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Let \( K/F \) be a separable extension, and let \( L/F \) be a purely inseparable extension. Then \( K \) and \( L \) are linearly disjoint over \( F \). | To prove this, note that if \( \operatorname{char}\left( F\right) = 0 \), then \( L = F \), and the result is trivial. So, suppose that \( \operatorname{char}\left( F\right) = p > 0 \) . We first consider the case where \( K/F \) is a finite extension. By the primitive element theorem, we may write \( K = F\left( a\rig... | Yes |
Corollary 20.21 Suppose that \( K = F\left( {{a}_{1},\ldots ,{a}_{n}}\right) \) is finitely generated and separable over \( F \) . Then there is a subset \( Y \) of \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) that is a separating transcendence basis of \( K/F \) . | Proof. This corollary is more accurately a consequence of the proof of (3) \( \Rightarrow \) (1) in Theorem 20.18, since the argument of that step is to show that if \( K \) is finitely generated over \( F \), then any finite generating set contains a separating transcendence basis. | Yes |
Corollary 20.22 Let \( F \) be a perfect field. Then any finitely generated extension of \( F \) is separably generated. | Proof. This follows immediately from part 3 of Theorem 20.18, since \( {F}^{1/{p}^{\infty }} = F \) if \( F \) is perfect. | Yes |
Corollary 20.23 Let \( F \subseteq E \subseteq K \) be fields.\n\n1. If \( K/F \) is separable, then \( E/F \) is separable.\n\n2. If \( E/F \) and \( K/E \) are separable, then \( K/F \) is separable.\n\n3. If \( K/F \) is separable and \( E/F \) is algebraic, then \( K/E \) is separable. | Proof. Part 1 is an immediate consequence of condition 2 of Theorem 20.18. For part 2 we use Theorems 20.18 and 20.12. If \( E/F \) and \( K/E \) are separable, then \( E \) and \( {F}^{1/p} \) are linearly disjoint over \( F \), and \( K \) and \( {E}^{1/p} \) are linearly disjoint over \( E \) . However, it follows f... | Yes |
Example 21.7 Let \( V = \left\{ {\left( {{t}^{3},{t}^{4},{t}^{5}}\right) : t \in C}\right\} \) . Then \( V \) is a \( k \) -variety, since \( V \) is the zero set of \( \left\{ {{y}^{2} - {xz},{z}^{2} - {x}^{2}y}\right\} \) . | To verify this, note that each point of \( V \) does satisfy these two polynomials. Conversely, suppose that \( \left( {a, b, c}\right) \in {C}^{3} \) is a zero of these three polynomials. If \( a = 0 \), then a quick check of the polynomials shows that \( b = c = 0 \), so \( \left( {a, b, c}\right) \in V \) . If \( a ... | Yes |
Lemma 21.11 The sets \( \left\{ {Z\left( S\right) : S \subseteq k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack }\right\} \) are the closed sets of a topology on \( {C}^{n} \) ; that is,\n\n1. \( {C}^{n} = Z\left( {\{ 0\} }\right) \) and \( \varnothing = Z\left( {\{ 1\} }\right) \) .\n\n2. If \( S \) and \( T \) a... | Proof. The first two parts are clear from the definitions. For the third, let \( P \in Z\left( S\right) \) . Then \( f\left( P\right) = 0 \) for all \( f \in S \), so \( \left( {fg}\right) \left( P\right) = 0 \) for all \( {fg} \in {ST} \) . Thus, \( Z\left( S\right) \subseteq Z\left( {ST}\right) \) . Similarly, \( Z\l... | Yes |
Lemma 21.14 If \( V \) is any subset of \( {C}^{n} \), then \( I\left( V\right) \) is a radical ideal of \( k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . | Proof. Let \( f \in k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) with \( {f}^{r} \in I\left( V\right) \) for some \( r \) . Then \( {f}^{r}\left( P\right) = 0 \) for all \( P \in V \) . But \( {f}^{r}\left( P\right) = {\left( f\left( P\right) \right) }^{r} \), so \( f\left( P\right) = 0 \) . Therefore, \( f ... | Yes |
Lemma 21.15 The following statements are some properties of ideals of subsets of \( {C}^{n} \) . | Proof. The first two parts of the lemma are clear from the definition of \( I\left( V\right) \) . For the third, let \( V \) be a subset of \( {C}^{n} \) . If \( f \in I\left( V\right) \), then \( f\left( P\right) = 0 \) for all \( P \in V \), so \( P \in Z\left( {I\left( V\right) }\right) \), which shows that \( V \su... | Yes |
