g be two faces of 'P. If / and g have more than two

attachments, then at least two of them are of same type. By Lemma 2.4.1, the

corresponding dual circuit is 2-sided. Since any 2-sided simple curve in the projective

plane is null homotopic, it will bound a disk. Therefore, there is a non-trivial 2-

vertex cut. Hence either G U {xy} is not 3-connected or this 2-vertex cut together

with x (or y) forms a good 3-vertex separation, which contradicts our assumptions.

Thus any two faces have at the most two attachments, and the embedding must be

a 2-attachment embedding. □

The next three lemmas will show the existence of a good face chain which joins

two non-adjaccnt vertices under certain circumstances, provided the surface is the

Klein bottle.

Lemma 4.3.3. Let 4* he a 2-at,tachment closed 2-cell embedding of a 2 -connected graph G in a Klein bottle. Then there exists a good simple face chain joining any

two non-adjaccnt vertices.

Proof: The idea of the proof is similar to the one in the proof of Lemma 4.3.1.

Let x and y be two non-adjacent vertices and C = (/i,...,/n) be a simple face chain joining x and y. If C is not good, then by Lemma 2.5.9, there are three consecutive faces /,-i,/, and /,+) with the attachment as the case of Figure 8.

Suppose d/,_i fid/, = {». b} and d /,n /,+1 = {c, d}, such that a and b (as well as c and d) are of different type and a, b are separated by c, d on d/,-. Since both r af, and r cj are 1-sided and they intersect each other once in the face /,, by Lemma 3.1.2, both Ta6 and Pcci pass through the same cross-cap. Therefore A = P — U — A 1 UA 2 68

is a disjoint union of Ai and A 2, w ith A\ homeomorphic to a closed disk and A2

homeomorphic to a closed disk containing a cross-cap. If both x and y are in A\,

then any simple face chain in A\ joining x y is good since A\ is homeomorphic to a

disk. If both x and y are in A 2, then applying a similar argument to the proof of

Lemma 4.3.1, we can show the existence of a good simple face chain joining x and y.

If x is in A\ and y is in A 2, we find a good simple face chain C\ = (/n ,..., /i,m) >n

A\ to join x and $ /,_ i, and a good simple face chain C2 = ( / 21, . . . , / 2,n2) in A2 to join

y and d/,_j. Since ||<9/i,n, n || < 2, ||<9/ 2,,l2 fl d fi-i || < 2, and d f Uni D <9/,_ 1 and

8/2,n2 G dfi- 1 a r e separated on by Lemma 2.5.6, C\ U /,•_ 1 U C 2 is also a good simple face chain joining x and y. Thus the lemma is true. □

Lemma 4.3.4. Let be a closed 2-cell embedding of G in a Klein bottle. Let x and y be two non-adjaccnt vertices and the minimum non-orientable surface of G U {z y } be N 3 . Suppose G U {xy} is 3-connected, has no good 3-vertex separations, and there are two attached faces f and

D = N 2 — ( f U y) is a union of cylinders and disks.

Proof: Suppose / and y have at least two 2-sided dual circuits. There are only three types of 2-sided simple curves in N2, i.e., curves bounding a disk, curves sepa rating two cross-caps and nonseparating curves.

If one of the two 2-sided dual circuits is of the first two cases, it separates the surface and causes a 2-vertex cut. This 2-vertex cut together with x, forms a good

3-vertex separation of G'U{.rv/}, which is impossible. Therefore these two 2-sided dual circuits are both nonseparating. By Theorem 3.1.1, these two dual circuits must be 69

homotopic. They are either disjoint or have at least two trivial intersections. If they

have two trivial intersections, since these two dual circuits only intersect in the face /

and g, there must be a third 2-sided dual circuit which bounds a disk. Hence we are

back to the previous separating dual circuit case. If they are disjoint, then they bound

two cylinders with one cylinder twisted. Now it is easy to see that D = N2 — ( / U g)

is a union of cylinders and disks. □

Lemma 4.3.5. Let 'k be a closed 2 -cell embedding of G in N 2, x ,y € V(G), and

xy E(G). Let the minimum non-orientable surface of G U {x y } be N 3 . If G U {x y }

is 3-connected and has no good ^-vertex separations, then there is a good simple face

chain joining x and y.

Proof: If p(N2) > 3, then any vertex of G has a wheel-neighborhood, and any two

faces have at the most one attachment. Hence any simple face chain joining x and y

is good. Therefore we assume p(N2) = 2.

Case 1: ps(N2) = 2.

Since there is a essential separating simple curve T which meets G at two vertices, we have a good 3-vertex separation. This contradicts our assumption.

Case 2: p2 (N2)=2 and ps(N2) > 3.

There exists a nonseparating 2-sided simple curve T in the Klein bottle which m eets G at two vertices. Let the two faces that T passes through be / and g.

If ||df fl 0g\\ > 4 or \\df H <9y|| = 3 and all three attachments are of same type, then we have at least two 2-sided dual circuits. By Lemma 4.3.4, D = N2 — (f Ug) is a union of cylinders and disks. If x and y are in the same component of D, then 70

any simple face chain in that disk (or cylinder) joining x and y is good. If x and y are

in different component of D, then we can find a good simple face chain in each disk

(01 cylinder) to join x(y) and df. To avoid a good 3-vertex separation, these simple

face chains can only have one attachment with df. Now use / to combine these two

simple face chains and we obtain a good simple face chain joining x and y.

If \ \dfndg\\ = 2, then we have the case shown in Figure 23(a). If \\dff]dg\\ = 3 and

the three attachments are of different type, we have the case shown in Figure 23(b).

D = N2 — ( / U g) is either a cylinder or a disk. In both cases D is orientable and any

face chain joining x and y is good.

Case 3: p\{N2) = 2 and p2 (N2), p3 (N2) > 3.

First we claim that any two faces have at the most two attachments, and in the case of two attachments, they must be of different type. If two faces have more than two attachments, or two faces have two of the same type attachments, then we have a

2-sided dual circuit. This 2-sided dual circuit must be null-homotopic since we have

P2 (N2), pa(N2) > 3. It turns out that G U {xy} has a good 3-vertex-separation, which contradicts our assumption. So the claim is true. Thus ^ is a 2-attachment closed

2-cell embedding. By Lemma 4.3.3, there is a good simple face chain joining x and y. □

4.4 Face chains in

In this section we show that if the surface is Nj and the embedding has representativity of at least 2, then under some circumstances there exists a good face chain to join 71

------— / \ r------1 f I------g ------*— (b)

Figure 23:

any two non-adjacent vertices.

Lemma 4.4.1. Let 'if be a 2-attachment closed 2-cell embedding of a nonseparable

graph G in /V3. Then there exists a good simple face chain joining any two non-

adjacent vertices.

Proof: Let x and y be two non-adjacent vertices. Choose a simple face chain

C = (/i, ...,/n) joining x and y. Suppose C is not good, by Lemma 2.5.9 we have the

case shown in Figure 8. Suppose for some i, dfi-i C\dfi = { a ,6}, dfiDdfi+i = {c ,d },

and a and b are separated by c and d on 5/,. The dual circuits raj, and rcj have one

intersection in the face /,.

If one of r *6 and (defined in Section 3.3), say T*6, is null-homotopic, then by

Lemma 3.3.3, A = — {fi-i U /,) is homeomorphic to T~. By Theorem 2.5.1, any simple face chain in T~ is good. Since x,y € A, we can find a good simple face chain in A joining x and y.

Now we may assume that both r*b and are essential in T. By Lemma 3.3.6, 72

B = N 3 — (/j_i U /(U /j+i) = £?i U J32, where eitherB\ and B 2 are disks with a cross cap, or B\ is null-homotopic and B 2 is a disk with two cross-caps. By Lemma 4.3.1 and 4.3.3, we can choose a good simple face chain C\ = (/11,..., /i,ni) in B\ joining x and 5/i+i, and a good simple face chain C2 = ( / 21, •••, h,n2) in ^2 joining d/i+i and y.

