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1004_(GTM170)Sheaf Theory | 0 |
# GraduateTexts inMathematics
Glen E. Bredon
# Sheaf Theory
Second Edition Springer
Editorial Board
S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3... |
1004_(GTM170)Sheaf Theory | 1 | ology theories.
The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Alge... |
1004_(GTM170)Sheaf Theory | 2 | this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the autho... |
1004_(GTM170)Sheaf Theory | 3 | m of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) .
Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit
\[
{\mathcal{A}}... |
1004_(GTM170)Sheaf Theory | 4 | eq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mat... |
1004_(GTM170)Sheaf Theory | 5 | ight| = \{ x \in X \mid s\left( x\right) \neq 0\} \) .
The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 .
1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -man... |
1004_(GTM170)Sheaf Theory | 6 | thcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V... |
1004_(GTM170)Sheaf Theory | 7 | cal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}}
\]
\[
{\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}}
\]
\[
\text{...}
\]
\( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal... |
1004_(GTM170)Sheaf Theory | 8 | presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions.
4.2. Definition. If \( \mathcal{... |
1004_(GTM170)Sheaf Theory | 9 | sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -c... |
1004_(GTM170)Sheaf Theory | 10 | Y \) . Similarly, the total torsion product is defined to be
\[
\mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right)
\]
Clearly, we have natural isomorphisms
\[
{\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle ... |
1004_(GTM170)Sheaf Theory | 11 | ns over nonopen subspaces, and in particular that
\[
\operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right)
\]
in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack ... |
1004_(GTM170)Sheaf Theory | 12 | and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \)
If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while ... |
1004_(GTM170)Sheaf Theory | 13 | ups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology.
## Singular cohomology
Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \)... |
1004_(GTM170)Sheaf Theory | 14 | ightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by
\[
{Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldo... |
1004_(GTM170)Sheaf Theory | 15 | \mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories.
## Exercises
1. (c) If \( \mathcal{A} \) is a sheaf on... |
1004_(GTM170)Sheaf Theory | 16 | \left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \)
16. Show that there are natural isomorphisms
\[
\operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right)
\]
a... |
1004_(GTM170)Sheaf Theory | 17 | {2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \)
1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters).
We have th... |
1004_(GTM170)Sheaf Theory | 18 |
\]
Then the sequence
\[
0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{1}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{2}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }\ldots
\]
i... |
1004_(GTM170)Sheaf Theory | 19 | {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \) and \( {F}^{p}\left( {X;\mathcal{A}}\right) \) are exact functors of \( \mathcal{A} \) . We shall define a differential \( \delta : {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \rightarrow \) \( {\mathcal{F}}^{p + 1}\left( {X;\mathcal{A}}\right) \) that makes \( {\ma... |
1004_(GTM170)Sheaf Theory | 20 | h stalk \( {\mathcal{L}}_{x} \) is an injective \( {\mathcal{R}}_{x} \) -module. Thus 3.1 immediately yields the result that if \( I\left( x\right) \) is an injective \( {\mathcal{R}}_{x} \) -module for each \( x \in X \), then the sheaf \( \mathcal{I} \) on \( X \) defined by
\[
\mathcal{I}\left( U\right) = \mathop{\... |
1005_(GTM171)Riemannian Geometry | 0 |
Graduate Texts in Mathematics GTM
Peter Petersen
# Riemannian Geometry
Third Edition
Springer
## Graduate Texts in Mathematics
## Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
## Advisory Board:
Alejandro Adem... |
1005_(GTM171)Riemannian Geometry | 1 | the Cheeger-Gromoll splitting theorem and its consequences for manifolds with nonnegative Ricci curvature.
Chapter 8 covers various aspects of symmetries on manifolds with emphasis on Killing fields. Here there is a further discussion on why the isometry group is a Lie group. The Bochner formulas for Killing fields ar... |
1005_(GTM171)Riemannian Geometry | 2 | Scalar Curvature 86
3.1.6 Curvature in Local Coordinates. 89
3.2 The Equations of Riemannian Geometry 90
3.2.1 Curvature Equations 90
3.2.2 Distance Functions 95
3.2.3 The Curvature Equations for Distance Functions 97
3.2.4 Jacobi Fields 98
3.2.5 Parallel Fields 101
3.2.6 Conjugate Points. 101
3.3 Further Stu... |
1006_(GTM172)Classical Topics in Complex Function Theory | 0 |
Gradua in Math
# Reinho
Class ILON Graduate Texts in Mathematics 172
Editorial Board
S. Axler F.W. Gehring P.R. Halmos
Springer Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3 Schaefer. Topo... |
1006_(GTM172)Classical Topics in Complex Function Theory | 1 | ;
\( {}^{1} \) Cf. W. Sartorius von Waltershausen: Gauß zum Gedächtnis, Hirzel, Leipzig 1856; reprinted by Martin Sändig oHG, Wiesbaden 1965, p. 82.