Let \( f \in k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a polynomial, and let \( V = Z\left( f\right) \) . If \( f = {p}_{1}^{{r}_{1}}\cdots {p}_{t}^{{r}_{t}} \) is the irreducible factorization of \( f \), then \( I\left( V\right) = \sqrt{\left( f\right) } \) by the Nullstellensatz. However, we show th... | for, if \( g \in \sqrt{\left( f\right) } \), then \( {g}^{m} = {fh} \) for some \( h \in k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) and some \( m > 0 \) . Each \( {p}_{i} \) then divides \( {g}^{m} \) ; hence, each \( {p}_{i} \) divides \( g \) . Thus, \( g \in \left( {{p}_{1}\cdots {p}_{t}}\right) \) . Fo... | Yes |
Let \( V \) be an irreducible \( k \) -variety. By taking complements, we see that the definition of irreducibility is equivalent to the condition that any two nonempty open sets have a nonempty intersection. Therefore, if \( U \) and \( {U}^{\prime } \) are nonempty open subsets of \( V \), then \( U \cap {U}^{\prime ... | If \( U \) is a nonempty open subset of \( V \), and if \( C \) is the closure of \( U \) in \( V \), then \( U \cap \left( {V - C}\right) = \varnothing \) . The set \( V - C \) is open, so one of \( U \) or \( V - C \) is empty. Since \( U \) is nonempty, this forces \( V - C = \varnothing \) , so \( C = V \) . But th... | Yes |
Proposition 21.21 Let \( V \) be a \( k \) -variety. Then \( V \) is irreducible if and only if \( I\left( V\right) \) is a prime ideal, if and only if the coordinate ring \( k\left\lbrack V\right\rbrack \) is an integral domain. | Proof. First suppose that \( V \) is irreducible. Let \( f, g \in k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) with \( {fg} \in I\left( V\right) \) . Then \( I = I\left( V\right) + \left( f\right) \) and \( J = I\left( V\right) + \left( g\right) \) are ideals of \( k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\rig... | Yes |
Let \( V = Z\left( {y - {x}^{2}}\right) \). Then the coordinate ring of \( V \) is \( k\left\lbrack {x, y}\right\rbrack /\left( {y - {x}^{2}}\right) \), which is isomorphic to the polynomial ring \( k\left\lbrack t\right\rbrack \) by sending \( t \) to the coset of \( x \) in \( k\left\lbrack V\right\rbrack \). | Therefore, the function field of \( V \) is the rational function field \( k\left( t\right) \). | No |
Let \( V = Z\left( {{y}^{2} - {x}^{3}}\right) \). Then \( k\left( V\right) \) is the field \( k\left( {s, t}\right) \), where \( s \) and \( t \) are the images of \( x \) and \( y \) in \( k\left\lbrack V\right\rbrack = k\left\lbrack {x, y}\right\rbrack /\left( {{y}^{2} - {x}^{3}}\right) \), respectively. Note that \(... | Let \( z = t/s \). Substituting this equation into \( {t}^{2} = {s}^{3} \) and simplifying shows that \( s = {z}^{2} \), and so \( t = {z}^{3} \). Thus, \( k\left( V\right) = k\left( z\right) \). The element \( z \) is transcendental over \( k \), since if \( k\left( V\right) /k \) is algebraic, then \( k\left\lbrack V... | Yes |
If \( V \) is an irreducible \( k \) -variety, then \( V \) gives rise to a field extension \( k\left( V\right) \) of \( k \) . We can reverse this construction. Let \( K \) be a finitely generated field extension of \( k \) . Say \( K = k\left( {{a}_{1},\ldots ,{a}_{n}}\right) \) for some \( {a}_{i} \in K \) . Let\n\n... | Then \( P \) is the kernel of the ring homomorphism \( \varphi : k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \rightarrow K \) that sends \( {x}_{i} \) to \( {a}_{i} \), so \( P \) is a prime ideal. If \( V = Z\left( P\right) \), then \( V \) is an irreducible \( k \) -variety with coordinate ring \( k\left\lbr... | Yes |
5.3 Lemma.\n\n(a) For every fixed nonzero \( u \in {\mathbb{R}}^{n} \), the hyperplane\n\n(*)\n\n\[ {H}_{K}\left( u\right) \mathrel{\text{:=}} \left\{ {x\mid \langle x, u\rangle = {h}_{K}\left( u\right) }\right\} \]\n\nis a supporting hyperplane of \( K \) (Figure 9b).\n\n(b) Every supporting hyperplane of \( K \) has ... | Proof.\n\n(a) Since \( K \) is compact and \( \langle \cdot, u\rangle \) is continuous, for some \( {x}_{0} \in K \),\n\n\[ \left\langle {{x}_{0}, u}\right\rangle = {h}_{K}\left( u\right) = \mathop{\sup }\limits_{{x \in K}}\langle x, u\rangle .\n\]\nFor an arbitrary \( y \in K \), it follows that \( \langle y, u\rangle... | Yes |