Since ||d/i,ni n a /I+i|| < 2 ,||d /2,nj D d/,+i|| < 2 and d/,,n, n d /,+1 and d /2,nj n d /t+i are separated on d/,+i, by Lemma 2.5.6, C\ U U/,+i U C2 is a good simple face chain join ing x and y. □

Lemma 4.4.2. Let $ be a closed 2-cell embedding of G in N 3 , x ,y € V(G ) and x y g E(G). Suppose the minimum nonorientable surface of G U {.tj/} is N$. If

G U {xy} is 3-connected and has no good 3-vertex separations, then there exists a good simple face chain C in E joining x and y.

Proof: Let C = ( / 1, ..., f n) be a simple face chain in E joining x and y.

Case 1: There exists two consecutive faces /,• and / l+i such that ||9/, n d /,+i|| > 3 and at least three attachments of /,- and /<+1 are of same type.

In this case, there exists at least three 2-sided dual circuits. Without loss of generality, we may assume none of these dual circuits is null-homotopic or essential separating circuit. Otherwise we can embed G U {zj/} into by adding a handle to N 3 with the edge xy embedded in it, and therefore we can find a good 3-vertex separation of G U {xy}.

Case 1.1: There exists two 2-sided dual circuits having two intersections.

By Lemma 3.3.4, A = N3 — /, U /,+1 is a disjoint union of at least two connected 73

components. At most one of these components is homeomorphic to a disk with a

cross-cap, and all other components are null-homotopic. If x and y are in the same

component, we can easily find a good simple face chain in that component to join

x and y. If x and y are in different components, then we use the face /, (or /,+i)

to join two simple face chains which join .r,/, and y,/,- respectively. Since G U {x y }

is 3-connected, in order to avoid any good 3-vertex separation of G U {xy}, any two

faces in the null-homotopic component have at most one attachment, and any two

faces in the component of a disk with a cross-cap have at most two attachments, and

whenever two faces have two attachments, they must be of different type. By Lemma

4.4.1, we know the combined face chain is good.

Case 1.2: Any two of these three 2-sided dual circuits have exact one intersection.

By Lemma 3.3.7, A = Ns — f U g is a disjoint union of connected components. At

most one of these components is a disk with a cross-cap, and all other components

are null-homotopic. By the same argument as in Case 1.1, we have a good simple

face chain joining x and y.

Case 1.3: Two of these three 2-sided dual circuits, say Ti and r2, are disjoint.

By Theorem 3.3.1, Ti and r2 do not pass through the cross-cap, and therefore they are nonseparating simple curves in T~. Hence they are homotopic in T~ and bound a cylinder. Since Tj and T2 are dual circuits and they only pass through two faces, there must be a dual circuit, formed by part of Ti and part of T2, which bounds a disk (cut the cylinder bounded by Tj and f 2 into two parts). Therefore G U {xy} has a good 3-vertex separation, which is a contradiction. 74

Tbc

(a) (b)

Figure 24:

Case 2: There exists two faces /, and /;+1 with ||dfi D df,+ 1|| = 4 such that two of these attachments are of same type, and the other two attachments are of different type.

We now have two 2-sided dual circuits ro(, and 1%,*, where a, 6, c, d are four attach ments of /, and /,+1- If r a6 and 1%,* are disjoint, then r a6 and rcj bound a cylinder, and we are back to case 1.3. If r„f, and 1%,* have two intersections, then we are back to case 1.1. Therefore we may assume that ra& and rc(f have one intersection. The attachment descriptions are in the cases shown in Figure 24 (we may assume that the four attachments a . 6, c and d are in a counterclockwise order on dfi and a and b are inconsistent attachments, as well as c and d are consistent attachments. Fix the po sition of a and b on dfi+i. To assure that r af, and Vd have exact one intersection, the position of c and d on dfi+i only have two choices). The three 1-sided dual circuits rac, ric and rati in Figure 24(a) (1^, r6c, r ad in Figure 24(b)) are disjoint each other.

Each of these 1-sided dual circuits has a Mobius band neighborhood and all three 75

C g :>

\ / ✓ \ ------C 1

(a) (b)

Figure 25:

Mobius band neighborhoods are also disjoint. After cutting N3 along one of these

dual circuits, say r a6, and glue a disk on the boundary, T(,c and r a(* are still 1-sided

simple circuits in the resulting surface. The resulting surface is N2. Repeating this

cutting and gluing surgery on Vbc and rad, we will end up at a sphere. Therefore

A = N3 — fi U /,•+1 is a disk or union of disks. So we can find a good simple face

chain C to join x and y (similarly, we may use face /,■ to join two good simple face chains).

Case 3: There exists two consecutive faces /, and / I+1 such that ||5/, n 5 /t+i|| = 3, two of these attachments are of same type and the third attachment is of different type.

The two possible attachment descriptionons together with their corresponding embedding are shown in Figure 25. In the case of Figure 25(a), A = N3 — f U g is a cylinder which is orientable. In the case of Figure 25(b), A = N3 — / U <7 is a disk with a cross-cap. In both cases, by the same argument as before, we can find a good simple face chain C to join x and y. 76

Case 4: The embedding is a 2-attachment closed 2-cell embedding.

By Lemma 4.4.1 there is a good simple face chain joining x and y.

Thus Lemma 4.4.2 is true. □

4.5 Closed 2-cell embeddings of some projective graphs with new edges

In this section we will show that if a graph G is projective planar, then we can derive

a closed 2-cell embedding of G with two new edges.

We need the following lemma from [FHRR], which is a structure theorem of em

beddings of 3-connected graphs in the projective plane.

Lemma 4.5.1 ([FHRR], Proposition 1). Let 'P be an embedding of a 3-connected

graph G into P. Let P\ be any polygon in G such that ty(Pi) is null homotopic and

let D\ be the open disk contained in P bounded by $ (P i). Then there is a polygon P2

in G such that ty(P 2 ) is null homotopic and bounds a disk D D D\ with D D V(G).

Lemma 4.5.1 says that if a 3-connected graph is embedded in the projective plane,

then there is a disk, bounded by a circuit of the graph, which contains all vertices of

G. Now we will prove the main lemma of this section.

Lemma 4.5.2. Let G be a nonseparahle projective planar graph, x,y,u,v € V(G), and xy,uv are not edges of G. Suppose x,y,u and v are the only possible divalent vertices. Let G+ = G U {xy, uv) and the minimum nonorientable surface of G+ is N$.

Suppose G+ is 3-connected and has no good 3-vertex separation, then there exists a closed 2 -cell embedding of G+ in for some k. 77

Proof: If G is planar graph, this is exactly the same as Lemma 4.2.2. So we

assum e G is non-planar. Let ^ be one of the embedding of G in the projective

plane. Therefore p('P) > 2 (any graph with an embedding in the projective plane of

representativity one is a planar graph ([RV], page 303)). If 'P(G) has a cut vertex,

then G + has a good 3-vertex separation, which contradicts the assumption, Therefore,

by Proposition 1.1.3, ^ is a closed 2-cell embedding, and the boundary of each face

is a circuit.

If G has a non-trivial 2-vertex cut {i>i,u2} such that G = G'j U G 2 >G\ 0 G 2 —

{i>i, v2}. Then there is a sim ple curve T in P which meets G at v\ and v2. T m ust be

null-homotopic, since any essential simple curve in P is 1-sided and nonseparating.

A ssum e G\ is contained in the disk bounded by P. To avoid good 3-vertex separation

of G+ , x and y (as well as u and v) must be separated by this 2-cut, say x .u €

Viv £ V{Gi). Now by replacing G\ with a new edge joining Vi and v2 and

subdivided by x and u, we obtain a new graph G' with a new embedding 't' (see

Figure 26). If there are two good simple face chains joining x and y, as well as u and

v, in 4P, then we can also find two good simple face chains joining x and y, as well as

xl and v, in 'P (Let the sub-embedding of »P bounded by T be TL Note that to avoid

any good 3-vertex separation in G+, whenever there is a 2-vertex cut in P, x and u

must be in the same part, therefore, we can find a simple face chain C\ joining x and

the boundary. Since any two consecutive faces can only have one attachment, the rest part is connected and we can find another simple face chain C2 in the rest part joining u and the boundary.). So we may assume G is topologically 3-connected, and 78

////

Figure 26:

the only possible divalent vertices are x,y,u and v. By Lemma 4.5.1, we can find a polygon Pi in the graph such that 'J'(.Pi) is null-homotopic and 'F(Pi) 2 V(G), w ith the exception that some of x , y, u and v are divalent internal vertices in the middle of some edges passing through the cross-cap.

