preface - press.ustc.edu.cnpress.ustc.edu.cn/sites/default/files/fujian/field... · on the duality,...

Preface

The subject of this book reflects new developments mainly by theauthor himself in company with cooperators most of them his formerand present graduate students on the foundation established in Liu,Y.P.[33–34]. The central idea is to extract suitable parts of a topo-logical object such as a graph not necessary to be with symmetry, aslinear spaces which are all with symmetry for exploiting global proper-ties in construction of the object. This is a way of combinatorizationsand further algebraications of an object via relationship among theirsubspaces.

Graphs are dealt with three vector spaces over GF(2), the finitefield of order 2, generated by 0(dimensional)-cells, 1(dimensional)-cellsand 2(dimensional)-cells. The first two spaces were known from, e.g.,Lefschetz, S.[2] by taking 0-cells and 1-cells as, respectively, verticesand edges. Of course, a graph is only a 1-complex without two cells.

Since the fifties of last century, in Wu, W.J.[1] and Tutte, W.T.[4,16], the chain groups generated by 0-cells and 1-cells over, respectively,GF(2) and the real field were independently used for describing agraph. And they both then after ten years adopted nonadjacent pairof edges as a 2-cell for which the cohomology on a graph was allowedto be established.

Their results especially in Wu, W.J.[1–6] enabled the present au-thor to create a number of types of planarity auxiliary graphs inducedfrom the graph considered for the study of the efficiency of theoremsin Liu, Y.P.[1,2,19,22,42] as one approach. Another approach can beseen in Liu, Y.P.[23–25,43].

More interestingly, two decades later than Liu, Y.P.[1], in Archdea-con, D. and J. Siran[1] a theta graph(network) was used for charac-

ii Preface

terizing the planarity of a given graph. The theta graph can be seento be a type of planarity auxiliary graph(network) because our pla-narity auxiliary graphs are subgraphs of the theta graph. However, invirtue of the order of theta network upper bounded by an exponentialfunction of the size of given graph and that of planarity auxiliary net-work by a quadratic polynomial of the size of given graph, theoremsdeduced from a theta network are all without efficiency while thosefrom a planarity auxiliary network are all with efficiency.

The effects of planarity auxiliary graphs are reflected in Chapters8, 10, 11, 12 and 13 with a number of extensions.

On the other hand, in Liu, Y.P.[31] a graph was dealt with a setof polyhedra via double covering the edge set by travels under certaincondition so that travels were treated as 2-cells. These enable usto discover homology and another type of cohomology for showingthe sufficiency of Eulerian necessary condition in this circumstance.Further, all the results for the planarity of a graph in Whitney, H.[7]on the duality, MacLane, S.[1–2] on a circuit basis and Lefschetz, S.[1]on a circuit double covering have a universal view in this way. In fact,our polyhedra are all on such surfaces, i.e., 2-dimensional compactmanifolds without boundary. If a boundary is allowed on a surface,the Eulerian necessary condition is not always sufficient in general.Some person used to have missing the boundary condition in Abrams,L. and C.D. Slilaty[1].

The effects of this theoretical thinking are reflected in Chapters4,5,7 and 14.

Because of the clarification of the joint tree model of a polyhedronin Liu, Y.P.[35–36] by the present author recently on the basis of Liu,Y.P.[8–9], we are allowed to write a chapter for brief description ofthe theories of surfaces and polyhedra each in Chapters 2 and 3 withrelated topics in Chapters 6, 9 and 15.

Although quotient embeddings(current graph and its dual, voltagegraph) were quite active in constructing an embedding of a graph ona surface with its genus minimum in a period of decades, this bookhas no space for them. One reason is that some books have mentionedthem such as in White, A.T.[1], Ringel, G.[3] and Liu, Y.P.[33–34], etc.Another reason is that only graphs with higher symmetry are suitablefor quotient embeddings, or for employing the covering space methodwhence this book is for general graphs without such a limitation of

Preface iii

symmetry.In spite of refinements and simplifications for known results, this

book still contains a number of new results such as in §5.2, the suffi-ciency in the proof of Theorem 5.2.1, §9.4, §11.3–4, §13.1–2, §13.4–5etc., only name a few. Researches were partially supported by NNSFin China under Grants No.60373030 and No.10571013.

Y. P. LiuBeijing

December 2007

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Chapter 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Sets and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Partitions and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Graphs and networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Groups and spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 2 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Polygon double covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Supports and skeletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Orientable polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Nonorientable polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Classic polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Polyhegons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Surface closed curve axiom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .453.3 Topological transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Complete invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553.5 Graphs on surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573.6 Up-embeddability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 4 Homology on Polyhedra . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Double cover by travels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vi Contents

4.2 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Bicycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 5 Polyhedra on the Sphere . . . . . . . . . . . . . . . . . . . . . . 91

5.1 Planar polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Jordan closed curve axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4 Straight line representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Convex representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter 6 Automorphisms of a Polyhedron . . . . . . . . . . . . 114

6.1 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 V -codes and F -codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.3 Determination of automorphisms . . . . . . . . . . . . . . . . . . . . . . .129

6.4 Asymmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter 7 Gauss Crossing Sequences . . . . . . . . . . . . . . . . . . . .151

7.1 Crossing polyhegons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Dehn’s transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.3 Algebraic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.4 Gauss Crossing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Chapter 8 Cohomology on Graphs . . . . . . . . . . . . . . . . . . . . . . .171

8.1 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2 Realization of planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.4 Planarity auxiliary graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.5 Basic conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Contents vii

Chapter 9 Embeddability on Surfaces . . . . . . . . . . . . . . . . . . . 195

9.1 Joint tree model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Associate polyhegons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.3 The exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009.4 Criteria of embeddability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Chapter 10 Embeddings on the Sphere . . . . . . . . . . . . . . . . . 207

10.1 Left and right determinations . . . . . . . . . . . . . . . . . . . . . . . . . 20710.2 Forbidden configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21210.3 Basic order characterization . . . . . . . . . . . . . . . . . . . . . . . . . . .21910.4 Number of planar embeddings . . . . . . . . . . . . . . . . . . . . . . . . 22810.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Chapter 11 Orthogonality on Surfaces . . . . . . . . . . . . . . . . . . 235

11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23511.2 On surfaces of genus zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24311.3 Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26911.4 On surfaces of genus not zero . . . . . . . . . . . . . . . . . . . . . . . . . 27311.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Chapter 12 Net Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

12.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27712.2 Face admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28412.3 General criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29112.4 Special criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29812.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Chapter 13 Extremality on Surfaces . . . . . . . . . . . . . . . . . . . . .308

13.1 Maximal genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30813.2 Minimal genus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31213.3 Shortest embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31613.4 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

viii Contents

13.5 Crossing number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33113.6 Minimal bend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33313.7 Minimal area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34013.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Chapter 14 Matroidal Graphicness . . . . . . . . . . . . . . . . . . . . . . 349

14.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34914.2 Binary matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35014.3 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.4 Graphicness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36114.5 Cographicness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36714.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .368

Chapter 15 Knot Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . .370

15.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37015.2 Knot diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37515.3 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.4 Pan-polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38615.5 Jones polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39515.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Chapter 1

Preliminaries

Throughout for the sake of brevity, the usual logical conventionsare adopted: disjunction, conjunction, negation, implication, equiva-lence, universal quantification and existential quantification denoted,respectively, by the familiar symbols:∨, ∧, ¬, ⇒, ⇔, ∀ and ∃. And,§x.y is for the section y in Chapter x.

In the context, (i.j.k) refers to item k of section j in chapter i.

A reference [k] refers to item k of the corresponding author(s) in thebibliography where k is a positive integer to distinguish publicationsof the same author(s).

1.1 Sets and relations

A set is a collection of objects with some common property whichmight be numbers, points, symbols, letters or whatever even sets ex-cept itself to avoid paradoxes. The objects are said to be elements ofthe set. We always denote elements by italic lower letters and sets bycapital ones. The statement “ x is (is not) an element of M” is writtenas x ∈ M(x /∈ M). A set is often characterized by a property. Forexample

M = x | x ≤ 4, positive integer = 1, 2, 3, 4.

The cardinality of a set M (or the number of elements of M if finite)is denoted by | M |.