6.4 Theorem. Let \( K \) be a convex body with \( 0 \in \operatorname{int}K \) . Then,\n\n(a)\n\[ \n{K}^{* * } = K \n\]\n\n(b) The distance function of \( K \) equals the support function of \( {K}^{ * } \), and, conversely,\n\n\[ \n{d}_{K} = {h}_{{K}^{ * }},\;{d}_{{K}^{ * }} = {h}_{K}. \n\] | Proof.\n\n(a) By definition of \( {H}_{u} \), for every \( u \neq 0 \) of \( K \) ,\n\n\[ \n{H}_{u}^{ - } = \{ x \mid \langle u, x\rangle \leq 1\} \n\]\n\nTherefore,(using the obvious notation \( \langle K, x\rangle \leq 1 \) )\n\n\[ \n{K}^{ * } = \{ x \mid \langle K, x\rangle \leq 1\} \;\text{ and }\;{K}^{* * } = \lef... | Yes |
6.8 Theorem. Every positive homogeneous and convex function \( h : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is the support function \( h = {h}_{K} \) of a unique convex body \( K \) (whose dimension is possibly less than \( n \) ). | Proof. Let us write \( {\mathbb{R}}^{n} = U \oplus {U}^{ \bot } \), where \( U \) is the maximal linear subspace of \( {\mathbb{R}}^{n} \) on which \( h \) is linear. Then, there exists \( a \in U \) such that, for \( \left( {x,{x}^{\prime }}\right) \in U \oplus {U}^{ \bot } \) ,\n\n(*)\n\n\[ h\left( {x,{x}^{\prime }}\... | Yes |
2.8 Theorem. Let \( F \) be a proper face of the polytope \( P \), and let \( {P}^{ * } \) be the polar polytope of \( P \) with respect to a polarity \( \pi \) . For an affine subspace \( U \) of \( {\mathbb{R}}^{n} \), let \( {\pi }_{U} \) denote the restriction of \( \pi \) to \( U \), and set \( {\pi }_{U}\left( {P... | Proof. Let \( k \mathrel{\text{:=}} \dim F \), so that \( \dim U = n - k - 1 \) for \( U \) as in 2.6. For any face \( G \) which contains \( F \) properly, we set\n\n\[G\underset{\varphi }{ \mapsto }\;{\pi }_{U}\left( {G \cap U}\right)\]\n\nIf \( g \mathrel{\text{:=}} \dim \left( {G \cap U}\right) \), so that \( g = \... | Yes |
Let \( U, W \) be subspaces of \( V = {\mathbb{R}}^{4} = U \oplus W \) where \( U = \operatorname{lin}\left\{ {{e}_{1},{e}_{2}}\right\} \) , \( W = \operatorname{lin}\left\{ {{e}_{3},{e}_{4}}\right\} ,{e}_{1},\ldots ,{e}_{4} \) the canonical basis of \( {\mathbb{R}}^{4} \) . We write the coordinates of the elements of ... | Let\n\n\[ X = \left( {{x}_{1}{x}_{2}{x}_{3}{x}_{4}}\right) = \left( \begin{array}{llll} 1 & 0 & 1 & 2 \\ 0 & 1 & 2 & 1 \end{array}\right) \]\n\n\[ \left( {{b}_{1}{b}_{2}{b}_{3}{b}_{4}}\right) = \left( \begin{array}{llll} 1 & 0 & 1 & 2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \]\n\n\( {L}_{1... | Yes |
In Example 2, replace the elements \( {x}_{3},{x}_{4} \) by \( \frac{1}{3}{x}_{3},\frac{1}{3}{x}_{4} \), respectively. Then the linear transform obtained by the same calculation as in Example 1 is | \[ \left( \begin{array}{l} {\bar{x}}_{1} \\ {\bar{x}}_{2} \\ {\bar{x}}_{3} \\ {\bar{x}}_{4} \end{array}\right) = \left( \begin{array}{l} {L}_{2}^{ * }\left( {b}_{1}^{ * }\right) \\ {L}_{2}^{ * }\left( {b}_{2}^{ * }\right) \\ {L}_{2}^{ * }\left( {b}_{3}^{ * }\right) \\ {L}_{4}^{ * }\left( {b}_{4}^{ * }\right) \end{array... | Yes |
Example 1. We consider a triangular prism \( P \) in \( {\mathbb{R}}^{3} \) and wish to find a Gale transform of \( X \mathrel{\text{:=}} \) vert \( P \) (see Figure 9). Since \( v = 6 \) and \( n = 3 \), the elements of a Gale transform lie in a two-dimensional space. By Theorem 5.3, it is sufficient to find two indep... | \[ {\left( {a}_{1}{a}_{2}\right) }^{t} = \left( \begin{array}{rrrrrr} - 1 & 1 & 0 & 1 & - 1 & 0 \\ - 1 & 0 & 1 & 1 & 0 & - 1 \end{array}\right) . \] By Theorem 5.3, the rows of \( \left( {{a}_{1}{a}_{2}}\right) \) provide a Gale transform. | Yes |
5.4 Theorem. A finite sequence \( X \) in \( U = \operatorname{aff}X \) consists of all points of the vertex set of a polytope \( P \) in \( U \) if and only if one (and, thus, every) Gale transform \( {\bar{X}}_{\widehat{U}} \) of \( X \) satisfies the following condition:\n\n(4) For every hyperplane \( H \) in \( \op... | Proof. In appropriate coordinates for \( \widehat{U} \), we may identify \( U \) with \( U \times \{ 1\} \subset \widehat{U} \) . First, suppose that \( X \) comes from a polytope. Then, each element \( {\widehat{x}}_{i} \) of \( {X}_{i\prime } \) is a face. By Theorem 4.14, \( {\bar{X}}_{\widehat{U}} \smallsetminus \l... | Yes |