D enote by D the disk bounded by 'P(Pi). Clearly, any two faces in D have at most one attachment to avoid 2-vertex-cuts. Therefore the embedding inside the disk is a wheel-neighborhood embedding.

First we assume p(^>) > 3. Hence the embedding ^ is a wheel-neighborhood embedding and any two faces have at most one attachment, we can find a simple face chain C\ to join x and y.

(i) If none of u and v is a divalent internal vertex in the middle of some attachment

edge in Ci, we can find a simple face chain C? in A = P — U/i6c,/,- to join u and v.

Apply Operation 2.2.3 on C\ and C2, respectively, we obtain a closed 2-cell embedding

of G+; 79

(ii) If both u and v are divalent internal vertices in the middle of some attachment

edges of C i, we choose a sub-face chain C2 from C\ to join u and u, and choose a

simple face chain C3 from P — C2 to join x and y, then apply Operation 2.2.3 on C2

and C3;

If u is a divalent internal vertex in the middle of an attachment edge e of C\ and v

is not, then choose a simple face chain C2 to join u and v. If none of x, y is a divalent

internal vertex in the middle of an attachment edge of C2, we choose another simple

face chain C3 to join x and y in the rest area and apply Operation 2.2.3 to C2 and C3.

If one of x and y (but not both of them), say x, is a divalent internal vertex in the

middle of an attachment edge of C2, by extending C2 to y, we obtain a simple face

chain C3 joining u and y with x and v being divalent internal vertices in the middle

of two attachment edges. The order of the appearance in C3 is u,x,u and y. Now

apply Operation 2.3.1 to obtain a closed 2-cell embedding.

Now we assume /)(^) = 2. This implies that there exists an essential simple curve

T which meets the graph at two vertices on Pi (see Figure 27). Let these two vertices

be vi and u2. For i = 1,2, let /?,• be the edges with u,- as one end, passing through

the cross-cap, and with some vertex on Pi as the other end.

Case 1: All of x,y,u,v are in D.

If at least one of x,y,u,u, say x, is not on Pi,then find a simple face chain C1 in

D to join x and y, and find another simple face chain C2 in D — Cj to join u and v.

If x, y, u, v C Pi and x and y are not separated by u and v, then the proof will be the same as previous case. If x,y,u,v C Pi and x,y are separated by u,v, then let P~ 80

Figure 27:

be the projective plane with the open disk D removed. Find a simple face chain C\

in D, which must be good, to join x and y, and find a simple face chain C 2 in P~, to

join u and v, which is also good by Lemma 4.3.2. Apply Operation 2.2.3 on C\ and

C2, and we obtain a closed 2-cell embedding of G+.

Case 2: All or part of x, y , u, v are divalent vertices in the middle of some edge in

E% U E2.

There are many sub-cases due to the various positions of x,y, u and v. A case by

case argument will give two good simple face chains joining x and y, as well as u and v. The argument is easy since we have very clear picture of the embedding given by

Lemma 4.5.1. We leave these for reader to check. □

An similar argument to the proof of Lemma 4.5.2 will prove the following lemma, which is an degenerate case of Lemma 4.5.2

Lemma 4.5.3. Let G be a nonseparable projective planar graph, x,y,v 6 V(G), 81 and xy , xv are not edges of G. Suppose x , y and v are the only possible divalent vertices. Let G+ = GU {xy, xv} and the minimum nonorientable surface of G+ is N$.

Suppose G+ is 3-connected and has no good 3-vertex separation, then there exists a closed 2-cell embedding of G+ in Nk for some k. □

4.6 Closed 2-cell embeddings of lower surface embeddable graphs

In this section, we prove the existence of closed 2-cell embeddings of graphs with x(G) > —3, i.e. the graphs which are either double toroidal or N$ embeddable.

Let be an embedding of G in E. If p(^) = 1, then by Proposition 1.1.5, there exists an essential simple curve T in E which intersects G only at a single vertex v.

Call such vertex v as intersection vertex ( of T and G).

Theorem 4.6.1. If G is a 2 -connected double toroidal graph, then G has a closed

2-cell embedding in some surface.

Proof: By the reductions given in Section 4.1, we may assume that G is simple,

3-vertex connected and without any good 3-vertex separation.

If G is planar graph, then by the well known result, G has a closed 2-cell em bedding in the sphere.

If the minimum orientable surface of G is the torus, choose an embedding 4/ of

G in the torus. If p('5) > 2, then the theorem is true by Proposition 1.1.3. Since the torus is the minimum orientable surface of G, we know that p / 0. So we may assume that p('I') = 1. By Proposition 1.1.5, there exists an essential simple curve V which meets G only at the intersecting vertex x. T must be nonseparating since G is

3-connected. Now cut the torus along T, and glue two disks to the two boundaries of

this cut. We obtain a new graph that is embedded in the sphere. The vertex x is split

into two vertices. We call these two vertices the counterpart(s) of x and denote them

by x and y. If the cut causes a pendant edge, we delete that edge and denote the

other end of that edge as y. Let H be the new graph and x,y be the two counterparts

of the intersecting vertex. Then H is planar and at least 2-connected. By Lemma

2.5.1, H U {.'ey} has a closed 2-cell embedding in some surface. If the cut causes a

pendant edge, then H \J {xy} — G. If the cut does not cause a pendant edge, since

the torus is the minimum orientable surface of (7, we know that x and y are not on

the boundary of the same face in the resulting planar embedding of H. Therefore we

can contract edge xy to obtain a closed 2-cell embedding of G.

If the minimum orientable surface of G is the double torus, choose an embedding

'k of G in the double torus. If > 2, the theorem is true. Since p ^ 0, we

may assume that p{Psi) = 1. By Proposition 1.1.5, there exists an essential simple curve T which meets G only at a single vertex x. Since G is 3-connected, T must be nonseparating simple curve. We proceed as before. Cut the double torus along T, glue

two disks to two boundaries of this cut, and we obtain a new graph embedded in the torus. If the cut causes a pendant edge, we delete that edge. Let H be the new graph and x,y be two counterparts of the intersecting vertex. If the resulting embedding of

II in the torus has representativity 2, then by Lemma 2.5.1, we can find a closed 2-cell embedding of IIU {xy}. By the minimum orientable surface assumption, x and y are not on the boundary of the same face. Therefore we can contract edge xy, if needed,

and the resulting embedding would still be a closed 2-cell embedding. Therefore G

has a closed 2-cell embedding. If the embedding in the torus has representativity 1,

then there exists a nonseparating simple curve Ti which meets H only at a single

vertex u. We cut the torus along Ti, delete the pendant edge, if any, and obtain a

new graph K that is embedded in the sphere. Let the counterparts of the intersecting

vertices x and u be x, y and u, v, respectively. If the two intersecting vertices x and u

are not distinct, we have x, u and v as the counterparts of the intersecting vertex. If

the planar embedding of I\ has no cut vertex, then by Lemma 4.2.2 (or Lemma 4.2.3),

K U {xy,uv} (or K U {xu,ui>}) has a closed 2-cell embedding. In the resulting plane

embedding of K, x and y (as well as u and v) are not on the boundary of the same

face. Therefore we can, by contracting an edge(s), if necessary, obtain a closed 2-cell

embedding of G. If K has a cut vertex 2, the vertices x, u and z will be a 3-vertex

separation. It is easy to see that this 3-vertex separation is good (by sending back

two handles to the sphere). Therefore, by the good 3-vertex separation reduction, one

side of this 3-vertex separation must be three edges with a vertex. If the other side

of this 3-vertex separation is 3-connected, then by Lemma 4.2.1, we can get a closed

2-cell embedding. If the other side of this 3-vertex separation has a 2-vertex cut, then

x and u will be in different parts due to the good 3-vertex separation reduction. Also

this is the only 2-vertex cut, and z is either with x or with u. Suppose z is with x.