Let A,B be two sets. If ( ∀a) ( a ∈ A ⇒ a ∈ B), then A is saidto be a subset of B which is denoted by A ⊆ B. Further, we maydefine the three main operations: union, intersection and subtraction

2 Chapter 1 Preliminaries

respectively as A ∪ B = x | (x ∈ A) ∨ (x ∈ B), A ∩ B = x | (x ∈A) ∧ (x ∈ B) and A \B = x | (x ∈ A) ∧ (x /∈ B).

If B ⊆ A, then A\B = A−B is denoted by B(A) which is said tobe the complement of B in A. If all the sets are considered as subsetsof Ω, then the complement of A in Ω is simply denoted by A. Theempty denoted by ∅ is the set without element. For those operationson subsets of Ω mentioned above, we have the following laws.

Idempotent law ∀A ⊆ Ω, A ∩ A = A ∪ A = A.

Commutative law ∀A,B ⊆ Ω, A∪B = B ∪A; A∩B = B ∩A.

Associative law ∀A,B,C ⊆ Ω, A ∪ (B ∪ C) = (A ∪ B) ∪ C;A ∩ (B ∩ C) = (A ∩B) ∩ C.

Absorption law ∀A,B ⊆ Ω, A ∩ (A ∪B) = A ∪ (A ∩B) = A.

Distributive law ∀A,B,C ⊆ Ω, A∪(B∩C) = (A∪B)∩(A∪C);A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C).

Universal bound law ∀A ⊆ Ω, ∅ ∩A = ∅, ∅ ∪A = A; Ω∩A =A, Ω ∪ A = Ω.

Unary complement law ∀A ⊆ Ω, A ∩ A = ∅; A ∪ A = Ω.

The unary complement law is also called the excluded middle lawin logic.

From the laws described above, we may obtain a large numberof important results. Here, only a few is listed for the usage in thecontext.

Theorem 1.1.1 ∀A ⊆ Ω,

(∀X ⊆ Ω)((A ∩X = A) ∨ (A ∪X = X)

)

⇒ A = ∅;(∀X ⊆ Ω)

((A ∩X = X) ∨ (A ∪X = A)

)

⇒ A = Ω.

(1.1.1)

Theorem 1.1.2 ∀A,B ⊆ Ω,

A ∩B = A ⇔ A ∪B = B. (1.1.2)

Theorem 1.1.3 ∀A,B,C ⊆ Ω,

(A ∩B = A ∩ C) ∧ (A ∪B = A ∪ C) ⇔ B = C. (1.1.3)

1.1 Sets and relations 3

Theorem 1.1.4 ∀A ⊆ Ω,

A = A. (1.1.4.)

Theorem 1.1.5 ∀A,B ⊆ Ω,

A ∪B = A ∩B; A ∩B = A ∪B. (1.1.5)

From those described above, it is seen that ∅ = Ω and Ω = ∅.Further, the symmetry (or duality) that any proposition related to∪,∩, ∅, Ω can be transformed into another by interchanging ∪ and∩, ∅ and Ω.

For A,B ⊆ Ω, an injection ( or 1 - to - 1 correspondence ) betweenA and B is a mapping α : A → B such that ∀a, b ∈ A, a 6= b ⇒ α(a) 6=α(b). A surjection between A and B is a mapping β : A → B suchthat (∀b ∈ B)(∃a ∈ A)(β(a) = b). If a mapping is both an injectionand a surjection, then it is called a bijection. Two sets are said to beisomorphic if there is a bijection between them. Two isomorphic setsA and B, or write A ∼ B, are always treated as the same. Of course,for finite sets, it is trivial to justify if two sets are isomorphic by thefact: ∀A,B ⊆ Ω, A ∼ B ⇔| A |=| B |.

For a set M , let M ×M = ≺ x, y Â| ∀x, y ∈ M which is said tobe the Cartesian product of M . Here, ≺ x, y Â6=≺ y, x Â in general.

A binary relation R on M is a subset of M ×M . The adjective“binary” of the relation will often be omitted in the context. If therelation R holds for x, y ∈ M , then we write ≺ x, y Â∈ R, or xRy.An order, denoted by ¹, is a relation R which satisfies the followingthree laws:

Reflective law ∀x ∈ M,xRx.

Antisymmetry law ∀x, y ∈ M,xRy ∧ yRx ⇒ x = y.

Transitive law ∀x, y, z ∈ M,xRy ∧ yRz ⇒ xRz.

The set M with the order ¹ is said to be a poset (or partial orderset) denoted by (M,¹).

Theorem 1.1.6 In a poset (M,¹), ∀x1, x2, · · · , xn ∈ M ,

x1 ¹ x2 ¹ · · · ¹ xn ¹ x1 ⇒ x1 = x2 = · · · = xn. (1.1.6)


The theorem is sometimes called the anti-circularity law. If a re-lation only satisfies Reflective law and Transitive law but not Anti-symmetry law, then it is called the quasi-order which is denoted by• ≺. A set M with • ≺ is said to be a quoset denoted by (M, • ≺).

Theorem 1.1.7 Any subset S of a quoset (M, • ≺), is itself aquoset with the restriction of the quasi-order to S.

If a quasi-order R on M satisfies the symmetry law described be-low, then it is called an equivalent relation, or simply an equivalencedenoted by ∼.

Symmetry law ∀x, y ∈ M , xRy ⇒ yRx.

For the equivalence ∼ on M , we are allowed to define the setx(M) = y | ∀y ∈ M, y ∼ x which is said to be the equivalent classfor x ∈ M . The set which consists of all the equivalent classes is calledthe quotient set of (M,∼) denoted by M/ ∼. In a quoset (M, • ≺),let ∼•≺ be defined by

∀x, y ∈ M,x ∼•≺ y ⇔ (x• ≺ y) ∧ (y• ≺ x). (1.1.7)

Then, it is easily seen that ∼•≺ is an equivalence on M and that (M/∼•≺, • ≺) is also a quoset.

Theorem 1.1.8 A quoset (M, • ≺) is a poset if, and only if,M/ ∼•≺= M, or say, it satisfies the anti- circularity law.

In a poset (M,¹), we define the strict inclusion, denoted by ≺, ofthe order by the anti-reflective law: ¬x ∈ M,x ≺ x and the transitivelaw: (x ≺ y) ∧ (y ≺ z) ⇒ x ≺ z while noticing that x ¹ y ⇔ (x ≺y)∨(x = y). If an order ¹ on M satisfies the alternative law describedbelow, then it is called a total order, or a linear order.

Alternative law ∀x, y ∈ M , x 6¹ y ⇒ y ¹ x.

A set with a total order is said to be a chain. The length of achain with n elements is defined to be n−1. From Theorem 1.1.7 andthe definitions, we may have

Theorem 1.1.9 Any subset of a poset is a poset and any subset

1.1 Sets and relations 5

of a chain is a chain.

The converse of a relation R on M is, by definition, the relationR∗ : ∀x, y ∈ M,xR∗y ⇔ yRx. It is obvious from inspection of thethree laws for order to have

Theorem 1.1.10 (Duality principle) The converse of any orderis itself an order.

In a poset (M,¹), there may have an element a : ∀x ∈ M,a ¹ x.Because of Antisymmetry law, such an element, if it exists, is a uniqueone which is called the least element denoted by O. In dual case, thegreatest element, if it exists, denoted by I. The elements O and I,when they exist, are called universal bound of the poset.

Theorem 1.1.11 A chain has the universal bounds if it is finite.

In a poset (M,¹), an element a ∈ M : ∀x ∈ M,x ¹ a ⇒ x = a iscalled a minimal element. Dually, a maximal element is defined asa ∈ M : ∀x ∈ M,a ¹ x ⇒ a = x.

Theorem 1.1.12 Any finite nonempty poset (M,¹) has minimaland maximal elements.

A mapping τ : M → N from a poset (M,¹) to a poset (N,¹) iscalled order- preserving, or isotone if it satisfies

∀x, y ∈ M, x ¹ y ⇔ τ(x) ¹ τ(y). (1.1.8)

Further, if an isotone τ : M → N satisfies

∀x, y ∈ M, τ(x) ¹ τ(y) ⇒ x ¹ y, (1.1.9)

then it is called an isomorphism. Two posets (M,¹) and (N,¹)are said to be isomorphic, that is (M,¹) ∼= (N,¹), if there is anisomorphism between them. All isomorphic posets are treated as thesame. However, it is not trivial as for sets to justify if two posets areisomorphic in general.