The vertex set \( X = \left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) \) of an \( n \) -simplex in \( {\mathbb{R}}^{n} \) is characterized by \( {\overline{\widehat{x}}}_{1} = \cdots = {\overline{\widehat{x}}}_{n} = 0 \) for a Gale transform \( {\bar{X}}_{\widehat{U}} \) of \( X \). | This also follows directly from \( \dim \operatorname{lin}{\bar{X}}_{\widehat{U}} = n + 1 - n - 1 = 0 \) by Lemma 4.5. | Yes |
6.5 Theorem. An \( n \) -dimensional polytope \( P = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{v}}\right\} \) is simplicial if and only if, for a Gale diagram \( \bar{X} \) of \( X = \left( {\operatorname{vert}P}\right) \), the following condition holds: (1) For any hyperplane \( H \) in \( \operatorname{lin}\ba... | Proof. Suppose \( 0 \in \) relint conv \( \left( {\bar{X} \cap H}\right) \) for some hyperplane \( H \) . Since \( \dim P = n \) and \( P \) lies in a hyperplane of \( U \) which does not contain 0, we have \( \dim U = \operatorname{rank}X = n + 1 \), so, \( \operatorname{rank}\bar{X} = v - \operatorname{rank}X = v - n... | Yes |
6.5 Theorem. An \( n \) -dimensional polytope \( P = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{v}}\right\} \) is simplicial if and only if, for a Gale diagram \( \bar{X} \) of \( X = \left( {\operatorname{vert}P}\right) \), the following condition holds: (1) For any hyperplane \( H \) in \( \operatorname{lin}\ba... | Proof. Suppose \( 0 \in \) relint conv \( \left( {\bar{X} \cap H}\right) \) for some hyperplane \( H \) . Since \( \dim P = n \) and \( P \) lies in a hyperplane of \( U \) which does not contain 0, we have \( \dim U = \operatorname{rank}X = n + 1 \), so, \( \operatorname{rank}\bar{X} = v - \operatorname{rank}X = v - n... | Yes |
1.16 Lemma. Let \( F,{F}^{\prime } \) be cones in \( {\mathbb{R}}^{n} \) such that \( F \cdot {F}^{\prime } \) is defined. | Proof. (a) For \( x \in F \cdot {F}^{\prime } \), we may set\n\n\[ x = {\lambda }_{1}{y}_{1} + \cdots + {\lambda }_{k}{y}_{k} + {\lambda }_{k + 1}{y}_{k + 1} + \cdots + {\lambda }_{m}{y}_{m} \]\n\nwhere \( {y}_{1},\ldots ,{y}_{k} \in F,{y}_{k + 1},\ldots ,{y}_{m} \in {F}^{\prime } \), and \( {\lambda }_{1},\ldots ,{\la... | Yes |
1.16 Lemma. Let \( F,{F}^{\prime } \) be cones in \( {\mathbb{R}}^{n} \) such that \( F \cdot {F}^{\prime } \) is defined.\n\n(a) \( F \cdot {F}^{\prime } = F + {F}^{\prime } \) (vector sum).\n\n(b) \( F \cdot {F}^{\prime } \) is a cone with apex 0 .\n\n(c) If \( S \) is the unit sphere of \( {\mathbb{R}}^{n} \), then,... | Proof. (a) For \( x \in F \cdot {F}^{\prime } \), we may set\n\n\[ x = {\lambda }_{1}{y}_{1} + \cdots + {\lambda }_{k}{y}_{k} + {\lambda }_{k + 1}{y}_{k + 1} + \cdots + {\lambda }_{m}{y}_{m} \]\n\nwhere \( {y}_{1},\ldots ,{y}_{k} \in F,{y}_{k + 1},\ldots ,{y}_{m} \in {F}^{\prime } \), and \( {\lambda }_{1},\ldots ,{\la... | Yes |
1.16 Lemma. Let \( F,{F}^{\prime } \) be cones in \( {\mathbb{R}}^{n} \) such that \( F \cdot {F}^{\prime } \) is defined.\n\n(a) \( F \cdot {F}^{\prime } = F + {F}^{\prime } \) (vector sum).\n\n(b) \( F \cdot {F}^{\prime } \) is a cone with apex 0 .\n\n(c) If \( S \) is the unit sphere of \( {\mathbb{R}}^{n} \), then,... | Proof. (a) For \( x \in F \cdot {F}^{\prime } \), we may set\n\n\[ x = {\lambda }_{1}{y}_{1} + \cdots + {\lambda }_{k}{y}_{k} + {\lambda }_{k + 1}{y}_{k + 1} + \cdots + {\lambda }_{m}{y}_{m} \]\n\nwhere \( {y}_{1},\ldots ,{y}_{k} \in F,{y}_{k + 1},\ldots ,{y}_{m} \in {F}^{\prime } \), and \( {\lambda }_{1},\ldots ,{\la... | Yes |
Let \( P \) be a polytope and \( s\left( {p;F}\right) \mathcal{B}\left( P\right) \) a stellar subdivision of its boundary complex. Then, there exists a polytope \( {P}^{\prime } \) such that \[ s\left( {p;F}\right) \mathcal{B}\left( P\right) \approx \mathcal{B}\left( {P}^{\prime }\right) \] | We may assume \( \dim P = n \) . For \( \dim F = n - 1 \), we obtain a \( {P}^{\prime } \) by placing a sufficiently \ | No |
2.2 Theorem. Let \( P \) be a polytope and \( s\left( {p;F}\right) \mathcal{B}\left( P\right) \) a stellar subdivision of its boundary complex. Then, there exists a polytope \( {P}^{\prime } \) such that \n\n\[ \n s\left( {p;F}\right) \mathcal{B}\left( P\right) \approx \mathcal{B}\left( {P}^{\prime }\right) \n\] | Proof. We may assume \( \dim P = n \) . For \( \dim F = n - 1 \), we obtain a \( {P}^{\prime } \) by placing a sufficiently \ | No |