It is easy to find a simple face chain C\ joining x and 2, and then find a simple face chain C2 to j°in u and C\. With the same argument given in the proof in Lemma 84

4.2.1, we can obtain a closed 2-cell embedding for G+ and this finishes our proof. □

Theorem 4.6.2. If G is a 2 -connected N 4 embeddable graph, then G has a close

2 -cell embedding in some surface.

Proof: By the reductions in Section 4.1, we may assume G is simple, 3-connected and without a good 3-vertex separation.

Let be an embedding of G satisfying the above assumption. If the embedding

is 1-representative but not 2-representative, by Proposition 1.1.5, we can find an essential circuit T in E, which meets the graph G only once. T cannot be a separating curve in E because of 2-connectivity. We may assume T is 2-sided, since if T is 1-sided, we can apply Operation 2.2.1 to get a new embedding of G in a lower surface with one more face.

C ase 1: G is a projective planar graph.

Let ^ be any embedding of G in P. If p('Jr) = 1, then G is a planar graph, and by the well known result G has a closed 2-cell embedding in the sphere. If p('i') > 2, then G(\&) is already a closed 2-cell embedding.

Case 2: The minimum non-orientable surface of G is the Klein bottle (N?).

Let ^ be an embedding of G in N%. If p(\&) > 2, we are done. Since /9(\&) ^ 0

(jV2 is the minimum non-orientable surface of G) we m ay assum e p(\&) = 1. By

Proposition 1.1.5, there exists a 2-sided nonseparating simple curve T which meets

G only at a single vertex x. Now we cut along T, and by gluing two disks to boundaries caused by this cut, we get a new graph embedded in the sphere. Name the new graph H and the two counterparts of the cut vertex x and y. H is planar 85 and at least 2-connected. We can get G back by adding a new handle, therefore G is a toroidal graph and this case has been solved in Theorem 4.6.1.

Case 3: The minimum non-orientable surface of G is 7V3.

We proceed as we did for N2 (cut and glue) and obtain a new embedding of

H in the projective plane. Either G — H U {xy} or G is the graph obtained by contracting the edge xy in H U {xy}. By Lemma 4.3.2 there is a good simple face chain joining x and y. Since Nj is the minimum non-orientable surface of G, x and y are not on the same face boundary of ty'. Now we can apply Operation 2.2.3 to this face chain (if necessary, we may contract the edge xy) to Obtain a closed 2-cell embedding for G.

Case 4: The minimum non-orientable surface of G is N4.

Let $ be an embedding of G into N4 and we assume p('I') = 1. There exists a

2-sided non-separating simple curve T in N4 which meets G only at a single vertex x. Proceeding as before, we cut N.t along T, glue two disks to two boundaries of this cut, and get a new graph embedded in iV2 or T (torus). Name the new graph H and the two counterparts of the cut vertex x and y. If the resulting surface is T, then

G is double toroidal as was solved in Theorem 4.6.1. Hence we assume the resulting surface is N2. If the resulting embedding of H in N2 has representativity of 2, by

Lemma 4.3.5 we can find a closed 2-cell embedding of HU{xy}. By the minimum non- orientable surface assumption, x and y are not on the boundary of the same face of the embedding in N2. We can contract edge xy, if needed, and the resulting embedding is still a closed 2-cell embedding. Therefore G has a closed 2-cell embedding. If the 86

embedding in N 2 has representativity of 1, we cut /V2, delete the pendant edge, if any,

and get a new graph K embedded in the sphere. If this happens, we know that G is

doubly toroidal as has been solved in Theorem 4.6.1. Therefore the theorem is true.

□

Theorem 4.6.3. If G is a 2 -connected N 5 embeddable graph, then G has a closed

2 -cell embedding in some surface.

Corollary 4.6.4. The above graph has a cycle double cover.

Proof of Theorem 4.6.3: By the reduction in Section 4.1 and Theorems 4.6.1

and 4.6.2, we may assume G is simple, 3-vertex connected, without good 3-vertex

separation, the minimum non-orientable surface is Ns, and G is not doubly toroidal.

Let be one of the embedding of G in N5 which satisfies the above assumption.

If the embedding 'P is 1-representative but not 2-representative, by Proposition 1.1.5,

there exists an essential simple curve T in N$ which intersects G only at a single

vertex x. T cannot be a separating curve because of the 3-connectivity. T must

be 2-sided, because if T is 1-sided, then we can apply Operation 2.2.1 to obtain a

new embedding of G with more faces, and therefore either G is doubly toroidal or the

minimum non-orientable surface of G has less number of cross-caps, which contradicts

the assumption. Cut N 5 along T, and by gluing two disks to two boundaries of this cut, we obtain a new graph H with embedding ty' in N3 . If the cut causes a pendant edge, we delete this edge. Call the two counterparts of intersecting vertex x as x and y (in the case of deleting a pendant edge, call the other endvertex of this pendant 87 edge as y).

If V has representativity 2, by Lemma 4.4.2, we can find a good simple face chain

C join in g x and y. By applying Operation 2.2.3, we obtain a closed 2-cell embedding o f H U {xy}. Since N$ is the minimum non-orientable surface of G, x and y are not on the boundary of same face of 4*', and we can contract edge xy, if needed, and the resulting embedding is still a closed 2-cell embedding. Therefore G has a closed 2-cell embedding in some surface.

If \f>' has representativity 1, then, by the same argument, we can find a 2-sided simple curve Ti in N3 which intersects H at a single vertex u. Ti cannot be separating curve since otherwise G will have a good 3-vertex separation, we cut N3 along Tj, delete the pendant edge, if any, and get a new graph K with embedding $" in P (the projective plane). Call the counterparts of the intersecting u as u and v (in the case of deleting a pendant edge, call the other endvertex of this pendant edge as v). N ote that two intersecting vertices x and u may be same. If this occurs, we have x, y and v as the counterparts of the intersecting vertex x.

If the representativity of is 1, then K is planar. Therefore G is a doubly toroidal graph, which contradicts the assumption again. So we may assume that p{W) = 2. If A' has no cut vertex, by Lemma 4.5.2 (or Lemma 4.5.3), K U {xy, uv)

(or K U {xy,xv}) has a closed 2-cell embedding. In W, x and y (as well as u and v) cannot be on the boundary of same face because of the minimum non-orientable surface assumption. Therefore we can contract edge xy and uv, if necessary, to obtain a closed 2-cell embedding of G. If K has a cut vertex z, then the vertices x, u and 88 z will form a good 3-vertex separation which also contradicts the assumption. This completes the whole proof. □ C H A PTER V

Minimal Surface Embeddings

Recall that the minimal surface of a graph is the lowest surface that G can be embed ded in. The minimal surface can be either orientable or non-orientable. The embed ding of a connected graph in its minimum surface must be an open ‘2-cell embedding and have highest Euler characteristic. Even though there are examples, as mentioned in Chapter I, which show that the orientable genus embeddings of some graphs are not closed 2-cell embeddings, we still believe that the minimum surface embedding, not necessarily orientable, will bring us some useful information. In this chapter we study minimal surfaces embeddings. We will prove some structural properties of (1) the minimal surface embeddings, which are either orientable genus embeddings or non-orientable genus embeddings, and (2) embeddings with a minimum number of repeated vertices or edges in facial walks (if the closed 2-cell embedding conjecture is true, then such embeddings do not exist). An upper bound on the number of homo topic classes of principal ^-minimal curves is given for 3-connected graphs embedded in minimal surfaces. This is 3g — 3 on Sg, for g > 2 (or 1 if g = 1) and 3[|J — 3, for k > 4 (or = 1,2,3). Examples are given to show that these bounds are the best possible for minimal surface embeddings. These examples also provide a

89 90 complete list of surfaces where the genus embedding conjecture fails.

5.1 Notation and some technical results

Let v be a vertex of G and e \, e2 be two consecutive edges in the edge rotation at v. These two consecutive edges form a section of a facial walk of a face / and we define such a pair of edges as a com er (e \, e2) of /. Suppose v is a multiple vertex of a bad face /. There are at least two corners of / at v. Note that different corners may have a common edge and this corresponds to the case that the common edge is a monofacial edge of /. Let (ei,e2) and (eJ/j) be two corners of / at v. Call (01,62),

(e\,e'2) a pair of monofacial corners. A pair of monofacial corners is consistent if these two corners are traversed in the same direction in the facial walk, otherwise, it is inconsistent (this definition is similar to the one of consistent and inconsistent monofacial edges).