\( {}^{2} \) Just let the first one come up with a wrong reference, twenty others will copy his error without ever consulting the text. [The translator is grateful to Mr... |
1006_(GTM172)Classical Topics in Complex Function Theory | 2 | ings of Holomorphic Functions 135
1. Ideals in \( \mathcal{O}\left( G\right) \) that are not finitely generated 136
2. Wedderburn's lemma (representation of 1) 136
3. Linear representation of the gcd. Principal ideal
theorem 138
4. Nonvanishing ideals 138
5. Main theorem of the ideal theory of \( \mathcal{O}\left... |
1006_(GTM172)Classical Topics in Complex Function Theory | 3 |
\]
for \( \pi \) (cf. [Z], p. 104 and p. 118). In 1655 J. Wallis discovered the famous product
\[
\frac{\pi }{2} = \frac{2 \cdot 2}{1 \cdot 3} \cdot \frac{4 \cdot 4}{3 \cdot 5} \cdot \frac{6 \cdot 6}{5 \cdot 7} \cdot \ldots \cdot \frac{{2n} \cdot {2n}}{\left( {{2n} - 1}\right) \cdot \left( {{2n} - 1}\right) } \cdot ... |
1006_(GTM172)Classical Topics in Complex Function Theory | 4 | mediately from the continuity theorem I.3.1.2.
a) If \( \prod {f}_{\nu } \) converges compactly to \( f \) in \( X \), then \( f \) is continuous in \( X \) and the sequence \( {f}_{\nu } \) converges compactly in \( X \) to 1 .
b) If \( \prod {f}_{\nu } \) and \( \prod {g}_{\nu } \) converge compactly in \( X \), th... |
1006_(GTM172)Classical Topics in Complex Function Theory | 5 | } - 1}}\right) \left( {1 + {g}_{2\nu }}\right) = 1 \) ; hence \( {p}_{1, n} = 1 \) for even \( n \) and \( {p}_{1, n} = 1 + \frac{1}{n} \) for odd \( n \) . The product \( \mathop{\prod }\limits_{{\nu > 1}}\left( {1 + {g}_{\nu }}\right) \) thus converges compactly in \( \mathbb{C} \) to 1 . In this example the subprod... |
1006_(GTM172)Classical Topics in Complex Function Theory | 6 | ) for \( \nu \geq m \) . Since \( \sum {\left| {g}_{\nu }\right| }_{L} < \infty \) by hypothesis, \( \left( *\right) \) follows.
The differentiation theorem is an important tool for concrete computations; for example, we use it in the next subsection to derive Euler's product for the sine, and we give another applicat... |
1006_(GTM172)Classical Topics in Complex Function Theory | 7 | \in \mathcal{O}\left( G\right) \) has no zeros in \( \lbrack 0, r) \) and satisfies a multiplicative duplication formula
\[
\text{(*)}g\left( {2z}\right) = {cg}\left( z\right) g\left( {z + \frac{1}{2}}\right) \;\text{when}\;z, z + \frac{1}{2},{2z} \in \lbrack 0, r)\;\text{(with}c \in {\mathbb{C}}^{ \times }\text{).}
\... |
1006_(GTM172)Classical Topics in Complex Function Theory | 8 | }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + {2w}}}\right) \cdot \mathop{\prod }\limits_{{\nu = - \infty }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + 1 + {2w}}}\right)
\]
\[
= E\left( {w, z}\right) E\left( {w + \frac{1}{2}, z}\right) \text{.}
\]
口
Eisenstein used the (trivial, but astonishing!) formula
\( \left( *\r... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 0 |
# Reinhard Diestel Graph Theory
## Electronic Edition 2005
(C) Springer-Verlag Heidelberg, New York 1997, 2000, 2005
This is an electronic version of the third (2005) edition of the above Springer book, from their series Graduate Texts in Mathematics, vol. 173. The cross-references in the text and in the margins a... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 1 | ing more over presentation than I might otherwise have done.