An upper bound of a subset X of a poset (M,¹) is an elementa : ∀x ∈ X, x ¹ a. The least upper bound ( or l.u.b.) is an upper


bound b : a ¹ b ⇒ a = b, where a is another upper bound of X.Dually, a lower bound and the greatest lower bound( g.l.b.). Thelength of a poset is the l.u.b. of the lengths of chains in the poset. Alattice is a poset any two x and y of whose elements has a g.l.b. ormeet denoted by x∧y and an l.u.b. or join denoted by x∨y. A latticeL = (M,¹;∨,∧) is complete if each of its subset X has an l.u.b. anda g.l.b.. Moreover, we have known that all finite length lattices arecomplete.

Let 2Ω be the set which consists of all subsets of Ω. From §1.1, wemay see that (2Ω,⊆;∪,∩) is a lattice. In fact, we have

Theorem 1.1.13 A poset is a lattice if, and only if, it satisfiesthe idempotent, commutative, associative and absorption laws.

Two lattices (M,¹;∨,∧) and (N,¹;∨,∧) are isomorphic if thereis an isomorphism τ between (M,¹) and (M,¹) such that, ∀x, y ∈ M ,

(τ(x ∨ y) = τ(x) ∨ τ(y)) ∧ (τ(x ∧ y) = τ(x) ∧ τ(y)). (1.1.10)

Of course, it is nontrivial as well to justify if two lattices are isomorphicin general.

1.2 Partitions and permutations

A partition of a set X is such a set of subsets of X that any twosubsets are without common element and the union of all the subsetsis X.

Theorem 1.2.1 A partition P (X) of a set X determines anequivalence on X such that the subsets in P (X) are the equivalentclasses. 2

Let P (X) = p1, p2, · · · , pk1 and Q(X) = q1, q2, · · · , qk2 be twopartitions of X. If for any qj, 1 ≤ j ≤ k1, there exists a pi, 1 ≤ i ≤ k2

such that qj ⊂ pi, then Q(X) is called a refinement of P (X) and P (X),an enlargement of Q(X) except only for P (X) = Q(X). The partitionof X with each subset of a single element, or only one subset which is

1.2 Partitions and permutations 7

X in its own right is, respectively, called the 0-partition, or 1-partitionand denoted by 0(X), or 1(X).

Theorem 1.2.2 For a set X and its partition P (X), the 0-partition 1(X) (or 1-partition 1(X)) can be obtained by refinements(or enlargements) for at most O(log |X|) times in the worst case.

Proof In the worst case, it suffices to consider P (X) = 1(X)(or0(X)) and only one more subset produced in a refinement. Because of

1 + 2 + 22 + · · ·+ 2log |X| =21+log |X| − 1

2− 1= O(|X|), (1.2.1)

the times of refinements(or enlargements) needed for getting 0(X) (or1(X)) is O(log |X|). The theorem is obtained. 2

For two partitions P = p1, p2, · · · , ps and Q = q1, q2, · · · , qt ofa set X, the family intersection of P and Q is defined to be

P ∩Q =s⋃

i=1

pi ∩ q1, pi ∩ q2, · · · , pi ∩ qt. (1.2.2)

Actually, pi ∩ q1, pi ∩ q2, · · · , pi ∩ qt for i = 1, 2, · · · , l are partitionsof pi.

Theorem 1.2.3 The family intersection satisfies the commuta-tive and associate laws. And further, P ∩Q is a refinement of both Pand Q. 2

A permutation of a set X is a bijection of X to itself. Because ele-ments in a set are no distinction, they are allowed to be distinguishedby natural numbers as X = x1, x2, · · ·, or simply X = 1, 2, · · ·.So, a permutation of set L = 1, 2, · · · , l can be expressed as

(1 2 3 · · · li1 i2 i3 · · · il

). (1.2.3)

If ij = j for all 1 ≤ j ≤ l, the the permutation is call the identity.From Theorem 1.1.4, the identity is unique.

Theorem 1.2.4 Let π be a permutation of set L = 1, 2, · · · , l,then for any i ∈ L there is an integer n ≥ 0 such that pni = i.


Proof By contradiction. If no such an integer, by the 1–to–1property it is a contradiction to the finiteness of l. 2

On the basis of this theorem, the set Xi = i, πi, π2i, · · ·, πn−1i iscalled the orbit of i. Because any element in Xi has the same orbit asi, it can also be called an orbit of π, denoted by Orbπi, or simplyiπ. Because any two orbits of a permutation are either same ordisjoint, all orbits form a partition of L.

An orbit with the order in its own right is called a cyclic permu-tation, or in brief, a cycle. The cycle corresponding to Orbπi isdenoted by Orbπ(i), or simply (i)π. Because of the disjointness amongorbits, by considering that the composite of disjoint cycles satisfies thecommutative law and the associate law, a permutation can always beexpressed as a product of cycles. The order of a cycle is one greaterthan its length, i.e., the number of elements in the cycle. A cycle oforder 1 is called a fixed point of the permutation. All the fixed pointsin a permutation are always omitted in its cyclic expression.

As an example,

(1 2 3 4 5 6 72 5 1 6 3 4 7

)= (1, 2, 5, 3)(4, 6)(7)

= (1, 2, 5, 3)(4, 6).

However, the product of two cycles with a common element is notcommutative in general. For example, P1 = (1, 3, 2) and P2 = (1, 2, 4),

P1P2 = (2, 4, 3) 6= (1, 3, 4) = P2P1.

Because of Csk = 1 for the order k of a cycle C and any pos-itive integer s, it can from Theorem 1.2.4 be seen that if permuta-tion π = C1C2 · · ·Cn where Ci, 1 ≤ i ≤ n, are all the disjoint cy-cles of order ni, then π has its order [l1, l2, · · · , ln], the least commonmultiple(lcml1, l2, · · · , ln = [l1, l2, · · · , ln]) of l1, l2, · · ·, and ln.

Theorem 1.2.5 The unique inverse of a cycle C = (c1, c2, · · · , cn)is the cycle C−1 = (cn, cn−1, · · · , c1). 2

Let Σ|L| be the set of all permutations on L. The cardinality of Lis also called the degree of permutations. For two permutations π andσ in Σ|L|, if there is a permutation ρ ∈ Σ|L| such that π = ρσρ−1, thenπ and σ are conjugate for ρ.

1.2 Partitions and permutations 9

Let γ = (x1, x2, · · · , xr) be a cycle and τ , another permutation inΣ|L|. For y ∈ L, if x = τ−1y is not in γ, then τγτ−1y = τx = τ(τ−1y) =y. Otherwise, if x = τ−1y = xi(1 ≤ i ≤ r), then τγτ−1y = τxi = xi+1.This implies that

τγτ−1 = τ(x1, x2, · · · , xr)τ−1

= (τx1, τx2, · · · , τxr).(1.2.4)

For π ∈ Σ|L|, let c(π) be the number of cycles in its cyclic partitionand li, the number of cycles of length i, 1 ≤ i ≤ c(π). The cyclictype of permutation π is defined to be the decreased sequence of li,1 ≤ i ≤ c(π).

Theorem 1.2.6 Two permutations are conjugate if, and only if,they have a same cyclic type.

Proof The necessity is obvious because of (1.2.4) for cyclic parti-tion representation of permutations. Conversely, for any two permu-tations with a same cyclic type, assume with one cycle each withoutgenerality as π = (x1, x2, · · · , xr) and σ = (y1, y2, · · · , yr), it is from(1.2.4) seen that let

τ =(

x1 x2 · · · xry1 y2 · · · yr

)

then τπτ−1 = σ. Therefore, π and σ are conjugate. 2

Two particular cases should be mentioned for conjugate pair π, σof permutations. One is for π = σ and the other, π = σ−1. Theformer is called self-conjugate and the later, inverse conjugate. Ifπ = (x1, x2, · · · , xr) and σ = (y1, y2, · · · , yr), then the self-conjugate isonly for τ = 1r, the identity of degree r and the inverse conjugate isfor

τ = (x1, xr)(x2, xr−1) · · · (xbr/2c, xbr/2c+1).

Let D = 〈a1, a2, · · · , ad〉 be a set with linear order a1 < a2 <· · · < ad. An order pair 〈ai, aj〉 is called an inversion if 1 ≤ j < i ≤d. Let sgn(π) denote the total number of inversions in the sequence≺ x1, x2, · · · , xd Â with linear order x1 ≺ x2 ≺ · · · ≺ xd for

π =(

d1 d2 · · · drx1 x2 · · · xr

).