Let R

(5.1.1) R» = (EvJR*(v,f))-2M

The summation in (5.1.1) is over all bad faces and multiple vertices, i?# is called the repetition number of 'P and measures how far an embedding 'P is from a closed 2-cell embedding. Since both endvertices of a monofacial edge are multiple vertices and each appearance of a monofacial edge in the facial walk results in two repetitions of its endvertices, one repetition of each endvertex, we identify these two repetitions of 91 multiple vertices in the counting of R y. This accounts for the term of —2 M y (E ) in

(5.1.1). Clearly, if all repetitions of multiple vertices are caused by monofacial edges, then /2q> = 2 M y (E ). An embedding of a trivalent graph with some monofacial edges is an example of this case.

Let be an embedding of G in E with p(ty) = 1. By definition, there are some ty-minimal curves in E and each meets G only once. This will be the case throughout this chapter. Every ^-minimal curve passes through a pair of monofacial corners.

Since 'I' is an open 2-cell embedding, every face is an open disk and the following two propositions follow immediately.

Proposition 5.1.1. Let ^ be an embedding with />('P) = 1 and v be a m u ltiple vertex. Let (e ! ,e 2), (e^ e^ ) be a pair of monofacial corners at v. Suppose Ti and T2 are tw o 'P-minimal curves passing through the same pair of corners (ei,e2), (e'n e'2).

Then Ti and T2 are homotopic.

Proposition 5.1.2. Let V be an embedding with p(W) = 1 and e be a monofacial edge with two endvertices V\ and v2. Then any ty-minimal curve passing through the pair of corners at u1 with e as a common edge and any V-minimal curve passing through the pair of corners at v2 with e as a common edge are homotopic.

Theorem 5.1.3. Let ^ be an embedding of a graph G with p(V) = 1. Suppose v is a multiple vertex. Then all pairs of monofacial corners at v of a face f and all (homotopy classes of) V-minimal curves passing through v and f have a one-to- one correspondence. Moreover, let ( e i,e 2), ( e ' j , ef2 ) be a pair of monofacial corners of 92

f . Then ( e i,e 2), (c\, c2) is a pair of consistent monofacial corners if T(Cl,e2),(ej

I-sided and a pair of inconsistent monofacial corners j7T(eiiej)t(e'ie') is 2 -sided.

Proof: By Proposition 5.1.1, the one-to-one correspondence between the pairs of

monofacial corners at v and the ^-minimal curves passing through v is obvious.

Let (e i, e2), (e'j, e'2) be a pair of monofacial corners of /. Let W be one of the

sections of the facial walk of / between the corresponding two appearances of v.

Without loss of generality, we may assume that F is in the neighborhood of W.

Since (ej, e2), (e'j, e2) is a pair of consistent monofacial corners if and only if W passes

through an odd number of cross-caps (here, the surface must be non-orientable and we

may view the surface as a sphere with a number of cross-caps) and so does T(ei,ej)(ej,e;)>

the second half of Theorem 5.1.3 also holds. □

Lemma 5.1.4. Let ^ be an embedding of a 2-connected graph G with p('k) = 1

either in its minimal surface or with minimum Ry. Then all -minimal curves are

2 -sided and nonseparating.

Proof: Suppose there exists a ^-minimal curve which is 1-sided and meets G at vertex v. By Theorem 5.1.3, there exists a pair of consistent monofacial corners

(e!,e2)» (e'^ej) at v. By applying Operation 2.2.1 to (ei,e2), {e-\,e'2) we obtain a new embedding with = x(^) + 1 and Ry> < R y , which contradicts the assumptions. Therefore all 'k-minimal curves must be 2-sided. Since any ^-minimal curve meets G only once, it cannot be a separating curve as G is 2-connected. □ 93

5.2 A structure theorem

In this section, we prove a structure theorem of minimal surface embeddings and embeddings with minimum R y. We need the following two operations.

Let £ be a surface and P i,r2 be two 2-sided nonseparating simple curves in £.

Define r2 to be orthogonal to Tj if r2 intersects Tj only once up to homotopy.

Operation 5.2.1. Let be an embedding of G in £ with p(^f) = 1, and T i,]^ be two 2-sided nonseparating 'k-minimal curves such that r2 is orthogonal to Tj.

Suppose r, n r2 = r , n<7= {u}. By cutting E along Ti and capping it off with two disks, we obtain an embedding ty' of a graph G' in £' with = x(^) + 2, where

C is obtained from G by an appropriate splitting of v into two vertices v\ and V2 .

Since r2 intersects once, T2 becomes a path from V\ to u2 in £'. Identify V\ and t>2 by sliding V\ along T2 to u2 in E', together with all edges incident with uj. We obtain an embedding of G in S' with 7iV» < /?*. Note that if E is an orientable surface then so is £'.

Operation 5.2.2. Let ^ be an embedding of G in E with ^('P) = 1, and T i,r2 be two 2-sided nonseparating 'k-minimal curves such that T2 is orthogonal to IV

Suppose Ti H G = {?;}, r2n (j = {u} and Ti fl T2 is in a face /. By cutting E along

Pi and capping it off with two disks, we obtain an embedding of a graph G' on

E \ where G' is obtained from G by an appropriate splitting of v into two vertices Vj and v 2, and x('^/) = x(^) + 2- By cutting off a small disk near u and identifying the antipodal points on the boundary, we obtain a surface E;/, which is a connected 94

✓ S / \

(a) Cb)

Figure 28:

sum of E' and a projective plane. Identify v\ and v-i by sliding v\ along T2 in £", together with all edges incident with V\, then passing through the cross-cap as shown in Figure 28. We obtain an embedding 'F'/ of G in E" with R y» < /?*. N ote that E" is a non-orientable surface with x { ^ n) = — l = +

With these two operations, we now prove the following important lemma.

Lemma 5.2.3. L et \F be an embedding of G with p('F) = 1, either in its minimal surface or with minimum Rq. Let Ti be a \F-minimal curve and T2 be a simple closed 2-sided curve in the surface which is orthogonal to T i . Then r 2 cannot be a

'F-minima] curve.

Proof: We prove this lemma by way of contradiction. Let Ti, T2 be two 'F-minimal curves such that T2 is orthogonal to Ti. Suppose Ti fl (? = {u}.

Case 1: Ti and T2 pass through different faces. 95

V V

Figure 29:

In this case Ti fl r 2 must be a vertex on the common boundary of these two faces.

Let r in r 2 = TiDG = {i>}. Apply Operation 5.2.1 to *F and we obtain an embedding

*F' in a surface with x('F') = y('F) + 2 and R yt < R y , which is impossible by our assumption.

Case 2: Ti and r 2 pass through the same face.

If r2 n rx = Ti fl G = {w}, then we can apply Operation 5.2.1 to *F to obtain an embedding with higher Euler characteristic and lower repetition number. This is again a contradiction. If Ti fl T2 is in /, then Ti fl G ^ T2 H G. Let r2 flG= {u }.

Apply Operation 5.2.2 to 'F and we obtain an embedding \F' with x (^ ') = x ( ^ ) + 1 and R y < R y , which also contradicts our assumption. This completes the proof. □

Let \F be an embedding with />($) = 1 and / be a bad face of *F. Let (e^ e2) and

(e3>64) be two corners of / at a multiple vertex v such that the two corresponding appearances of v are consecutive on the facial walk of / among all appearances of v 96 on this facial walk. The edges e2 and e3 may be the same when e-i is a monofacial edge. The ^-minimal curve r(ei,e2)(es,e4) is called a prin cipal ^-minimal curve at v.

Clearly, if a multiple vertex v appears n (n > 3) times in the facial walk of a bad face, then there are n principal ^-minimal curves meeting G at v\ and if v is a double vertex, then there is only one ^-minimal curve meeting G at v. Denote by £(^) the set of all principal ^-minimal curves of an embedding ’P. Note that if an edge e is an inconsistent monofacial edge of $ with two end vertices v\ and U2, then there are two principal ^-minimal curves; one is at , passing through the pair of corners at vi with e as a common edge, and the other is at V2 , passing through the pair of corners at V2 also with e as a common edge. By Proposition 5.1.2, these two principal

^-minimal curves are homotopic, and >C(^) contains only one of them.