There are two major changes. The last chapter on graph minors now gives a complete proof of one of the major results of the Robertson-Seymour theory, their theorem that excluding a graph as a minor bounds the tree-width if and only if that graph is planar. T... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 2 | Homogeneous and universal graphs* 212
8.4 Connectivity and matching 216
8.5 The topological end space 226
Exercises 237
Notes 244
9. Ramsey Theory for Graphs 251
9.1 Ramsey's original theorems* 252
9.2 Ramsey numbers \( {}^{\left( *\right) } \) . 255
9.3 Induced Ramsey theorems 258
9.4 Ramsey properties and c... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 3 | rsection; the vertices 2,3,4 induce (or span) a triangle in \( G \cup {G}^{\prime } \) but not in \( G \)
We set \( G \cup {G}^{\prime } \mathrel{\text{:=}} \left( {V \cup {V}^{\prime }, E \cup {E}^{\prime }}\right) \) and \( G \cap {G}^{\prime } \mathrel{\text{:=}} \left( {V \cap {V}^{\prime }, E \cap {E}^{\prime }}\... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 4 | arepsilon \left( G\right) \) , none of the graphs in our sequence is trivial, so in particular \( H \neq \varnothing \) . The fact that \( H \) has no vertex suitable for deletion thus implies \( \delta \left( H\right) > \varepsilon \left( H\right) \) , as claimed.
## 1.3 Paths and cycles
path A path is a non-empty g... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 5 | + \frac{d}{d - 2}\left( {{\left( d - 1\right) }^{k} - 1}\right) < \frac{d}{d - 2}{\left( d - 1\right) }^{k}.
\]
Similarly, we can bound the order of \( G \) from below by assuming that both its minimum degree and girth are large. For \( d \in \mathbb{R} \) and \( g \in \mathbb{N} \) let
\[
{n}_{0}\left( {d, g}\right)... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 6 | ). It does, however, imply the existence of a highly connected subgraph:
## Theorem 1.4.3. (Mader 1972)
Let \( 0 \neq k \in \mathbb{N} \) . Every graph \( G \) with \( d\left( G\right) \geq {4k} \) has a \( \left( {k + 1}\right) \) -connected subgraph \( H \) such that \( \varepsilon \left( H\right) > \varepsilon \le... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 7 | eil x\rceil \cap \lceil y\rceil \cap V\left( P\right) \), the result follows.
(ii) Since \( S \) is down-closed, the upper neighbours in \( T \) of any vertex of \( G - S \) are again in \( G - S \) (and clearly in the same component), so the components \( C \) of \( G - S \) are up-closed. As \( S \) is down-closed, ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 8 | class.
---
If \( G = {MX} \) is a subgraph of another graph \( Y \), we call \( X \) a minor of \( Y \)
minor; \( \preccurlyeq \) and write \( X \preccurlyeq Y \) . Note that every subgraph of a graph is also its minor; in particular, every graph is its own minor. By Proposition 1.7.1, any minor of a graph can be ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 9 | t) \), we write
\[
{\mathcal{F}}^{ \bot } \mathrel{\text{:=}} \{ D \in \mathcal{E}\left( G\right) \mid \langle F, D\rangle = 0\text{ for all }F \in \mathcal{F}\} .
\]
\( {\mathcal{F}}^{ \bot } \)
This is again a subspace of \( \mathcal{E}\left( G\right) \) (the space of all vectors solving a certain set of linear equ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 10 | ension as \( {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \), since \( \left( \dagger \right) \) implies
\[
\dim {\mathcal{C}}^{ * } + \dim {\mathcal{C}}^{* \bot } = m = \dim {\mathcal{C}}^{* \bot } + \dim {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot }.