The permutation π is said to be even or odd according as sin(π) is evenor odd. The mapping (−1)sgn(π) from a permutation π to 1,−1 iscalled the parity of π. A cycle of length 2 is called transposition.A transposition (xi, xj), assume xi < xj and i < j without loss ofgenerality, is always an odd permutation because of odd number ofinversions as 〈xj, xi〉 with pairs (xj, xk) and (xk, xi) for i < k < j.

By observing that a cycle

(a1, a2, · · · , al) = (a1, al)(a1, al−1) · · ·(a1, a3)(a1, a2),

(1.2.5)

any permutation can be represented by composite of transpositions.Because for 1 ≤ j < k < l,

(aj, ak+1) = (ak, ak+1)(aj, ak)(ak, ak+1) (1.2.6)

the transposition representations of a permutation may have differentnumbers of transpositions. A transposition in form as ai, ai+1, 1 ≤ i <l, is said to be adjacent.

Theorem 1.2.7 Any permutation π of degree at least two has anadjacent transposition representation of the same congruent numberof transpositions modulo 2 as sgn(π).

Proof First, we show the existence of such a representation. Invirtue of (1.2.5) and (1.2.6), an adjacent transposition representationcan be found. Then, by considering that a transposition and the twosides of (1.2.6) have all an odd number of inversions, such a repre-sentation has its total number of inversions the congruent number oftranspositions as sgn(π). 2

Theorem 1.2.8 For any two permutations π and σ,

sgn(πσ) = sgn(π) + sgn(σ) (mod 2). (1.2.7)

Proof Since each transposition involves odd number of inversions,from Theorem 1.2.7 expression (1.2.7) holds. 2

In virtue of (1.2.6), we have

(−1)sgn(πσ) = (−1)sgn(π)(−1)sgn(σ) (1.2.8)

1.3 Graphs and networks 11

i.e., the parity of composite of two permutations is the product oftheir parities.

Theorem 1.2.9 All transposition representations of a permuta-tion have the same parity of the permutation.

Proof A direct conclusion of Theorem 1.2.8 in the case that oneof π and σ is the identity. 2

1.3 Graphs and networks

A graph denoted by G = (V, E) is a set V , the vertex set whoseelements are called vertices, with a binary relation E ⊆ V ∗ V =(u, v) | ∀u, v ∈ V, u 6= v. Here, (u, v) = (v, u). E is said to bean edge set whose elements are called edges. Occasionally, (u, u) andrepetition of an element in E are allowed to be called a loop and amulti-edge respectively. | V | is the order of G, which is denoted by ν,and | E |, the size denoted by ε. Of course, only finite graphs, whichare those of finite order, are considered without specific explanation inthis book. The graph whose edge set is V ∗V is called a complete graphdenoted by Kν , or simply K when without confusion. If a graphH = (V (H), E(H)) satisfies V (H) ⊆ V and E(H) ⊆ E, then it iscalled a subgraph of G denoted by H ⊆ G. It is easily seen that allgraphs are subgraphs of a complete graph and that the empty graphdenoted by ∅ as well is a subgraph of any graph. A graph withoutedge is an isolated graph and the graph with a single vertex, trivialgraph.

Theorem 1.3.1 ∀V1 ⊆ V2, E1 ⊆ E2,

(V1, E1) = G1 ⊆ G2 = (V2, E2) ⇔ E1 ⊆ V1 ∗ V1. (1.3.1)

Similarly to the case for sets in §1.1, we can define the operations:union and intersection as follows: ∀G1 = (V1, E1), G2 = (V1, E2) ⊆K,

G1 ∪G2 = (V1 ∪ V2, E1 ∪ E2); (1.3.2)

G1 ∩G2 = (V1 ∩ V2, E1 ∩ E2). (1.3.3)


It is easily shown that (2K ,⊆), 2K is the set of all subgraphs of K, is aposet with the idempotent, commutative, associative and absorptionlaws for ∪ and ∩ defined above. Therefore, from Theorem 1.1.13,(2K ,⊆;∪,∩) is a lattice.

For an edge e = (u, v) ∈ E, u and v are said to be adjacent, orsimply write “u adj v”, and e is said to be incident with u or v, or write“e ind u” or “e ind v”. Conversely, u or v are said to be incident to e,or write “u ind e” or “v ind e” as well. An edge can be considered toconsists of two semiedges: [u, v) and (u, v]. The valency of vertex v,denoted by ρ(v), is the number of semi-edges incident with v. A vertexis odd if ρ(v) = 1 (mod 2); otherwise, even. A vertex of valency k issaid to be k- valent for k ≥ 0. A 0-valent vertex is called an isolatedvertex. An articulate vertex is 1-valent.

Theorem 1.3.2 In a graph, the number of odd vertices is even.

A subgraph H of G is called a vertex induced subgraph denotedby H = G[V (H)] if E(H) = (u, v) | ∀u, v ∈ V (H), (u, v) ∈ E. Ifa subgraph H of G satisfies that V (H) = v | ∃e ∈ E(H), v ind e,then it is called an edge-induced subgraph denoted by H = G[E(H)].We may see that ∀H ⊆ G,

H = G[V (H)] ⇔ ∀u, v ∈ V (H), ¬e = (u, v) ∈ E \ E(H)

andH = G[E(H)] ⇔ ¬v ∈ V (H), ρH(v) = 0.

Let 2[G;v] and 2[G;e] be the sets of all vertex- and edge-induced sub-graphs of G respectively. It is easily shown from inspection of thethree laws for partial order in §1.1 that both (2[G;v],⊆) and (2[G;e],⊆)are posets. Further, both (2[G;v],⊆) and (2[G;e],⊆) are lattices althoughthe union and the intersection of induced subgraphs are not closed onthem in general.

A trail between two vertices u and v in G denoted by Trl(u, v) is asequence of edges e1, e2, · · · , el such that ei = (vi, vi+1), i = 1, 2, · · · , l, u =v1, v = vl+1. Here, l is called the length. When u = v, the trailTrl(u, v) is called a travel denoted by Trl(u), or simply Trl. If all theedges in Trl(u, v) are distinct, then the trail is called a walk denotedby Tr(u, v). When u = v, the walk Tr(u, v) is called a tour denoted by


Tr(u), or simply Tr. If the edge-induced subgraph H = G[E(Tr(u, v))]satisfies that (ρH(u) = ρH(v) = 1) ∧ (ρH(vi) = 2, i = 1, 2, · · · , l − 1),then the walk is called a path denoted by P (u, v). When u = v, thepath P (u, v) is a circuit denoted by C(u), or C. Of course, walks andpaths can be both seen as edge-induced subgraphs. Two vertices aresaid to be connected if there is a path between them. If all pairs ofvertices in G are connected, then G is a connected graph. It is easy tocheck by the reflective, symmetry and transitive laws in §1.1 that theconnectedness between two vertices is an equivalence on the vertexset, which is denoted by ∼c.

Theorem 1.3.3 A graph G = (V, E) is connected if, and only if,| V/ ∼c|= 1.

Let σ =| V/ ∼c| which is called the number of components of G.For a vertex v, we define G− v = (V \ v, E \ Ev), where Ev = e |∀e ∈ E, e ind v. A vertex v is called a cut-vertex if σ(G − v) > σ.Similarly, an cut-edge e : σ(G− e) > σ, G− e = (V, E \ e). A treeis such a graph that it is connected and is of least size. We may showthat all trees of order ν have the same size which is ν − 1. A graphwhose components are all trees is called a forest.

Theorem 1.3.4 A graph of order ν is a tree if, and only if, itssize is ν − 1 and all its edges are cut-edges.

A graph which has neither isolated vertex nor cut-vertex is calleda block, or non-separable one. It is obvious from inspection of O1, O2and O3 in §1.2 that the statement “two edges are on the same circuit” defines an equivalence denoted by ∼b on the edge set of a graph.

Theorem 1.3.5 A graph without isolated vertex is non- separa-ble if, and only if, | E/ ∼b|= 1.

A subgraph H of G is said to be of spanning if V (H) = V . Aspanning circuit is called a Hamiltonian circuit and a spanning touron which each edge of the graph occurs, a Eulerian tour in the graph.If a graph has a Hamiltonian circuit, or Eulerian tour, then it is aHamiltonian, or Eulerian graph respectively.