Lem m a 5.2.4. Let \& be an embedding of a graph G w ith p (\&) = 1, either in its minimal surface or with R * minimum. Then all principal ^-minimal curves in are disjoint up to homotopy, namely, they do not intersect each other transversely.

Proof: Let T 1 and T2 be two \&-minimal curves. If they pass through different faces, then |T 1 H T2| < 1. Suppose Ti fl T2 = {n}. If the edges in a corner through which T1 passes are separated in the edge rotation at v by the edges in corners through which T2 passes, then Ti and P2 are orthogonal, which is impossible by Lemma 5.2.3.

Therefore, v is only a “touching point” of Tj and T2- Hence Ti and T2 are disjoint up to homotopy.

Now we assume Ti and 1^ pass through the same face. Any two ^-minimal curves passing through the same face intersect each other at most twice, once on the 97 boundary and once in the face. Since a principal ^-minimal curve passes through two consecutive monofacial corners at a multiple vertex in the facial walk, any two principal ^-minimal curves do not intersect each other in the face. Hence they in tersect each other at most once. By Lemma 5.2.3, they cannot intersect each other transversely, and therefore the lemma is true. □

Theorem 5.2.5. Let be an embedding of a graph G with p (’J') = 1, either in its minimal surface or with Ry minimum. Let v be a multiple vertex of a bad face f and e \ ,e 2 , ..., et , t > 3, be all edges incident with v in an anti-clockwise order according to the rotation projection (see Figure 30(a)) such that e, is on the facial walk of f, for i = 1 ,2 , ...,t. Then the ei’s appear in the facial walk of f in sequence; i.e., the facial walk of f is

(5.2.1) ...ei...ei...e2...e5...et...ej'...

In (5.2.1), we omit the appearances of v. If e, is a monofacial edge of f, then e* is the second appearance of e, in the facial walk. If e, is not a mono facial edge, namely e,- only appears once in the facial walk of f, then delete e\ from (5.2.1). The rotation projection at v is shown in Figure30(b).

Proof: We examine the pairs of corners at each vertex v. During our proof we as sume that there are no monofacial edges at v of /. The proofs of the cases which have some monofacial edges are similar. By Lemma 5.1.5, we know that all the ^-minimal curves through v are 2-sided, and all pairs of monofacial corners are inconsistent.

Case 1: The vertex v is a double vertex.

The theorem is true in this case since we only have one pair of corners which is 98

% /

(a) (b)

Figure 30:

inconsistent.

C ase 2: T he vertex v appears in the facial walk of / three times.

There are two subcases. One of them is as required already. The other subcase has the rotation projection as shown in Figure 31. The edge sequence in the facial walk of / is as the following:

(5.2.2) e \v e 2 ...esveQ...ezve4....

In this subcase, the principal ^-minimal curves r(ei)e2)(e3,e4) and F(ei)e2)(e5ie6) intersect each other once and this intersection occurs at v. To see this, one can draw r(ej,e2)(e3,e<) in the neighborhood of the section of the facial walk ve\...e^ v, as shown in Figure 31, and draw F(ei)ej)(C5)(.6) similarly. This is a contradiction to Lemma 5.2.4. Therefore the second subcase does not exist and the theorem is true in Case 2.

Case 3: T he vertex v appears in the facial walk of f more than three times, and therefore there are more than 3 pairs of corners. Suppose (5.2.1) does not hold. Let 99

e 2 )

Figure 31:

(ei> e2), (^3,64) , (et_i, et) be all the corners of / at v. Start from corner (01,62) and trace the facial walk. Let (e,-,et+i) be the first corner such that the following corner at v along the facial walk is (ej,eJ+1) with j ^ i + 2. Therefore the three corners (e,-, e,+ i), (e,+2, e,+3), (ej, eJ+i) appear in the facial walk in the same order as

(5.2.2). Therefore r (ei)ei+1)(ei+2)ei+3) intersects r (eite.+1)(eiiej+1) only once, which is a contradiction. Hence the Theorem 5.2.5 is also true in this case. This completes the proof. □

5.3 Application

As an application of Lemma 5.2.4 and Theorem 5.2.5, in this section we give an upper bound of |£('I,)|, the maximum number of principal ^-minimal curves of an embedding ^ of a graph G with p('I') = 1 either in its minimal surface or with R # minimum. We know that the repetition number and the number of principal 'F- 100 minimal curves are closely related as we mentioned earlier. These two numbers vary by at most 1 at each multiple vertex. The deviation is due to whether a multiple vertex is a double vertex or a vertex with repetition number > 3, and whether a multiple vertex is an endvertex of a monofacial edge or not.

Let S n denote the orientable surface of genus n and TV* denote the non-orientable surface of genus k. A maximal set of simple closed curves C = {Joo,— ,J$} inS n is called a decomposition system o f Sn if all c,-’s are disjoint, 2-sided, essential and no two of c, ’s are homotopic. To define the decomposition system o f for k > 2 we need an additional condition that we require that, for i = l,...,m , c,- is not an essential separating simple closed curve which separates TV* into two connected components with one of the components being a projective plane. This condition will guarantee that if there is an essential separating simple curve in C, then each component contains at least one essential 2-sided nonseparating simple closed curve.

Let E be a non-sphere surface. Let A(E) = max{\C\ : C is a decomposition system of E} and c 6 C. If c is an essential separating curve, then c separates E into two connected components, each being a non-sphere surface with a disk removed. Denote by E ^ and E£2) the two surfaces obtained by cutting E along c and capping off the boundaries with two closed disks. The curve c is split into two curves d and c", where d C E^1) and d ' C E^2); If c is a 2-sided nonseparating curve, then denote by Ec the surface obtained by cutting E along c and capping it off with two closed disks. The curve c is split into two curves d and d ' in Ec. We have the following two lemmas.

Lemma 5.3.1. L et E be a surface and Ci,C2 be two disjoint, 2-sided, essential and 101 non-homotopic simple closed curves in £ . If cx is separating, then c2 is 2-sided and essential in (or £ ^ ) .

Proof: We may assume that c2 is contained in £(j). Since c2 is disjoint from c\ in

E, c2 has an annular neighborhood in E which is also disjoint from cx. This annular region is unchanged during the surgery to obtain £ ^ . Hence c2 is 2-sided in E ^ .

Suppose c2 is null-homotopic in £<}). Let D be the disk bounded by c2 and dx be the counterpart of cx in E^ with D\ being the disk in E ^ bounded by dx. If dx is not contained in D, then c2 is also null-homotopic in E, which is a contradiction. If

Ci is contained in D, then cx and c2 are homotopic in E, which is also a contradiction.

Therefore c2 must be essential in E ^. Thus Lemma 5.3.1 is true. □

Lemma 5.3.2. Let Ti be a surface and c i,c 2 be two disjoint 2-sided, essential and non-homotopic simple curves in E. Suppose cx is nonseparating. Then

( 1) c2 is 2 -sided in £ Cl;

(2) i f c2 is null-homotopic in E Cl, then c2 is separating in E an d c2 cuts o ff one han dle o f E containing cx;

(3) i f c2 is nonseparating in E, then c2 is essential in E Cl.

Proof: Lemma 5.3.2(1) can be proved similarly to Lemma 5.3.1.

In the surgery to obtain ECl, ci is split into two simple curves dx and d[, which are two null-homotopic simple curves in Ec. Let D\ and D" be two disks bounded by

Ci and d( in ECl, respectively. Clearly, c2 is not contained in either D[ or D".

If c2 is null-homotopic in Ec,, then bounds a disk D2 in ECl and separates 102

ECl into two components, £>2 and Ec, \D 2 . If dx and d[ are contained in different components, since E can be obtained from Ec, by removing D [ , D x and identifying dx and d[, then it is easy to see that c\ and C2 are homotopic in E, which is a contradiction. Therefore dx and d[ are contained in the same component. If both dx and d[ are contained in ECl \ D 2, then c2 is null-homotopic in E, which is impossible.