2.6 Theorem. Let \( \mathcal{C} \) be a cell complex and let \( {Z}_{1},\ldots ,{Z}_{r} \) be the 1-cells (in case \( \mathcal{C} \) is a complex of cones) or 0-cells otherwise (in some order). For \( {v}_{i} \in \) relint \( {Z}_{i}\left( \left\{ {v}_{i}\} = {Z}_{i}\text{in case of 0 -cells}), i = 1,\ldots, r\text{, w... | Proof. By definition, \( s\left( {{v}_{i};{Z}_{i}}\right) \) does not add a 1-cell (in case of a cone complex) or a 0-cell (otherwise) to \( {\mathcal{C}}_{i - 1}, i = 1,\ldots, r \) . Suppose \( F \in {\mathcal{C}}_{r} \) is not a simplex cone or a simplex. Then, there exist \( {Z}_{j} \subset F,{Z}_{k} \subset F \) s... | Yes |
2.6 Theorem. Let \( \mathcal{C} \) be a cell complex and let \( {Z}_{1},\ldots ,{Z}_{r} \) be the 1-cells (in case \( \mathcal{C} \) is a complex of cones) or 0-cells otherwise (in some order). For \( {v}_{i} \in \) relint \( {Z}_{i}\left( \left\{ {v}_{i}\} = {Z}_{i}\text{in case of 0 -cells}), i = 1,\ldots, r\text{, w... | Proof. By definition, \( s\left( {{v}_{i};{Z}_{i}}\right) \) does not add a 1-cell (in case of a cone complex) or a 0-cell (otherwise) to \( {\mathcal{C}}_{i - 1}, i = 1,\ldots, r \) . Suppose \( F \in {\mathcal{C}}_{r} \) is not a simplex cone or a simplex. Then, there exist \( {Z}_{j} \subset F,{Z}_{k} \subset F \) s... | Yes |
In \( {\mathbb{R}}^{2} \), let \( K \mathrel{\text{:=}} \operatorname{conv}\left\{ {2{e}_{1}, - 2{e}_{1},2{e}_{2}}\right\}, L \mathrel{\text{:=}} \operatorname{conv}\left\{ {{e}_{1}, - {e}_{1},{e}_{1} + }\right. \) \( \left. {2{e}_{2}, - {e}_{1} + 2{e}_{2}}\right\} \) . As is seen from Figure \( 4, d\left( {K, L}\right... | Proof of Theorem 2.3.\n\n(1) and (3) are true by definition of \( d \) .\n\n(2) If \( d\left( {K, L}\right) = 0, K + 0 \cdot B \supset L \), and \( L + 0 \cdot B \supset K \) ; hence, \( K = L \) . Clearly, \( d\left( {K, K}\right) = 0 \) .\n\n(4) We set \( r \mathrel{\text{:=}} d\left( {K, M}\right), s \mathrel{\text{... | Yes |
Let \( S \mathrel{\text{:=}} \operatorname{conv}\left\{ {0,{e}_{1},{e}_{2},{e}_{3}}\right\} \) be a simplex in \( {\mathbb{R}}^{3} \), and \( {I}_{1} \mathrel{\text{:=}} \left\lbrack {0,{e}_{1}}\right\rbrack \) , \( {I}_{2} \mathrel{\text{:=}} \left\lbrack {0,{e}_{2}}\right\rbrack \) line segments. We wish to calculate... | By Theorem 3.7,\n\n\[ \n{6V}\left( {S,{I}_{1},{I}_{2}}\right) = V\left( {S + {I}_{1} + {I}_{2}}\right) - V\left( {S + {I}_{1}}\right) - V\left( {S + {I}_{2}}\right) \n\]\n\n\[ \n- V\left( {{I}_{1} + {I}_{2}}\right) + V\left( S\right) + V\left( {I}_{1}\right) + V\left( {I}_{2}\right) . \n\]\n\nClearly \( V\left( {I}_{1}... | Yes |
4.6 Lemma. Let \( P, Q \) be \( n \) -polytopes in \( {\mathbb{R}}^{n} \). Then, for any \( \varepsilon > 0 \), there exists a polytope \( {Q}^{\prime } \) such that (1) \( d\left( {Q,{Q}^{\prime }}\right) < \varepsilon \), and (2) \( P \) and \( {Q}^{\prime } \) are in skew position. | Proof. Let \( {G}_{1},\ldots ,{G}_{r} \) be those faces of \( Q \) for which faces \( {F}_{1},\ldots ,{F}_{r} \) of \( P \) exist, respectively, such that \( {U}_{{F}_{i}} \cap {U}_{{G}_{i}} \neq \{ 0\}, i = 1,\ldots, r \). Up to renumbering, let \( \dim {G}_{1} = \cdots = \dim {G}_{{r}_{1}} = : {k}_{1},\ldots ,\dim {G... | Yes |
4.13 Theorem. Let \( {K}_{1},\ldots ,{K}_{n} \) be convex bodies. Then,\n\n(a) \( V\left( {{K}_{1},\ldots ,{K}_{n}}\right) \geq 0 \) ,\n\n(b) \( V\left( {{K}_{1},\ldots ,{K}_{n}}\right) > 0 \) if and only if each \( {K}_{i} \) contains a line segment \( {I}_{i} = \) \( \left\lbrack {{a}_{i},{b}_{i}}\right\rbrack \) suc... | Proof.\n\n(a) If we replace each \( {K}_{i} \) by one of its points, \( {p}_{i}, i = 1,\ldots, n \), and apply Theorem 4.12, we obtain (a) from \( V\left( {{p}_{1},\ldots ,{p}_{n}}\right) = 0 \) (which follows, for example, by Theorem 3.7).\n\n(b) From Theorem 3.7, we obtain\n\n\[ n!V\left( {{I}_{1},\ldots ,{I}_{n}}\ri... | Yes |