Theorem 1.3.6 A connected graph is Eulerian if, and only if, allthe valencies of its vertices are even.

For a graph G, if V = A + B ( i.e., A ∪ B provided A ∩ B =∅) and both G[A] and G[B] are isolated graphs, then G is called abipartite graph denoted by G = (A,B; E). If E = e = (u, v) | ∀(u ∈A)(v ∈ B), then the bipartite graph (A,B; E) is called a completeone denoted by Kα,β where α =| A | and β =| B |.

Theorem 1.3.7 A graph is bipartite if, and only if, it is withouta circuit of odd length.

If any pair of elements in a subset of V or E is not adjacent, thenthe subset is said to be independent. An independent subset of E fora graph G = (V, E) is also called a matching. If a matching inducesa spanning subgraph of G, then it is said to be perfect. For a ∈ V ,let Na = v | ∀v ∈ V, v adj a and for A ⊆ V , let

N(A) =⋃

a∈A

Na\A.

Theorem 1.3.8 A bipartite graph G = (X,Y ; E) has a perfectmatching if, and only if, ∀A ⊆ X and ∀A ⊆ Y, | N(A) |≥| A |.

It is known that any graph can be realized as a subset of 3-Euclidean space such that edges are represented by sections of curves(in fact, straight segments here) any of whose pairs is without commonpoint except only for the end points of the sections, which representthe common end of the corresponding edges. Such a representation ofa graph is called an embedding in the space. However, not all graphshave an embedding in the plane, or 2-Euclidean space. If a graph hasan embedding in the plane, then it is said to be planar.

A bisection is an operation of transforming G = (V, E) into a graph(V +w, (E \(u, v))+(u,w), (w, v)). If a graph can be obtainedfrom another one by a series of bisections and/or the inverses, thenthe two graphs are said to be homeomorphic.

Theorem 1.3.9 A graph is planar if, and only if, it has no sub-


graph homeomorphic to K5 or K3,3.

Two graphs G1 = (V1, E1) and G2 = (V2, E2) are said to beisomorphic if there is a bijection τ : V1 → V2 such that

∀u, v ∈ V1, (u, v) ∈ E1 ⇔ (τ(u), τ(v)) ∈ E2. (1.3.4)

The bijection τ defined by (1.3.4) is called an isomorphism betweenG1 and G2. An automorphism of G is an isomorphism between Gand itself. It would be the most difficult problem among those arementioned to justify if two graphs are isomorphic in general.

Similarly, a digraph ( or directed graph) denoted by D = (V, A)is a set V ,which is also called the vertex set, with a binary relationA ⊆ V × V = ≺ u, v Â| ∀u ∈ V, ∀v ∈ V , which is called thearc set. All the discussions above have analogues in the directed case.Particularly, a poset P = (M ;¹) can be represented by a digraphDos = (M,Aos), where ≺ x, y Â∈ Aos ⇔ (x ¹ y) ∧ (¬z, x ≺ z ≺ y),or say x is covered by y for x, y ∈ M . If a graph of order ν is associatedwith an injection (almost in any case, a bijection) from its vertex setto (onto) the integer set (1, 2, · · · , ν), then it is said to be labelled.The injection is called the labelling. The image of a vertex underthe labelling is called its label. Of course, an isomorphism betweenlabelled (directed) graphs has to be considered with the labels onvertices (directions on edges).

A network N is such a graph G = (V, E) with a real functionw(e) ∈ R, e ∈ E on E, and hence write N = (G; w). Usually, anetwork N is denoted by the graph G itself if no confusion occurs.

Finite recursion principle On a finite set A, choose a0 ∈ A asthe initial element at the 0th step. Assume ai is chosen at the ith,i ≥ 0, step with a given rule. If not all elements available from ai arenot yet chosen, choose one of them as ai+1 at the i + 1st step by therule, then a chosen element will be encountered in finite steps unlessall elements of A are chosen.

Finite restrict recursion principle On a finite set A, choosea0 ∈ A as the initial element at the 0th step. Assume ai is chosen atthe ith, i ≥ 0, step with a given rule. If a0 is not available from ai,choose one of elements available from ai as ai+1 at the i + 1st step by


the rule, then a0 will be encountered in finite steps unless all elementsof A are chosen.

The two principles above are very useful in finite sets, graphs andnetworks, even in a wide range of combinatorial optimizations.

Let N = (G; w) be a network where G = (V, E) and w(e) =−w(e) ∈ Zn = 0, 1, · · · , n − 1, i.e., mod n, n ≥ 1, integer group.For examples, Z1 = 0, Z2 = B = 0, 1 etc. Suppose xv = −xv ∈Zn, v ∈ V , are variables. Let us discuss the system of equations

xu + xv = w(e) (mod n), e = (u, v) ∈ E (1.3.5)

on Zn.

Theorem 1.3.10 System of equations (1.3.5) has a solution onZn if, and only if, there is no circuit C such that

∑

e∈C

w(e) 6= 0 (mod n) (1.3.6)

on N .

Proof Necessity. Assume C is a circuit satisfying (1.3.6) on N .Because the restricted part of (1.3.5) on C has no solution, the wholesystem of equations (1.3.5) has to be no solution either. Therefore, Nhas no such circuit. This is a contradiction to the assumption

Sufficiency. Let x0 = a ∈ Zn, start from v0 ∈ V . Assume vi ∈ Vand xi = ai at step i. Choose one of ei = (vi, vi+1) ∈ E withoutused(otherwise, backward 1 step as the step i). Choose vi+1 withai+1 = ai + w(ei) at step i + 1. If a circuit as e0, e1, · · · , el, ej =(vj, vj+1), 0 ≤ j ≤ l, vl+1 = v0, occurs within a permutation of indices,then from (1.3.6)

al+1 = al + w(el)

= al−1 + w(el−1) + w(el)

· · · · · ·

= a0 +l∑

j=0

w(ej) = a0.

Because the system of equations obtained by deleting all the equationsfor all the edges on the circuit from (1.3.5) is equivalent to the original


system of equations (1.3.5), In virtue of the finite recursion principlea solution of (1.3.5) can always be extracted. 2

When n = 2, this theorem has a variety of applications. In Liu,Y.P.[5] where Theorem 1.3.7 is a special case, some applications canbe seen. Further, its extension on a nonAbelian group can also bedone while the system of equations are not yet linear but quadratic.

A graph is said to be even if the valency of each vertex is even.

Theorem 1.3.11 A graph is even if, and only if, its edges sethas a cycle partition.

Proof Since what obtained from an even graph by deleting all theedges on a cycle is still an even graph, based on the finite recursionprinciple, the theorem is done. 2

Let G = (V, E) be a graph where V = ¶(X) and E = Bx|x ∈ Xwhere ¶(X) is a partition on

B(X) =⋃

x∈X

Bx

and Bx = x(0), x(1) for a set X. Two graphs G1 = (V1, E1) andG2 = (V2, E2) are isomorphic if, and only if, there exists a bijection ι:X1 → X2 such that the diagrams

X1ι−−−−−−−−−→ X2

σ1

y

yσ2

X1 −−−−−−−−−→ι

X2

(1.3.7)

for σi = Bi,¶i, i = 1, 2, are commutative. Let Aut(G) be the auto-morphism group of G.

On the other hand, a semi-arc isomorphism between two graphsG1 = (V1, E1) and G2 = (V2, E2) is defined to be such a bijection τ :B1(X1) → B2(X2) that

B1(X1)τ−−−−−−−−−→ B2(X2)

σ1

y

yσ2

B1(X1) −−−−−−−−−→τ

B2(X2)

(1.3.8)


for σi = Bi,¶i, i = 1, 2, are commutative. Let Aut1/2(G) be thesemi-arc automorphism group of G.

Theorem 1.3.12 If Aut(G) and Aut1/2(G) are, resp., the auto-morphism and semi-arc automorphism groups of graph G, then

Aut1/2(G) = Aut(G)× Sl2 (1.3.9)

where l is the number of self-loops on G and S2 is the symmetric groupof degree 2.

Proof Since each automorphism of G just induces two semi-arcisomorphisms of G for a self-loop, the theorem is true. 2

1.4 Groups and spaces

A group denoted by Γ = (X,♦) is a set X with a binary operationγ: X × X → X, it would be better to write x♦y for ≺ x, y Â γreferring to “♦” as the operation, such that the laws Γ1, Γ2 and Γ3described below are satisfied.