\]
Hence \( {\mathcal{C}}^{ * } = {\left( {\mathcal{C}}^... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 11 | ding an edge between its two neighbours. \( {}^{11} \) (If its two incident edges are identical, i.e. form a loop at \( v \), we add no edge and obtain just \( G - v \) . If they go to the same vertex \( w \neq v \), the added edge will be a loop at \( w \) . See Figure 1.10.2.) Since the degrees of all vertices other ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 12 | 1.9 does not convey an adequate impression. A good introduction is N.L. Biggs, Algebraic Graph Theory (2nd edn.), Cambridge University Press 1993. A more comprehensive account is given by C.D. Godsil & G.F. Royle, Algebraic Graph Theory, Springer GTM 207, 2001. Surveys on the use of algebraic methods can also be found ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 13 | ge theorem by alternating paths
By the marriage condition, our sequence cannot end in a vertex of \( A \) : the \( i \) vertices \( {a}_{0},\ldots ,{a}_{i - 1} \) together have at least \( i \) neighbours in \( B \) , so we can always find a new vertex \( {b}_{i} \neq {b}_{1},\ldots ,{b}_{i - 1} \) that satisfies (ii)... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 14 | \( G \) contains an Euler tour \( {v}_{0}{e}_{0}\ldots {e}_{\ell - 1}{v}_{\ell } \) , with \( {v}_{\ell } = {v}_{0} \) . We replace every vertex \( v \) by a pair \( \left( {{v}^{ - },{v}^{ + }}\right) \), and every edge \( {e}_{i} = {v}_{i}{v}_{i + 1} \) by the edge \( {v}_{i}^{ + }{v}_{i + 1}^{ - } \) (Fig. 2.1.4). ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 15 | \in C \), and consider \( T \mathrel{\text{:=}} S \cup \{ c\} \) . As \( C - c \) has odd order it has at least one odd component, which is also a component of \( G - T \) . Therefore
\[
q\left( {G - T}\right) \geq q\left( {G - S}\right) + 1\;\text{ while }\;\left| T\right| = \left| S\right| + 1,
\]
so \( d\left( T\r... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 16 | \leq 2\log \left| H\right| \) by Corollary 1.3.5 (or by \( \left| H\right| \geq {s}_{k} \), if \( \left| C\right| = g\left( H\right) \leq 2 \) ), and \( \left| H\right| \geq {s}_{k} \geq 6 \), we have
\[
\left| {H}^{\prime }\right| \geq \left| H\right| - 2\left| C\right| \geq \left| H\right| - 4\log \left| H\right| \... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 17 | ) \) and contains the ends of \( {e}^{0} \) .
Proof. As \( {F}^{0} \in {\mathcal{F}}^{0} \), we have \( {e}^{0} \in {E}^{0} \) ; let \( {C}^{0} \) be the component of \( {G}^{0} \) \( {C}^{0} \) containing \( {e}^{0} \) . We shall prove the assertion for \( U \mathrel{\text{:=}} V\left( {C}^{0}\right) \) .
Let \( i \... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 18 | cted graph \( G \) is a set of disjoint paths in \( G \) which together contain all the vertices of \( G \) .
Theorem 2.5.1. (Gallai & Milgram 1960)
Every directed graph \( G \) has a path cover \( \mathcal{P} \) and an independent set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) of vertices such that \( {v... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 19 | on. So \( V \) has a non-trivial partition \( \left\{ {{V}_{1},\ldots ,{V}_{r}}\right\} \) with exactly \( k\left( {r - 1}\right) \) cross-edges. Assume that \( \left| {V}_{1}\right| \geq 2 \) . If \( {G}^{\prime } \mathrel{\text{:=}} G\left\lbrack {V}_{1}\right\rbrack \) has \( k \) disjoint spanning trees, we may com... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 20 | at of its blocks. So what can we say about the blocks themselves? The following proposition gives a simple method by which, in principle, a list of all 2-connected graphs could be compiled:
Proposition 3.1.3. A graph is 2-connected if and only if it can be
\( \left\lbrack {\;{4.2.6}\;}\right\rbrack \)
constructed fr... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 21 | the fundamental triangles of \( G \) ; they are clearly (induced and) non-separating, as otherwise \( \left\{ {u,{v}_{e}}\right\} \) would separate \( {G}^{\prime } \), contradicting its 3-connectedness.