When does equality hold in (AF)? | Clearly a sufficient condition is \( K \) and \( L \) to be homothetic: If \( K = a + {tL} \) , \( t > 0 \), then, \( V\left( {K, L,\mathcal{C}}\right) = V\left( {a + {tL}, L,\mathcal{C}}\right) = {tV}\left( {L, L,\mathcal{C}}\right) \), so that equality in (AF) follows. However, homothety of \( K \) and \( L \) is not... | Yes |
Example 2. Let \( {K}_{1} \) be a 3-simplex in \( {\mathbb{R}}^{3} \), and let \( K, L \) be obtained from \( {K}_{1} \) by chopping off the vertices but leaving at least one point of each edge remaining (Figure 15). Then, \( \left( {2}^{\prime }\right) \) and equality in \( \left( {3}^{\prime }\right) \) are trivially... | Proof. We can express (S) in terms of \( {\Theta }_{F}^{K},{\Theta }_{F}^{L} \) . Consider the set \( {\widehat{\Theta }}_{F}^{K} \) (see Definition 4.8). By the assumptions of the theorem and by using (1) in Lemma 5.4, we may suppose \( K, L \) to be strictly combinatorially isomorphic and both in skew position to \( ... | Yes |
3.5 Lemma. If \( \sigma \) has an apex, then, the monoid \( \sigma \cap {\mathbb{Z}}^{n} \) has (up to renumbering) precisely one minimal system of generators. | Proof. Let \( {a}_{1},\ldots ,{a}_{t} \) and \( {b}_{1},\ldots ,{b}_{m} \) be different minimal systems of generators and \( {b}_{1} \notin \left\{ {{a}_{1},\ldots ,{a}_{t}}\right\} \), say. There exist linear combinations, say\n\n\[ \n{b}_{1} = \mathop{\sum }\limits_{{j = 1}}^{r}{\lambda }_{j}{a}_{j}\text{ for }{\lamb... | Yes |
For every monoid \( \sigma \cap {\mathbb{Z}}^{n} \), where the cone \( \sigma = \operatorname{pos}\left\{ {{a}_{1},\ldots ,{a}_{k}}\right\} \) is generated by primitive lattice vectors \( {a}_{j} \), is \( {S}_{0} \mathrel{\text{:=}} \left\{ {{\alpha }_{1}{a}_{1} + \cdots + {\alpha }_{k}{a}_{k} \mid {\alpha }_{1},\ldot... | As is seen from Example 1, \( {S}_{0} \) need not be equal to \( \sigma \cap {\mathbb{Z}}^{n} \). | No |
In \( {\mathbb{R}}^{3} \) let \( X = \left( {{a}_{1},\ldots ,{a}_{6}}\right) \) come from the vertices of a regular prism \( P \) with \( 0 \in \) int \( P \) ,(see Figure 8). Let the fan \( \sum \) be defined by the following facets:\n\n\[ \operatorname{pos}\left\{ {{a}_{1},{a}_{2},{a}_{3}}\right\} ,\operatorname{pos}... | provide a basis for the space of affine dependences of \( X \) . A Gale transform \( {\bar{X}}_{\widehat{U}} \) is formed by the rows of the transposed of the matrix\n\n\[ \left( \begin{array}{rrrrrr} 1 & 0 & - 1 & - 1 & 0 & 1 \\ 1 & - 1 & 0 & - 1 & 1 & 0 \end{array}\right) \]\n\n(compare Figure 9). | Yes |
In \( {\mathbb{R}}^{3} \) choose a 3-simplex \( T \), a vertex \( v \) of \( T \), and a triangle \( \Delta \subset \) \( \left( {{\mathbb{R}}^{3} \oplus \mathbb{R}}\right) \smallsetminus \) aff \( T.P \mathrel{\text{:=}} \operatorname{conv}\left( {T \cup \Delta }\right) \) is then a polytope with 3 double-simplices an... | We may place \( \Delta \) such that we obtain facets \( {F}_{1},\ldots ,{F}_{7} \), up to renumbering as follows:\n\n\[ \n{F}_{1} = \left\lbrack {1,2,4,5,6}\right\rbrack \n\]\n\n\[ \n{F}_{2} = \left\lbrack {2,3,4,6,7}\right\rbrack \n\]\n\n\[ \n{F}_{3} = \left\lbrack {1,3,4,5,7}\right\rbrack \n\]\n\n\[ \n{F}_{4} = \left... | Yes |
4.12 Theorem (Oda’s criterion). A complete simplicial fan \( \sum \) is regular if and only if the following conditions are satisfied:\n\n(a) There exists in \( \sum \) at least one regular \( n \) -cone.\n\n(b) If \( \sigma = \operatorname{pos}\left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\} ,{\sigma }^{\prime } = \o... | Proof. Let \( \sum \) be regular, \( \det \sigma = \det \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) = 1 \) . Then, \( \det \left( {x}_{1}^{\prime }\right. \) , \( \left. {{x}_{2},\ldots ,{x}_{n}}\right) = - 1 \), and, hence,\n\n\[ \n\det \left( {{x}_{1} + {x}_{1}^{\prime },{x}_{2},\ldots ,{x}_{n}}\right) = 0.\n\]\n... | Yes |
5.17 Theorem. Let \( \sum = \sum \left( P\right) \) be strongly polytopal. . (a) The polytope group \( \widetilde{\mathcal{G}} \) is the smallest group into which the semi-group of all polytopes strictly combinatorially isomorphic to \( P \) can be embedded. (b) Pic \( \sum \) can be generated by \( {f}_{n - 1}\left( P... | Proof. (a) is a consequence of Theorem 5.15; (b) follows from Theorems 5.9 and 5.15. | No |