Γ1(Associative law) ∀x, y, z ∈ X, (x♦y)♦z = x♦(y♦z).

Γ2(Identity law) (∃1Γ (or simply 1) ∈ X)(∀x ∈ X, x♦1Γ = x).

Γ3(Inverse law) (∀x ∈ X)(∃y ∈ X, x♦y = 1Γ).

The element 1 in Γ2 is called a right identity and the element yin Γ3 a right inverse of x. We may also define a left identity and aleft inverse of an element. However, it is easily shown that they areall unique and the left one equals to the right. So, we are allowed tocall 1 the identity and x−1 the inverse of x.

The order of a group Γ = (X,♦) is defined to be | Γ |= |X|. Wemay see that (1,♦) is a group which is called the trivial group or theidentity group. In this book, a group Γ = (X,♦) is always written asΓ = X without specific indication. If a group Γ satisfies the conditionΓ4 below, then it is said to be Abelian.

Γ4(Commutative law) ∀x, y ∈ Γ, x♦y = y♦x.

There are two commonly used ways of writing the group operationof Γ. One is the additive notation by writing x♦y as a “ sum ” x + y

1.4 Groups and spaces 19

with the identity 0Γ ( or 0 ) and the inverse −x of x especially forAbelian groups. The other is the multiplicative notation by using a “product ” x • y ( or xy), 1Γ ( or 1 ) and x−1 as x♦y, the identity andthe inverse of x respectively for general groups.

Let Γ = (X, •) be a group and let Y ⊆ X. If Λ = (Y, •) is a group,then Λ is called a subgroup of Γ, denoted by Λ ⊆ Γ. Of course, theidentity group is a subgroup of any group and a group is a subgroupof itself.

Theorem 1.4.1 ∀Y, ∅ 6= Y ⊆ X, Λ = (Y, •) ⊆ Γ = (X, •) ⇔(∀x, y ∈ Y )(xy−1 ∈ Y ).

Let Γi = (Xi, •) ⊆ Γ = (X, •), i ∈ I. It is easily seen that ∩i∈IΓi =(∩i∈IXi, •) ⊆ Γ, which is called the intersection. For an S ⊆ X,the intersection, denoted by 〈S〉, of all subgroups which contains Sis called the subgroup generated by S in Γ. The subgroup 〈∪i∈IXi〉denoted by ∪i∈IΓi is called the join of subgroups Γi, i ∈ I. Let Γbe the set which consists of all subgroups of Γ. Then, it is obviousfrom inspection of the laws:O1–O3 in §1.1 and Theorem 1.1.13 that(Γ,⊆;∪,∩) is a lattice, more precisely, a complete lattice because anysubset of Γ has the l.u.b. which is the intersection , and the g.l.b.,which is the join, in Γ.

A subgroup Λ of a group Γ is said to be normal, or write Λ / Γ, ifit satisfies one of the following three equivalent conditions:

∀x ∈ Γ, xΛ = Λx ⇔ ∀x ∈ Γ, x−1Λx = Λ

⇔ ∀x ∈ Γ,∀y ∈ Λ, x−1yx ∈ Λ.(1.4.1)

It is easily seen that any subgroup of an Abelian group is normal.However, it does in general exist subgroups which are not normal fora non - Abelian group. One may also see that the set of all normalsubgroups of a group forms a complete lattice with the inclusion asthe order and with the intersection and the join as the two operations.

Because it can be shown that the relation, denoted by ∼N :

x ∼N y ⇔ ∃h ∈ N, x = hy, (1.4.2)

provides an equivalence on the set X of the group Γ(X, •) for N / Γ.We are allowed to define the quotient (or factor) group of N in Γ to


beΓ/N = (X/ ∼N , •), (1.4.3)

where (Nx)(Ny) = N(xy). The order of Γ/N is called the index ofN in Γ.

Let Γ and Λ be two groups. A function α : Γ → Λ is called ahomomorphism from Γ to Λ if

∀x, y ∈ Γ, α(xy) = α(x)α(y). (1.4.4)

Because o : Γ → 1Λ is a homomorphism which is called zerohomomorphism, the set Hom (Γ, Λ) of all homomorphisms from Γ toΛ is always non-empty. A homomorphism from Γ to Γ itself is saidto be an endomorphism of Γ. The identity function ι : Γ → Γ is anendomorphism of Γ.

For a homomorphism α from Γ to Λ, let

Im α = α(Γ) = α(x) | ∀x ∈ X;Ker α = x | ∀x ∈ X,α(x) = 1Λ

(1.4.5)

which is said to be the image; the kernel of α respectively. It is easyto check by Theorem 1.4.1 that Im α ⊆ Λ and Ker α / Γ. If ahomomorphism α from Γ to Λ satisfies Ker α = 1Γ, then α is calleda monomorphism. If a homomorphism α from Γ to Λ has Im α = Λ,then α is called an epimorphism. A homomorphism which is both amonomorphism and an epimorphism is said to be an isomorphism.Two groups Γ and Λ are said to be isomorphic, or written as Γ ∼= Λ,when there is an isomorphism between them. An isomorphism from Γto Γ itself is called an automorphism of Γ. It can be easily shown frominspection of the laws: Γ1 − Γ3 that the set of all automorphisms ofΓ is a group which is called the automorphism group of Γ, denoted byAut Γ.

Theorem 1.4.2(First isomorphism law) ∀α ∈ Hom(Γ, Λ),

Γ/Ker α ∼= Im α.

Based on Theorem 1.4.2, we are allowed to call Γ/Ker α thecoimage of α. If N / Γ, then the mapping φ : x 7→ Nx is an


epimorphism from Γ to Γ/N with Ker φ = N . We call φ thecanonical homomorphism.

For two groups Λ = (Y, •) ⊆ (X, •) = Γ, let ΓΛ = (XY, •), whereXY = xy | ∀x ∈ X,∀y ∈ Y . One may see that ∀Λ ⊆ Γ, N / Γ ⇒Λ ∩N / Λ.

Theorem 1.4.3(Second isomorphism law) ∀Λ ⊆ Γ,∀N / Γ,

Λ/N ∩ Λ ∼= NΛ/N.

Let N and Q be two normal subgroups of a group Γ and let N ⊆ Q.Then, it is known that Q/N / Γ/N.

Theorem 1.4.4(Third isomorphism law) ∀N,Q / Γ,

N ⊆ Q ⇒ (Γ/N)/(Q/N) ∼= Γ/Q.

Let Φ be a group, S a nonempty set and σ : S → Φ, a function.Then, Φ, or precisely (Φ, σ), is said to be free on S if for each functionα : S → Γ, there is a unique homomorphism β : Φ → Γ such thatα = βσ. A group which is free on some set is called a free group.From the definition it can be derived that σ is injective and thatIm σ generates Φ. In fact, it can be shown that for any nonemptyset S there exists a group Φ and a function σ : S → Φ such that Φ isfree on S and Φ = 〈 Im σ〉.

Theorem 1.4.5 If Φ1 is free on S1 and Φ2 is free on S2, thenΦ1∼= Φ2 ⇔| S1 | = | S2 | .

This theorem allows us to define the rank of a free group as thecardinality of any set on which it is free. Further, we have known thatany group is an image of a free group. Such an image is called a pre-sentation of the group. More precisely, a free presentation of a group Γis an epimorphism π : Φ → Γ, Φ is a free group. From Theorem 1.4.2,we have Φ/Ker π ∼= Γ. The elements of Ker π are called relators ofthe presentation. Therefore, any group can be characterized by gen-erators and relaters. Although a presentation of a group is known, to


justify if two groups are isomorphic in general is still not easy becausea group may have different kinds of presentations.

A space (or precisely a vector space or linear space ) over F denotedby (X , F ; +, •) (or simply write X ) is an Abelian group (X , +), orX as well, associated with a field (F , +, •), or simply F , and twobinary operations: “+”, called the sum and “•”, the scalar product,satisfying the following four axioms: Vects.1–4. The sum is with thesame symbol as the addition on the group X and the addition on thefield F . The scalar product a • A , or simply aA, is defined for a ∈ Fand A ∈ X and is with the same symbol as the multiplication on F .Members of X are called vectors, and those of F , scalars.