Now consider an induced cycle \( C \subseteq G \) that is not a fundamental triangle. If \( e \in C \), then \( C/... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 22 | y \in X \), every \( A - X \) separator in \( G - e \) is also an \( A - B \) separator in \( G \) and hence contains at least \( k \) vertices. So by induction there are \( k \) disjoint \( A - X \) paths in \( G - e \) , and similarly there are \( k \) disjoint \( X - B \) paths in \( G - e \) . As \( X \) separates ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 23 | {{x}_{P} \mid P \in \mathcal{P}}\right\}
\]
meets every \( A - B \) path in \( G \) ; then \( X \) is an \( A - B \) separator on \( \mathcal{P} \) .
Suppose there is an \( A - B \) path \( Q \) that avoids \( X \) . We know that \( Q \)
\( Q \)
meets \( V\left\lbrack \mathcal{P}\right\rbrack \), as otherwise it w... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 24 | 3.4.1). The number of independent \( H \) -paths in \( G \) is therefore bounded above by
\( {M}_{G}\left( H\right) \)
\[
{M}_{G}\left( H\right) \mathrel{\text{:=}} \min \left( {\left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\left\lfloor {\frac{1}{2}\left| {\partial C}\right| }\right\rfloor }\rig... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 25 | \left( {3.3.1}\right) \)
Proof. We prove the assertion for \( f\left( k\right) = h\left( {3k}\right) + {2k} \), where \( h \) is a \( G \) function as in Lemma 3.5.1. Let \( G \) be an \( f\left( k\right) \) -connected graph. Then \( K \) \( d\left( G\right) \geq \delta \left( G\right) \geq \kappa \left( G\right) \geq... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 26 | H\left\lbrack A\right\rbrack \), whose value of \( \varepsilon \) is still as big as before, because the \( B \) -part of \( H \) was ’light’. In fact, we may even turn \( A \cap B \) into a complete subgraph of \( H\left\lbrack A\right\rbrack \), because such new edges, if used by our linking paths, can be replaced by... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 27 | \) has all edges on \( X \) except possibly the edges \( {s}_{i}{t}_{i} \) : as no other
\( G\left\lbrack X\right\rbrack \)
edges on \( X \) may be used by the paths \( {P}_{i} \), we may add them without affecting either the premise or the conclusion in \( \left( *\right) \) .
After this modification, we can now pr... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 28 | .
To complete our proof of \( \left( *\right) \), pick a vertex \( {v}_{0} \in V \smallsetminus X \) of degree \( {d}^{ * } \) , and consider the subgraph \( H \) induced in \( G \) by \( {v}_{0} \) and its neighbours. By (2) we have \( \delta \left( H\right) \geq {8k} \), and by (3) and the choice of \( {v}_{0} \) we... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 29 | ess 1978, in R. Halin, Graphentheorie, Wissenschaftliche Buchgesellschaft 1980, and in A. Frank's chapter of the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. A survey specifically of techniques and results on minimally \( k \) -connected graphs (see below) is given by W. ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 30 | minus O \) .
Theorem 4.1.1. (Jordan Curve Theorem for Polygons)
For every polygon \( P \subseteq {\mathbb{R}}^{2} \), the set \( {\mathbb{R}}^{2} \smallsetminus P \) has exactly two regions. Each of these has the entire polygon \( P \) as its frontier.
With the help of Theorem 4.1.1, it is not difficult to prove the... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 31 |
Fig. 4.2.1. Faces \( {f}_{1},{f}_{2} \) of \( G \) in the proof of Lemma 4.2.2
and thus links two disjoint point sets \( {X}_{1},{X}_{2} \) as in Lemma 4.1.3, with \( {X}_{1} \cup {X}_{2} = G \smallsetminus \overs... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 32 | ) be the other face of \( C \) . Since the vertices \( {v}_{1} \) and \( {v}_{3} \) lie on the boundary of \( f \), they can be linked by an arc whose interior lies in \( {f}_{C} \) and avoids \( G \) . Hence by Lemma 4.1.2 (ii), the plane edge \( {v}_{2}{v}_{4} \) of \( G\left\lbrack H\right\rbrack \) runs through \( ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 33 | -1} \) induces \( \sigma \) on \( V \cup E \) . (More formally: we ask that \( \psi \) agree with \( \sigma \) on \( V \) , and that it map every plane edge \( {xy} \in H \) onto the plane edge \( \sigma \left( x\right) \sigma \left( y\right) \in \) \( {H}^{\prime } \) . Unless \( \varphi \) fixes the point \( \left( {... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 34 | C\right) \cup \widetilde{\sigma }\left( f\right) \) . We finally take the union of all these homeomorphisms, one for every face \( f \) of \( \widetilde{H} \), as our desired homeomorphism \( \varphi : {S}^{2} \rightarrow {S}^{2} \) ; as before, continuity is easily checked.