Hence both dx and d[ are contained in D 2. This also implies that E\£>2 is not null- homotopic in ECl, and hence not null-homotopic in E. Therefore C2 is an essential separating circuit in E, which separates the handle containing c\ from the rest of the surface. Thus Lemma 5.3.2(2) is also true.

Lemma 5.3.2(3) is a consequence of Lemma 5.3.2(2). □

Theorem 5.3.3. L et A (E ),5 n, Nk be as defined at the beginning of this section.

Then

(1) A(iVi) = 0;

(2) A(S,) = X(N2) = X(N3) = 1;

(3) A(Sn) = Sn— 3, for n> 2 ;

(4) X(Nk) = 3 [ l \ - 3, for k > 4 .

Remark: We are actually interested in a special class of maximal decomposition systems, the systems which contain no essential separating simple curves. If E is an orientable surface Sn, for n > 2, or a non-orientable surface Nk, for A: > 4 and k even, then such a decomposition system gives a “pair of pants” decomposition of Sn or

Nk. A pair of pants is a sphere or a projective plane with the interior of three closed disks removed (The notation “pair of pants” is commonly used in surface topology 103 to refer to the sphere with three disks removed.). If E is a non-orientable surface

Nk w ith k odd and k > 5, then such a decomposition system gives a pair of pants decomposition of Nk in which one of these pairs of pants contains a cross-cap in it.

This is illustrated by Figure 33. Note that the decomposition system of a surface is not unique. The result that A (S n) = 3n — 3, for n > 2, can be found in [Abi] (see page 87) with some interpretation. However, it is not hard to give a direct proof here. We give an induction argument in the following proof which also applies to non-orientable surfaces. The reason that we allow some essential separating simple curves to be contained in a decomposition system is so that we are able to carry out the induction proof, since a 2-sided nonseparating simple curve c in E may become an essential separating curve in Ec, , for some disjoint nonseparating simple curve c\ in E.

Proof of Theorem 5.3.3: First let the surface be S\ (the torus) and C be a decom position system of Si. Suppose C contains more than one element. Let ci,c2 G C.

Since all essential simple curves in Si are nonseparating, by Lemma 5.3.2(3), c2 is essential in (Si)Cl = So, the sphere. This is a contradiction. Hence A(Si) = 1.

Claim: A(Sn) < 3n — 3 for n > 2.

We will prove this by induction. Let C = {ci, ...,cm} be a decomposition system of S„. Suppose n = 2. If C contains an essential separating curve c, then it is clear that \C\ < 3, since c separates S2 into two components, each being a torus with a disk removed. If C contains no essential separating curve, by Lemma 5.3.1(3), c2,...,crn are essential in SCi(= *S'i). Since A(5i) = 1, the curves c2,...,cm are homotopic in 104

ECl. By Lemma 1.1.1, Ec, is decomposed into m — 1 cylinders. Let dx and d{ be as defined at the beginning of this section. In order to avoid homotopic pairs of c;’s in

52, each cylinder must contain one of dx and d(. Hence Si can only be decomposed into two cylinders and therefore m < 3. Thus A(52) < 3.

Now suppose the surface is 5n, for n > 3.

Case 1: One of the c, ’s, say cx, is an essential separating curve in 5n.

By Lemma 5.3.1, all c,’s, i 1, are 2-sided and essential either in or Eg>. where E^ = Eg' — Sffi2 w ith n ij ,ni2 — 1 and mi 4- m2 — m-. At most two of the C .’s, i 7^ 1, are homotopic to each other in E ^ (or E ^ ), and if this happens, dx must be contained in the cylinder bounded by these two homotopic curves. Therefore

A(5n) < (A(5mi) + 1) + (A(5ma) + 1) + 1

< (3mi — 2) + (3 rri2 — 2) + 1 = 3(mi + m2) — 3 = 3m — 3.

The second inequality is by the induction hypothesis. Hence the Claim is true in Case

Case 2: All c,’s are nonseparating in E.

By Lemma 5.3.2(2), all c,’s, i ^ 1, are essential in Ec, = 5n_i. If c,- and cj are homotopic, then c,- and Cj bound a cylinder. This cylinder must contain at least one of dx and d{, otherwise c, and cj would be homotopic in E. Therefore there are at most two pair of homotopic c,’s. Hence A(5n) < A(5„_i) + 3 < 3(n — 1) — 3 + 3 = 3n — 3.

The second inequality is by the induction hypothesis. Hence the above claim is true.

On the other hand, when n > 2, a decomposition system Cn of 5n contains at 105 least two elements. Choose two elements cj and C2 in Cn o f Sn, and a simple closed curves o' which is homotopic to Cj, for j = 1,2, and disjoint from all the c,’s. By

Lemma 1.1.1, the pair of circuits c, and cj bound a cylinder H i, i = 1,2. These cylinders can be chosen pairwise disjoint. Cut off a disk D, from //,, for i = 1,2, and identify the two boundary circuits (i.e., add a new handle to S n). We obtain a surface S n+i and a decomposition system Cn+1 o f S n+j, where Cn+1 contains all the elements in Cn plus dx, and a circuit winding around the new handle (homotopic to the circuit obtained by identifying the boundaries of the disks D\ and D 2 ). Therefore

A(Sn+i)> A(S„) +3.

Thus the theorem is true for all orientable surfaces.

We now suppose the surfaces are non-orientable. For the surface Nk w ith k > 3, if k is odd, we view Nk as Nk = Sk - 1 # P T the connected sum ofS k-t and a projective 2 2 plane, and if k is even, we view Nk as S k - i # P # P . the connected sum of S k -i and 2 2 two projective planes.

The proof for non-orientable surfaces can be carried out similarly by Lemmas

5.3.1, 5.3.2 and the following facts:

(a) all essential simple closed curves in iVi (the projective plane) are 1-sided;

(b) Let c be an essential 2-sided nonseparating simple curve in Nk. Then (A^)c =

So. If k is odd, k > 3, then (N k)c = N k - 2 , and if k is even, k > 4, then (N k)c can be either orientable or non-orientable; i.e., S k -t or N k - 2 . 2

(c) The surface Nk, for k > 3 can be obtained from Nk - 2 by adding a handle on it

(removing the interiors of two disjoint closed disks and identifying their boundaries, 106

namely, adding a handle to Nk - 2 )• If A; is even and k > 2 , then Nk can also be obtained from S k -i by adding a handle to it in a twisted way (cut off two disjoint disks and identify their boundaries in a reversed order).

(d) By definition, any decomposition system in or N 3 contains no essential separating simple closed curves. □

Theorem 5.3.4. L et G be a 3-connected graph and be an embedding of G in E w ith p (^ ) = 1 either in its minimal surface or with minimum Ry. Let £($) be the se t of all principal minimal curves of Then |£(^)| < A(E). This upper bound is the best possible for minimal surface embeddings.

Remark: The condition in Theorem 5.3.4 that G is 3-connected is necessary. We know that two disjoint homotopic 2-sided essential simple curves bound a cylinder. If we choose such two simple curves Ti and T2 as ^-minimal curves, then {TinG , I^nG} forms a 2-cut of the embedded graph G. In fact, we can construct embeddings of

2-connected graphs in their minimal surfaces, which can be any non-sphere or non- projective surfaces, orientable or non-orientable, with an arbitrarily large number of disjoint principal \k-minimal curves.

Proof of Theorem 5.3.4: By Lemma 5.1.4, all ^-minimal curves are 2-sided. By

Lemma 5.2.4, all principal \Er-minimal curves are disjoint up to homotopy. Note that two principal ^-minimal curves may touch at a vertex. Since G is 3-connected, all elements in £(\£) are non-separating. Suppose Ti and T2 be two homotopic principal

^-minimal curves. If Ti and IT^ are disjoint, then by Lemma 1.1.1, they bound a cylinder. Suppose Pi fl G = {u i}, r 2 D G = {*>2}- Then {vi, 1/2} forms a 2-cut of G. 107

Since G is 3-connected, {i>i, u2) must be a trivial 2-cut; i.e., v\ and v2 are two adjacent vertices and only the edge (vly w2) is contained in the cylinder bounded by Ti and T2.