Vect.1 ∀a ∈ F , ∀A,B ∈ X , a(A + B) = aA + aB.

Vect.2 ∀a, b ∈ F , ∀A ∈ X , (a + b)A = aA + bA.

Vect.3 ∀a, b ∈ F , ∀A ∈ X , (ab)A = a(bA).

Vect.4 ∀A ∈ X , 1A = A.

It seems that the only notational distinction we have to make be-tween vectors and scalars is to denote the zero elements of X and Fby 0X and 0F respectively. However, since it is easily shown, from theaxioms Vects.1–4, that ∀A ∈ X , 0F A = 0X and that ∀a ∈ F , a0X = 0X ,the distinction will almost always be dropped and 0F , 0X be writtensimply 0.

A subset Y ⊆ X of a space X over F is said to be a subspace,denoted by Y ⊆vect X (or simply Y ⊆ X without confusion), of X ifY is a space over F in its own right, but with respect to the sameoperations as X . The zero vector 0 belongs to any space and itselfis a space called the zero space or trivial space denoted by 0 as well.Any non-zero vector of order 2 with 0 here forms a subspace which isdenoted by J .

Theorem 1.4.6 ∀Y ⊆ X , Y ⊆vect X ⇔

(∀A,B ∈ Y , A + B ∈ Y) ∧ (∀a ∈ F , ∀A ∈ Y , aA ∈ Y).

Proof The necessity is straight forward. Conversely, because Y ⊆X , from the last statement, Vect.2-4 hold and from the first statement,Vect.1 holds for Y . The sufficiency is obtained. 2


Apparently, for spaces we are also allowed to introduce the twooperations ∩, the intersection and ∪, the join as described above forgroups and find that (2X ,⊆;∪,∩) forms a lattice, of course, a completeone.

In what follows, we are only concerned with the field F = GF(2),the finite field of two elements for spaces. In this case, the space iscalled a binary space. For any A ∈ X , we always have A + A = 0, thezero vector. That is of characteristic 2. Suppose X = 2X is the freeAbelian group 〈x| ∀x ∈ X〉 generated by all the elements of X. Then,a vector is also a subset of X. We always employ the same symbol todenote a vector of X and a subset in X. Let A ∈ X , then

A =∑

x∈X

Axx =∑

x∈A

x, (1.4.6)

where Ax is said to be the coefficient, or component of A on x. Ofcourse, Ax = 1, if x ∈ A; 0, otherwise.

On the space X , we define an inner product denoted by (A,B) forA,B ∈ X as

(A,B) =∑

x∈X

AxBx. (1.4.7)

By this notation, we have the relation:

Ax = (A, x),∀x ∈ X. (1.4.8)

If for A,B ∈ X , (A,B) = 0, then A and B are said to be orthogonaldenoted by A⊥B or B⊥A from the symmetry: (A,B) = (B, A). Here,one may see

∀A,B ∈ X , (A,B) = 0 ⇔ |A ∩B| = 0 (mod 2). (1.4.9)

If (A,A) = 0, then vector A is said to be even. Let A(X ) be the setof all even vectors in X . It can be seen from inspection of axiomsVect.1–4 that A(X ) is a subspace of X and is called the alternating(or symplectic ) space on X.

Further, we may also see that for A ∈ X given,

A = 0 ⇔ ∀B ∈ X , (A,B) = 0. (1.4.10)

Or in other words, the inner product is non-degenerate.


If a vector A satisfies that ∀B ∈ B, (A,B) = 0 then it is said to beorthogonal to B and denoted by A⊥B.

Let A and B be two subspaces of X . If

A = A| ∀A ∈ X , A⊥B, (1.4.11)

then A is said to be the orthogonal space of B in X , and is denoted byA = B⊥. Moreover, from the symmetry of the inner product, we have

(B⊥)⊥ = B. (1.4.12)

In Chapters 6 and 7, we shall see a number of spaces related tographs. Almost all results for them can be extended to general spacesover GF(2), the finite field of 2 elements.

1.5 Notes

1.5.1 This book is in principle designed to be selfcontained underthe background presented in this chapter. One might still like to readmore materials related topology. References can be chosen such asAlexandroff, P.S.[1–2], Greenberg, M.J.[1], Massey, W.S.[1], Stillwell,J.[1], Agoston, M.K.[1], or Lefschets, S.[3].

1.5.2 Permutations are established from partitions on a set. Suchan idea enables us to observe embeddings, or super maps of a graphas permutations from the graph as a partition. A description in acertain detail can be seen in Liu, Y.P.[35–36] and Liu, Y.P.[41]. Mostbooks on basic algebra involve permutations such as Jacobson, N.[1],Gilbert, W.J.[1], particularly Dixon, J.D. and B. Mortimer[1].

1.5.3 A graph turns on a partition of the ground set from a setby a binary group sticking on Liu, Y.P.[35]. Although a great numberof graphs have appeared in literature, only a few much related to thisbook are listed as Ore, O.[1], Tutte, W.T.[14], Ringel, G.[3], White,A.T.[1], Lefschets, S.[2] and Liu, Y.P.[34].

1.5.4 Those mentioned in §1.4 are all extracted from Liu, Y.P.[33]and Liu, Y.P.[34]. One might also like to read more about generalgroups and finite vector spaces such as MacLane, S. and G. Birkhoff[1],Roman, S.[1], Robinson, D.J.S.[1].

Chapter 2

Polyhedra

2.1 Polygon double covers

A polygon, denoted by (a, b, c, · · ·) is a finite set of letters in a cyclicorder. In general, such a polygon can be represented by the infiniteface of a connected plane graph conformed with convex polygons andarticulate edges, or the inner face of a regular polygon. Hence, theletters in a polygon are allowed with repetition of each letter at mosttwice(with the same power or different powers: 1 always omitted and−1) in the first case. For a letter a, a−1 is called the inverse of a. Theinverse satisfies the following two rules:

Inverse rule 1 For a letter a, (a−1)−1 = a.

Inverse rule 2 For two letters a and b, (ab)−1 = b−1a−1, or(a, b)−1 = (b−1, a−1).

Two polygons A1 and A2 are dealt with the same if one becomesthe other by one of the following alternatives:

No.diff.gon1 For a ∈ A1, A2 is different from A1 only in inter-changing the positions of the two occurrences of a, if any.

No.diff.gon2 For a, b ∈ A1, A2 is different from A1 only in in-terchange between a and b.

Let polygon A = (a1, a2, · · · , al), then polygons (a2, a3, · · · , a1), · · ·,(al, a1, · · · , al−1) are, respectively, called cyclic left shift of A in 1, 2,· · ·, l − 1 bits.

No.diff.gon3 A2 is any of all the cyclic left shifts of A1.

The polygon (al, · · · , a2, a1) is called a reversion, denoted by (a1, a2,· · · , al)

rv, of polygon (a1, a2, · · · , al).

No.diff.gon4 A2 = (A1)rv.

The polygon (a−11 , a−1

2 , · · · , a−1l ) is called a conversion, denoted by

26 Chapter 2 Polyhedra

(a1, a2, · · · , al)cv, of polygon (a1, a2, · · · , al).

No.diff.gon5 A2 = (A1)cv.

An inversion of polygon A = (a1, a2, · · · , al is defined to be Aiv =(a−1

l , · · · , a−12 , a−1

1 ).

Proposition 2.1.1 For any polygon A,

Aiv = (Arv)cv = (Acv)rv. (2.1.1)

Proof Easy to check by the definitions. 2

On the basis of this proposition, it is from the inverse rule 2 seenthat Aiv = A−1.

If a set of polygons has each letter occurs exactly twice, then it iscalled a double cover on the set of all letters in.

A polyhedron P is a set C = Ci|1 ≤ i ≤ k, k ≥ 1, of polygonswhich forms a double cover on a set A of letters where Ci is called aface of P such that no proper subset of C is a double cover of a subsetof A.

This is the combinatorial representation of Heffter’s in Heffter, L.[1](and more than half a century later, Edmonds’ in Edmonds, J.R.[1]as dual case) for a polyhedron.