Let us return now to our original goal, the... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 35 | n \( C \smallsetminus {P}_{i} \), so \( {f}_{i} \) meets none of those edges. Hence \( {f}_{i} \subseteq {\mathbb{R}}^{2} \smallsetminus {\widetilde{G}}^{\prime } \), that is, \( {f}_{i} \) is contained in (and hence equal to) a face of \( {\widetilde{G}}^{\prime } \) . We may therefore extend \( {\widetilde{G}}^{\prim... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 36 | t( v\right) \) only for \( v = x \) and for \( v = y \) .
Theorem 4.5.1. (MacLane 1937)
\( \left\lbrack {4.6.3}\right\rbrack \)
A graph is planar if and only if its cycle space has a simple basis.
Proof. The assertion being trivial for graphs of order at most 2 , we consider a graph \( G \) of order at least 3 . If... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 37 | ting \( {v}^{ * }\left( {{f}^{ * }\left( v\right) }\right) \mathrel{\text{:=}} v \) and \( {f}^{ * }\left( {{v}^{ * }\left( f\right) }\right) \mathrel{\text{:=}} f \) for \( {f}^{ * }\left( v\right) \in {F}^{ * } \) and \( {v}^{ * }\left( f\right) \in {V}^{ * } \), we see that conditions (i) and (iii) for \( {G}^{ * } ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 38 | terized in this way?
15. \( {}^{ - } \) Does every planar graph have a drawing with all inner faces convex?
16. Modify the proof of Lemma 4.4.3 so that all inner faces become convex.
17. Does every minimal non-planar graph \( G \) (i.e., every non-planar graph \( G \) whose proper subgraphs are all planar) contain a... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 39 | has a \( k \) -colouring, a vertex colouring \( c : V \rightarrow \{ 1,\ldots, k\} \) . This \( k \) is the (vertex-) chromatic number of \( G \) ; it is denoted by \( \chi \left( G\right) \) . A graph \( G \) with \( \chi \left( G\right) = k \) is called \( k \) -chromatic; if \( \chi \left( G\right) \leq k \), we cal... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 40 | {v}_{i}\right) + 1 \) of colours required will be smallest if \( {v}_{i} \) has minimum degree in \( G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack \) . But this is easily achieved: we just choose \( {v}_{n} \) first, with \( d\left( {v}_{n}\right) = \delta \left( G\right) \), then choose as \( {v}_{n - 1} \) a v... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 41 | not force an arbitrary graph \( G \) to contain a copy of \( H \) just by making \( \chi \left( G\right) \) large enough:
## Theorem 5.2.5. (Erdős 1959)
For every integer \( k \) there exists a graph \( G \) with girth \( g\left( G\right) > k \) and chromatic number \( \chi \left( G\right) > k \) .
Theorem 5.2.5 was... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 42 | obtaining a colouring of \( G \) (contradiction).
---
\( x{y}_{0} \)
\( {G}_{0},{c}_{0},\alpha \)
\( {y}_{1},\ldots ,{y}_{k} \)
---
Let \( x{y}_{0} \in G \) be an edge. By induction, \( {G}_{0} \mathrel{\text{:=}} G - x{y}_{0} \) has a colouring \( {c}_{0} \) . Let \( \alpha \) be a colour missing at \( x \) in t... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 43 | aph of \( G \) induced by the vertices lying on \( {C}_{i} \) or in its inner face (Fig. 5.4.1). Applying the induction hypothesis first to \( {G}_{1} \) and then with the colours now assigned to \( v \) and \( w \) -to \( {G}_{2} \) yields the desired colouring of \( G \) .
![ecdfe8f9-7805-48fc-b1e8-df5c40de94c0_134_... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 44 | ced subgraph \( {D}^{\prime } \) of \( D \) has a kernel. This, however, is immediate by the stable marriage theorem (2.1.4) for \( G \) , if we interpret the directions in \( D \) as expressing preference. Indeed, given a vertex \( v \in X \cup Y \) and edges \( e,{e}^{\prime } \in V\left( {D}^{\prime }\right) \) at \... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 45 | ight on the notion of perfection, we instead give two direct proofs of its most important consequence: the perfect graph theorem, formerly Berge's weak perfect graph conjecture:
Theorem 5.5.4. (Lovász 1972)
---
perfect graph theorem
---
A graph is perfect if and only if its complement is perfect.