Example 7. If \( \sum \) consists of the cones into which \( {\mathbb{R}}^{n} \) is split by the coordinate hyperplanes, we can set \( {P}_{i} = \left\lbrack {0,{e}_{i}}\right\rbrack \) (line segments), \( i = 1,\ldots, n \), and we obtain a system of \( n \) polytope elements generating | \[ \text{Pic}\sum \cong {\mathbb{Z}}^{n}\text{.} \] | No |
Example 8. In the example of Figure 15, we may choose \( {P}_{1} \mathrel{\text{:=}} \left\lbrack {0,{e}_{1}}\right\rbrack ,{P}_{2} \mathrel{\text{:=}} \) \( \left\lbrack {0,{e}_{2}}\right\rbrack ,{P}_{3} \mathrel{\text{:=}} \operatorname{conv}\left\{ {{e}_{1},{e}_{2},{e}_{1} + {e}_{2}}\right\} \) . Therefore, | \[ \text{Pic}\sum \cong {\mathbb{Z}}^{3}\text{.} \] | Yes |
1.34 Lemma. Let \( \varphi : X \rightarrow Y \) be a morphism of affine varieties, and let \( {\varphi }^{ * } \) : \( {R}_{Y} \rightarrow {R}_{X} \) be the corresponding morphism of rings. Then,\n\n(a) \( {\varphi }^{ * } \) is an isomorphism if and only if \( \varphi \) is an isomorphism, and\n\n(b) \( {\varphi }^{ *... | Proof.\n\n(a) We need only show the \ | No |
1.38 Lemma. If \( Y \) is an affine algebraic variety, then, the dimension of \( Y \) is equal to the dimension of its coordinate ring \( {R}_{Y} \) . | Proof. The irreducible affine algebraic sets contained in \( Y \subset {\mathbb{C}}^{n} \) correspond to those prime ideals in \( R \mathrel{\text{:=}} \mathbb{C}\left\lbrack {{\xi }_{1},\ldots ,{\xi }_{n}}\right\rbrack \) which contain \( {i}_{Y} \) . These ideals are in one-to-one correspondence \( \varphi \) to prim... | Yes |
The largest possible \( n \) -dimensional cone is \( \sigma \mathrel{\text{:=}} {\mathbb{R}}^{n} \) . Then, viewed as a monoid, \( \sigma \cap {\mathbb{Z}}^{n} = {\mathbb{Z}}^{n} \) has generators \( {e}_{1},\ldots ,{e}_{n}, - {e}_{1},\ldots , - {e}_{n} \), so the associated algebra is \( {R}_{\sigma } = \mathbb{C}\lef... | \[ {\xi }_{i}{\xi }_{n + i} = 1, i = 1,\ldots, n\text{.} \] Hence, \( {X}_{\sigma } = V\left( {{\xi }_{1}{\xi }_{n + 1} - 1,\ldots ,{\xi }_{n}{\xi }_{2n} - 1}\right) \) . For \( n = 1 \), we obtain a (complex) hyperbola with \( \left\{ {{\xi }_{1} = 0}\right\} \) and \( \left\{ {{\xi }_{2} = 0}\right\} \) as asymptotes... | Yes |
For \( \sigma \mathrel{\text{:=}} \operatorname{pos}\left( \left\{ {{e}_{1},{e}_{2}}\right\} \right) \), the monoid \( \sigma \cap {\mathbb{Z}}^{2} \) has linearly independent generators \( {e}_{1},{e}_{2} \) . Therefore, \( \mathfrak{a} = \mathfrak{o} \), the zero ideal, and \[ {X}_{\sigma } = {\mathbb{C}}^{2} \] | The same is true for each cone \[ \sigma = \operatorname{pos}\left( \left\{ {{e}_{1} + v{e}_{2},{e}_{2}}\right\} \right) ,\;\text{ for }v \in \mathbb{Z}. \] More generally, if \( \sigma = \operatorname{pos}\left( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \right) \) is a regular lattice cone in \( {\mathbb{R}}^{n} \), t... | No |
For \( \sigma \mathrel{\text{:=}} \operatorname{pos}\left( \left\{ {{e}_{1},{e}_{1} + 2{e}_{2}}\right\} \right) \), the monoid \( \sigma \cap {\mathbb{Z}}^{2} \) is generated by \( {a}_{1} = {e}_{1},{a}_{2} = {e}_{1} + 2{e}_{2},{a}_{3} = {e}_{1} + {e}_{2} \) . There is a linear relationship | \[ {a}_{1} + {a}_{2} = 2{a}_{3} \] and, hence, a monomial equation \[ {u}_{1}{u}_{2} = {u}_{3}^{2} \] \( {X}_{\sigma } \) is a quadratic cone with \ | Yes |
Example 4. For the half-plane \( \sigma = \operatorname{pos}\left( \left\{ {{e}_{1}, - {e}_{1},{e}_{2}}\right\} \right) \) and fixed \( v \in {\mathbb{Z}}_{ \geq 0} \) the monoid \( \sigma \cap {\mathbb{Z}}^{2} \) has \( {e}_{1}, - {e}_{1}, b = v{e}_{1} + {e}_{2} \) as generators,(see Figure 3 for \( v = 2 \) ). In par... | \[ {v}_{1} = {u}_{1}\;{u}_{1} = {v}_{1} \] \[ {v}_{2} = {u}_{2}\;{u}_{2} = {v}_{2} \] \[ {v}_{3} = {u}_{1}^{v}{u}_{3},\text{ and }\;{u}_{3} = {v}_{2}^{v}{v}_{3} \] are transformation formulae for the coordinates. | Yes |
Example 1. Let the projective plane \( {\mathbb{P}}^{2} = \left\{ {\left\lbrack {{\eta }_{0},{\eta }_{1},{\eta }_{2}}\right\rbrack \mid {\eta }_{i} \in \mathbb{C}}\right. \), not all \( {\eta }_{i} = \) \( 0\} \) be given, the homogeneous coordinates \( {\eta }_{0},{\eta }_{1},{\eta }_{2} \) being determined only up to... | It is covered by three affine planes \( {A}_{0} \mathrel{\text{:=}} \left\{ {\left( {1,{\eta }_{1}{\eta }_{0}^{-1},{\eta }_{2}{\eta }_{0}^{-1}}\right) \mid {\eta }_{0} \neq 0}\right\} \) , \( {A}_{1} \mathrel{\text{:=}} \left\{ {\left( {{\eta }_{0}{\eta }_{1}^{-1},1,{\eta }_{2}{\eta }_{1}^{-1}}\right) \mid {\eta }_{1} ... | Yes |