Let P = Ci|1 ≤ i ≤ k be a polyhedron and X = XP , the setof all letters in P . An element(or letter) in X is called an edge of P .The property that the two occurrences of a letter with the same ordifferent directions in a polyhedron is called the status of the edge. Bysticking the group B of two elements on X, each edge consists of twosemi-edges as x+, x−, or written as +x,−x, x, x−1, or x, x forcertain convenience. Each semi-edge +x, or −x, is compounded withits copy marked by a prime, i.e., +x′ or −x′ (x′, or x−1′) respectively.Then, an edge is further considered as

+x, +x′,−x,−x′ or simply, x, x′, x−1, x−1′

and hence, +x, +x′ or −x,−x′ as well, is now a semi-edge. Theset

X (P ) =∑

x∈P

(x+, x−1+ x+, x−1′) (2.1.2)

2.1 Polygon double covers 27

is called a ground set of P . An element of the ground set is also calleda quarter (of an edge).

Attention 2.1.1 (1) For x ∈ X and x ∈ XP , x has differentmeanings. The former is a letter and the later, a quarter of an edge.

(2) For x ∈ XP , both ′ and −1 are seen as permutations on theground set, i.e.,

′ =∏

x∈X+X−1

(x, x′) and −1 =∏

x∈X+X′(x, x−1) (2.1.3)

where X ′ = x′|∀x ∈ X and X−1 = x−1|∀x ∈ X for X ∈ XP .(3) For x, y ∈ XP , (xy)′ = y′x′, (xy)−1 = y−1x−1, and x′−1 = x−1′.

A face A in polyhedron P is seen in companion with A−1 on itsground set.

Proposition 2.1.2 Let P be a polyhedron with its face set A.Then, P is determined by the permutation πP on its ground set as

πP =∏

A∈A(A)(A−1) (2.1.4)

in which two occurrences of a letter with the same power are distin-guished by one with a prime.

Proof By observing that all cycles appearing in (2.1.4) form apartition, in view of §1.2 the conclusion is seen. 2

Let σ = ′ and δ = −1 be the permutations shown in (2.1.3) on theground set XP , i.e., for x ∈ XP ,

σ(x) =

y′, when x = y;

y, when x = y′(2.1.5)

and for x ∈ XP ,

δ(x) =

y−1, when x = y;

y, when x = y−1(2.1.6)

Then, π∗P = πP σδ is a permutation on XP as well.

Lemma 2.1.1 On XP , δπP = π−1P δ.


Proof In virtue of πP δx = πP x−1 = (π−1P x)−1 = δ(π−1

P x) =(δπP )x, by the arbitrariness of x ∈ XP the lemma is obtained. 2

Lemma 2.1.2 On XP , σπ∗P = π∗P−1σ.

Proof By considering that

σπ∗P = σπP σδ = σ(πP δ)σ (by Lemma 2.1.1)

= σ(δπ−1P )σ = (σ−1δ−1π−1

P )σ = π∗P−1σ,

the lemma is done. 2

Lemma 2.1.3 For x ∈ XP , two orbits (x)π∗P and (x′)π∗P are dis-joint and conjugate.

Proof In virtue of Lemma 2.1.2, the two orbits have the sametype. From Theorem 1.2.6, they are conjugate. 2

Theorem 2.1.1 Permutation π∗P on XP determines a polyhe-dron.

Proof On the basis of Lemma 2.1.3, each pair of conjugate orbitsdetermine a polygon when the prime is omitted. Then, the set of allsuch polygons form a polyhedron. 2

The polyhedron P ∗ obtained by omitting the power −1 and thenreplacing the prime by −1 from the permutation π∗P shown in thistheorem is called a dual of P . A face of the dual P ∗ is defined to bea vertex of P .

For a polyhedron P determined by permutation πP on the roundset XP , the transposition

(x−1, πP x) = (δx, πP x)

is called an angle. Two semiedges incident with the sane angle issaid to be V-adjacent. Then, an equivalence called V-adjacence byappending the transitive law on the V-adjacent relation is obtainedon the set of all semiedges.

2.2 Supports and skeletons 29

Theorem 2.1.2 A set of semiedges of a polyhedron forms a ver-tex if, and only if, it is an equivalent class under V-equivalence.

Proof In fact, a conjugate pair of cycles on π∗P determines a equiv-alent class under V-equivalence. This is the theorem. 2

Example 2.1.1 Only one polygon (ae−1b−1cdefdb−1afc−1) formsa polyhedron named by P . The permutation that determines P is

πP = (ae−1b−1cde′fdb′−1a′f ′c′−1

)

(a−1c′f ′−1a′−1

b′d−1f−1e′−1d−1c−1be).

Then,

π∗P = (ab′c)(a′c′b)(df−1c′−1)(d′c−1f ′−1

)

(a−1f ′e′−1)(a′−1

e−1f)(b−1d′−1e′)(b′−1

e′d−1).

By omitting the power −1 and then replacing the prime by −1 onπ∗P , we have

P ∗ = (ab−1c)(dfc−1)(af−1e−1)(bd−1e−1).

Theorem 2.1.3 For two polyhedra P and Q, P is a dual of Qif, and only if, Q is a dual of P . Or in other words, P ∗∗ = P .

Proof By observing that

π∗P∗ = (πP σδ)δσ = πP (σδδσ) = πP (σσ) = P,

the theorem is done from Theorem 2.1.1. 2

2.2 Supports and skeletons

A support of polyhedron P = Ci|1 ≤ i ≤ k is the network formedby graph U = (VU , EU) with a weight w on EU where VU = Ci|1 ≤i ≤ k, (Ci, Cj) ∈ EU if, and only if, Ci and Cj, 1 ≤ i, j ≤ k, have acommon letter, and

w(e) =

0, when two powers are different;

1, otherwise(2.2.1)


for e ∈ EU .The support of P ∗ is called a skeleton of P , the under graph of P

with edge weights.

Example 2.2.1 The polyhedron P and its dual P ∗ in Example2.1.1 have their supports as shown as in Fig.2.2.1

a

b

c

d

e

f

11

1

1 a

bc

d

f e11

(a) A support of P (b) A skeleton of P

Fig.2.2.1 Support and skeleton

A polyhedron is orientable if there is an orientation of each cycle,clockwise or anticlockwise, such that the two occurrences of each letterwith different powers; nonorientable, otherwise.

Attention 2.2.1 For a polygon in clockwise, its form in anti-clockwise is the inversion. From No.Diff.gon4–5 and Proposition 2.1.1,the two forms are with no difference.

For a network N = (G; w) (G = (V, E), w(e) ∈ GF(2), e ∈ E), theequation system about xv ∈ V

xu + xv = w(e) (mod 2) (2.2.2)

for all (u, v) ∈ E is called an associate equation on N .

Lemma 2.2.1 If a polyhedron P = Ci|1 ≤ i ≤ k is orientable,then the associate equation (2.2.1) on the support has a solution.

Proof First, let P be in form as the two occurrences of each letter

2.2 Supports and skeletons 31

with different powers. Since the weights of all edges are the constant 0,the equation (2.2.1) has a solution of xi = 0 for all Ci ∈ VP , 1 ≤ i ≤ k.

Then, by considering that the consistency of equation (2.2.1) is notchanged from switching the orientation of a cycle between clockwiseand anticlockwise while interchanging the weights between 0 and 1 ofall the edges in the cycle on the support, the conclusion is done. 2

The set of all edges with weight 1 in a network is called the 1-setof the network.

Lemma 2.2.2 If the equation (2.2.1) on a network has a solution,then the 1-set on the network forms a cocycle.

Proof Assume the equation (2.2.1) has a solution of. The verticesof the network can be divided into two classes. It is seen that thecocycle consisted of all the edges between the two classes is the 1-setof the network. This is the lemma. 2

Lemma 2.2.3 If the 1-set on a network forms a cocycle, thenthe network has no odd weight circuit.

Proof Since any circuit meets even number of edges with a co-cycle, all circuits are with even weight. This means no odd weightcircuit. 2

Lemma 2.2.4 If a network has no odd weight circuit, then thenetwork has no odd weight fundamental circuit for a given spanningtree.

Proof On account of that a fundamental circuit is a circuit in itsown right, the lemma follows. 2

Lemma 2.2.5 If the support of a polyhedron has no odd weightfundamental circuit for a spanning tree, then what obtained by con-tracting all edges of weight 0 on the support is a bipartite graph.

Proof By considering that the contraction of an edge with weight1 does not change the parity of circuits, the final graph with all edgesof weight 1 has no odd length circuit. From Theorem 1.3.7, it is

preface - press.ustc.edu.cnpress.ustc.edu.cn/sites/default/files/fujian/field... · on the duality,...

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