The first proof ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 46 | stead, i.e. to show that this is larger than \( \omega \left( \widetilde{G}\right) \) .
But is the bound of \( \left| \widetilde{G}\right| /\alpha \) likely to reflect the true value of \( \chi \left( \widetilde{G}\right) \) ? In one special case it is: if the sets \( {A}_{K} \) happen to be disjoint, we have \( \left... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 47 | ht) \leq k \) ;
(ii) \( G \) has an orientation without directed paths of length \( k - 1 \) ;
(iii) \( G \) has an acyclic such orientation (one without directed cycles).
17. Given a graph \( G \) and \( k \in \mathbb{N} \), let \( {P}_{G}\left( k\right) \) denote the number of vertex colourings \( V\left( G\right)... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 48 | ism, not only because of their use of a computer. The authors responded with a 741 page long algorithmic version of their proof, which addresses the various criticisms and corrects a number of errors (e.g. by adding more configurations to the 'unavoidable' list): K. Appel & W. Haken, Every Planar Map is Four Colorable,... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 49 | on, we assume that \( H \) is a group. We call \( f \) a circulation on \( G \) (with values in \( H \) ), or an \( H \) -circulation, if \( f \) satisfies the following two conditions:
F1) \( f\left( {e, x, y}\right) = - f\left( {e, y, x}\right) \) for all \( \left( {e, x, y}\right) \in \overrightarrow{E} \) with \( ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 50 | \) whose \( f \) -value was changed. This value, and hence that of \( {f}_{n + 1}\left( {s, V}\right) \) was raised. Therefore \( \left| {f}_{n + 1}\right| > \left| {f}_{n}\right| \) as desired.
If \( t \notin {S}_{n} \), then \( \left( {{S}_{n},\overline{{S}_{n}}}\right) \) is a cut in \( N \) . By (F3) for \( {f}_{n... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 51 | t crucial difficulties in existence proofs of \( H \) -flows are unlikely to be of a group-theoretic nature. On the other hand, being able to choose a convenient group can be quite helpful; we shall see a pretty example for this in Proposition 6.4.5.
Let \( k \geq 1 \) be an integer and \( G = \left( {V, E}\right) \) ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 52 | tion to the choice of \( f \) .
Therefore \( K\left( f\right) = 0 \) as claimed, and \( f \) is indeed a \( k \) -flow.
Since the sum of two \( {\mathbb{Z}}_{k} \) -circulations is always another \( {\mathbb{Z}}_{k} \) -circulation, \( {\mathbb{Z}}_{k} \) -flows are often easier to construct (by summing over suitable... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 53 | \( f \), our edge colouring is correct.
Conversely, since the three non-zero elements of \( {\mathbb{Z}}_{2}^{2} \) sum to zero, every 3-edge-colouring \( c : E \rightarrow {\mathbb{Z}}_{2}^{2} \smallsetminus \{ 0\} \) defines a \( {\mathbb{Z}}_{2}^{2} \) -flow on \( G \) by letting \( f\left( \overrightarrow{e}\right... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 54 | \( {G}^{ * }\left\lbrack \bar{X}\right\rbrack \) is connected.
Thus, \( X \) and \( \bar{X} \) are both connected in \( {G}^{ * } \), so \( {F}^{ * } \) is even a minimal cut in \( {G}^{ * } \) . Let \( C \subseteq G \) be the cycle with \( E\left( C\right) = F \) that exists by Proposition 4.6.1. By Lemma 6.5.1 (ii),... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 55 | n \( G \), we may assume that \( G \) is connected; as \( G \) is bridgeless, it is then 2-edge-connected. Note that any two vertices in a 2-edge-connected graph lie in some common even connected subgraph - for example, in the union of two edge-disjoint paths linking these vertices by Menger’s theorem (3.3.6(ii)). We s... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 56 | n a \( {\mathbb{Z}}_{6} \) -circulation on \( G \) that is nowhere zero. Hence, \( G \) has a 6 -flow by Theorem 6.3.3.