Hence the edge (vi,u2) is a monofacial edge and Tj, T2 cannot both be contained in jC(^). If Ti and r 2 touch at a point, then they bound a degenerate cylinder, which is a cylinder with an identified point on its two boundaries. Ti H T2 can only be a vertex. Suppose rinr2 = r iflG = r 2 nGJ = {w}. If Ti and T2 pass through the same face, then since they cannot pass through the same pair of monofacial corners, there are edges which are contained in this degenerate cylinder. This implies that v is a cut-vertex, which is a contradiction. If Ti and T2 pass through different faces, then v is also a cut-vertex, which is again a contradiction. Therefore all principal tf- minimal curves in >C(^) are not homotopic to each other. Since all elements in £ ($ ) are disjoint, 2-sided, nonseparating and not homotopic to each other, by Theorem

5.3.3 the theorem is true.

The upper bound is the best possible if the embedding is a minimal surface em bedding. This can be confirmed by Examples 5.4.1 and 5.4.2 in next section. □

5.4 Examples

In this section, we construct examples which imply the upper bound given in Theorem

5.3.4 is the best possible if the embedding is a minimal surface embedding. Some of these examples also complete the list of surfaces such that the genus embedding conjecture fails in these surfaces.

Exam ple 5.4.1. We construct an embedding of a 3-connected graph with p(^) = 1 108

(a) S5 (b) Nn

(0 Nio Figure 32:

for every surface Sg, for g > 2 and Nk, for k > 4. These embeddings have = 1 and are minimal surface embeddings of their embedded graphs. In these embeddings, all 'F-minimal curves are principal, and \C (^ )\ = = A(Sg) (or A(Nk)), attaining the upper bounds given in Theorem 5.3.4. The embeddings are constructed by the following steps:

Step 1: For each surface Sg, for g > 2, or iVfc, for k > 4, we construct a decom position system C as shown in Figure 32. If the surface is Sg, for g > 2 or Nk, for 109 k > 4 and k is even, then the decomposition system C gives a pair of pants decom position which contains 2 g — 2 (or k — 2 when the surface is Nk) pairs of pants (see

Figure 32(a),(c)). If the surface is Nk, for A: > 5 and k is odd, then the decomposition system C gives a pair of pants decomposition in which one of these pairs of pants contains a cross-cap (see Figure 32(b)).

Step 2: In each pair of pants (including the pair of pants containing a cross-cap), embed a subgraph as shown in Figure 33, such that the embedding is a triangulation except for the three faces containing boundaries. Make the subgraph in each pair of pants big enough so that the distance between any two vertices of u, , for i = 1,2,3, is greater than d(£), where the number d(£) is related to the surface Sg (or Nk) and will be determined later.

Step 3: For each element c in C, which is a component of common boundaries of two pairs of pants, draw an edge ec crossing c and joining two u,’s, each from a different pair of pants. This gives an embedding of a graph G.

Clearly, G is 3-connected, p(’f) = 1 and |£('I,)| = = A(Sg) or (X(Nk)). For any cGC, ec is a monofacial edge. Each edge ec is in a face of size 8. All other faces are triangles. The subgraph in each pair of pants is connected to the rest of the graph by three edges.

C laim 1: Any minimal surface embedding of G is not a closed 2-cell embedding.

Suppose the surface is Sg (the proof for non-orientable surface Nk can be carried out similarly) and ec, , c\ G C, is one of the edges as defined in Step 3. Let and eCj, for C2,C3 G C, be the other two edges such that the subgraph in one of the pair 110 of pants is connected to the rest of the graph by eCj, for i = 1,2,3. Suppose 'I'1 is a closed 2-cell embedding of G in a surface £ 1. Then the edge ec, must be on the boundary of a face whose facial walk traverses through one of eCj, eCj and through another pair of pants. Therefore, the size of this face is at least 2d(£). The number of 8-faces in is (3*7 — 3) and of 3-faces is F — (3g — 3). Let E , F (Ei,Fi) denote the number of edges and faces in \k(\I'i, respectively). We have

(5.4.1) 2 E = 8(3*7 - 3) + 3(F - (3 g - 3)).

In there is at least one face of size not less than 2d(E). Since all other faces have sizes at least 3, we have

(5.4.2) 2E > 2d(E) + 3(.F1 - 1).

Combine (5.4.1) and (5.4.2) and let d(E) > 4(3*7 — 3). We have F > Fi when

<7 > 1. Therefore has lower Euler characteristic than has. Thus Claim 1 is true.

C laim 2: is a minimal surface embedding.

Let \I>2 be any minimal surface embedding of G. By a similar argument to the proof of Claim 1, each edge ec, for c € C, must be a monofacial edge in ^ 2, otherwise we have x(^ 2) < x(^)- By Lemma 5.1.4, the \&2-minimal curve passing through ec is 2-sided and therefore ec is an inconsistent monofacial edge. Hence ec appears in a facial walk twice and the size of this facial walk is at least 8, since G is simple and the girth o f G is 3. Let c\ and C2 be any two elements in £ (^ 2), then a similar argument to the proof of Claim 1 shows that ec, and are not on the facial walk of the same face. Therefore there are at least (3n — 3) 8—faces in the embedding ^ 2- This implies that x(^) > x(^2)- Hence x(^) = x(^2) and ^ is a minimal surface embedding. I l l

(a) (b)

Figure 33:

C laim 3: For any minimal surface embedding ^3 of G, Ry3 > Ry.

Let ^3 be any other minimal surface embedding of G. In the proof of Claim 2 we have shown that every edge ec, for c € C, must be an inconsistent monofacial edge in

^ 3, and any two distinct edges ec, , e^, for C\, C2 € C, are not on the facial walk of the same face in '5 3. Since eCl and eCJ do not have a common vertex, all ^-m inim al curves are disjoint. Any two of these ^-m inim al curves are not homotopic, for otherwise they bound a cylinder and G has a non-trivial 2-cut, which is a contradiction. Thus

Claim 3 is also true.

By Claims 1,2 and 3, it follows that the bound in Theorem 5.3.4 is the best possible when the surfaces are S n, n > 2 and Nk, k > 4. □

Example 5.4.2. 112

(a) (b)

Figure 34:

We construct embeddings of 3-connected graphs with p (^ )= l for the surface S2 and Nk, for k = 2,3. The idea is similar to Example 5.4.1. Instead of a pair of pants, here we use a cylinder (or a cylinder with a cross-cap) to construct embeddings. Let

T be a cylinder and P be a cylinder with a cross-cap as shown in Figure 34. Embed a graph H in T (or P) as shown in Figure 34, so that the embedding is a triangulation except for the two faces containing the boundaries. Make H big enough so that the distance between V\ and Vi is greater than 4. Identify two boundaries of the cylinder in ordinary order (or reversed order), then draw an edge (vi,^) to connect the vertices vi and V2 to obtain an embedding 'I’i (# 2) of a graph G\ (G2 ) in S\ (N2 , respectively). Perform the same surgery on the cylinder containing a cross-cap to obtain an embedding \&3 of G3 in JV3. By a similar argument to that used in Example

5.4.1, 'I',-, for i = 1,2,3 are minimal surface embeddings and p($;) = 1, |£(^,)| =

= A(E,), where E,■ is Si, N2 , iV3, respectively. This indicates that the bound in Theorem 5.3.4 is the best possible if the surface is Si, or iV3.

As mentioned in the first section, the genus embedding conjecture fails when the 113 surface is the torus. The genus embedding conjecture is actually true if the surface is either the sphere or the projective plane. We will see by the following examples that the sphere and the projective plane are the only two surfaces where the genus embedding conjecture is true.

Example 5.4.3.

Let E be a surface which is not a sphere or a projective plane and 'P be the embedding in E of a graph G as given in Examples 5.4.1 or 5.4.2. In E we split all the vertices of G into trivalent vertices to obtain an embedding \P' of G' in E. Since

^ is a minimal surface embedding of G and G is a minor of G\ is also a minimal surface embedding of G' and |£('I;/)| = A(E). These examples indicate that the genus embedding (orientable or non-orientable) of some trivalent graphs cannot be closed

2-cell embeddings.

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