## Exercises
1. \( {}^{ - } \) Prove Proposition 6.2.1 by induction on \( \left| S\right| \) .
2. (i) \( {}^{ - } \) Given \( n \in \mathbb{N} \), find a capacity function for the network below suc... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 57 | interesting way to force an \( H \) minor is to assume that \( \chi \left( G\right) \) is large. Recall that if \( \chi \left( G\right) \geq k + 1 \), say, then \( G \) has a subgraph \( {G}^{\prime } \) with \( {2\varepsilon }\left( {G}^{\prime }\right) \geq \delta \left( {G}^{\prime }\right) \geq k \) (Corollary 5.2... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 58 | ght) \) have the most edges, and their degrees show that \( {T}^{r - 1}\left( n\right) \) has more edges than any \( {T}^{k}\left( n\right) \) with \( k < r - 1 \) . So it suffices to show that \( G \) is complete multipartite.
If not, then non-adjacency is not an equivalence relation on \( V\left( G\right) \) , and s... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 59 | 2}, G \) has an \( {r}^{2} \) -connected subgraph \( H \) with \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \) . To find a \( T{K}^{r} \) in \( H \), we start by picking \( r \) vertices as branch vertices, and \( r - 1 \) neighbours of each of these as some initial subdividing v... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 60 | d \( r = 4 \) (exercises), and equivalent to the four colour theorem for \( r = 5 \) and \( r = 6 \) . For \( r \geq 7 \) the conjecture is open, but it is true for line graphs (Exercise 35) and for graphs of large girth (Exercise 33; see also Corollary 7.3.9). Rephrased as \( G \succcurlyeq {K}^{\chi \left( G\right) }... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 61 | \parallel X, Y\parallel \) the number of \( X - Y \) edges of \( G \), and call
\( d\left( {X, Y}\right) \)
\[
d\left( {X, Y}\right) \mathrel{\text{:=}} \frac{\parallel X, Y\parallel }{\left| X\right| \left| Y\right| }
\]
density the density of the pair \( \left( {X, Y}\right) \) . (This is a real number between 0 ... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 62 | ight\} \), and for \( i = 1,\ldots, k \) let \( {\mathcal{C}}_{i} \) be the partition of \( {C}_{i} \) induced by \( {\mathcal{P}}^{\prime } \) . Then
\[
q\left( \mathcal{P}\right) = \mathop{\sum }\limits_{{i < j}}q\left( {{C}_{i},{C}_{j}}\right)
\]
\[
\underset{\left( \mathrm{i}\right) }{ \leq }\mathop{\sum }\limits... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 63 | cal{P}}^{\prime } = \left\{ {{C}_{0}^{\prime },{C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime }}\right\} \) is indeed a partition of \( V \) . Moreover, \( {\widetilde{\mathcal{P}}}^{\prime } \) refines \( \widetilde{\mathcal{C}} \), so
\[
q\left( {\mathcal{P}}^{\prime }\right) \geq q\left( \mathcal{C}\right) \geq q\le... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 64 | ry vertex \( {V}_{i} \) of \( R \) by a set \( {V}_{i}^{s} \) of \( s \) vertices, and every edge by a complete bipartite graph between the corresponding \( s \) -sets. \( {R}_{s} \) The resulting graph will be denoted by \( {R}_{s} \) . (For \( R = {K}^{r} \), for example, we have \( {R}_{s} = {K}_{s}^{r} \) .)
The f... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 65 | urán’s theorem. We thus have to check that \( R \) has enough edges, i.e. that enough \( \epsilon \) -regular pairs \( \left( {{V}_{i},{V}_{j}}\right) \) have density at least \( d \) . This should follow from our assumption that \( G \) has at least \( {t}_{r - 1}\left( n\right) + \gamma {n}^{2} \) edges, i.e. an edge... |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 66 | be integers. Let \( G \) be a bipartite graph with bipartition \( \{ A, B\} \), where \( \left| A\right| = \left| B\right| = n \), and assume that \( {K}_{r, r} \nsubseteq G \) . Show that
\[
\mathop{\sum }\limits_{{x \in A}}\left( \begin{matrix} d\left( x\right) \\ r \end{matrix}\right) \leq \left( {r - 1}\right) \le... |
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