Given \( {\mathbb{P}}^{1} \times {\mathbb{P}}^{1} = \left\{ {\left( {\left\lbrack {{\eta }_{0},{\eta }_{1}}\right\rbrack ,\left\lbrack {{\zeta }_{0},{\zeta }_{1}}\right\rbrack }\right) \mid \left( {{\eta }_{0},{\eta }_{1}}\right) \neq \left( {0,0}\right) }\right. \) , \( \left( {{\zeta }_{0},{\zeta }_{1}}\right) \neq \... | We may cover \( {\mathbb{P}}^{1} \times {\mathbb{P}}^{1} \) by four \ | No |
Example 3 Hirzebruch surfaces \( {\mathcal{H}}_{k} \) . We consider a hypersurface in \( {\mathbb{P}}^{1} \times {\mathbb{P}}^{2} = \) \( \left\{ {\left( {\left\lbrack {{\eta }_{0},{\eta }_{1}}\right\rbrack ,\left\lbrack {{\zeta }_{0},{\zeta }_{1},{\zeta }_{2}}\right\rbrack }\right) \mid \left( {{\eta }_{0},{\eta }_{1}... | By a modification of the arguments in Example 2, we find four affine planes as charts whose gluing together depends on\n\n  has \( \frac{1}{2}\mathop{\sum }\limits_{{v \in V}}d\left( v\right) \) edges, so \( \sum d\left( v\right) \) is an even number. | Yes |
Proposition 1.2.2. Every graph \( G \) with at least one edge has a subgraph \( H \) with \( \delta \left( H\right) > \varepsilon \left( H\right) \geq \varepsilon \left( G\right) \) . | Proof. To construct \( H \) from \( G \), let us try to delete vertices of small degree one by one, until only vertices of large degree remain. Up to which degree \( d\left( v\right) \) can we afford to delete a vertex \( v \), without lowering \( \varepsilon \) ? Clearly, up to \( d\left( v\right) = \varepsilon \) : t... | Yes |
Proposition 1.3.2. Every graph \( G \) containing a cycle satisfies \( g\left( G\right) \leq \) \( 2\operatorname{diam}G + 1 \) . | Proof. Let \( C \) be a shortest cycle in \( G \) . If \( g\left( G\right) \geq 2\operatorname{diam}G + 2 \), then \( C \) has two vertices whose distance in \( C \) is at least \( \operatorname{diam}G + 1 \) . In \( G \) , these vertices have a lesser distance; any shortest path \( P \) between them is therefore not a... | Yes |
Proposition 1.3.3. A graph \( G \) of radius at most \( k \) and maximum degree at most \( d \geq 3 \) has fewer than \( \frac{d}{d - 2}{\left( d - 1\right) }^{k} \) vertices. | Proof. Let \( z \) be a central vertex in \( G \), and let \( {D}_{i} \) denote the set of vertices of \( G \) at distance \( i \) from \( z \) . Then \( V\left( G\right) = \mathop{\bigcup }\limits_{{i = 0}}^{k}{D}_{i} \) . Clearly \( \left| {D}_{0}\right| = 1 \) and \( \left| {D}_{1}\right| \leq d \) . For \( i \geq 1... | Yes |
Theorem 1.3.4. (Alon, Hoory & Linial 2002)\n\nLet \( G \) be a graph. If \( d\left( G\right) \geq d \geq 2 \) and \( g\left( G\right) \geq g \in \mathbb{N} \) then \( \left| G\right| \geq {n}_{0}\left( {d, g}\right) \) . | 2.3.1] Corollary 1.3.5. If \( \delta \left( G\right) \geq 3 \) then \( g\left( G\right) < 2\log \left| G\right| \) .\n\nProof. If \( g \mathrel{\text{:=}} g\left( G\right) \) is even then\n\n\[ \n{n}_{0}\left( {3, g}\right) = 2\frac{{2}^{g/2} - 1}{2 - 1} = {2}^{g/2} + \left( {{2}^{g/2} - 2}\right) > {2}^{g/2},\n\]\n\nw... | No |
Proposition 1.4.1. The vertices of a connected graph \( G \) can always be enumerated, say as \( {v}_{1},\ldots ,{v}_{n} \), so that \( {G}_{i} \mathrel{\text{:=}} G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack \) is connected for every \( i \) . | Proof. Pick any vertex as \( {v}_{1} \), and assume inductively that \( {v}_{1},\ldots ,{v}_{i} \) have been chosen for some \( i < \left| G\right| \) . Now pick a vertex \( v \in G - {G}_{i} \) . As \( G \) is connected, it contains a \( v - {v}_{1} \) path \( P \) . Choose as \( {v}_{i + 1} \) the last vertex of \( P... | Yes |
Corollary 1.5.2. The vertices of a tree can always be enumerated, say as \( {v}_{1},\ldots ,{v}_{n} \), so that every \( {v}_{i} \) with \( i \geq 2 \) has a unique neighbour in \( \left\{ {{v}_{1},\ldots ,{v}_{i - 1}}\right\} \) | Proof. Use the enumeration from Proposition 1.4.1. | No |
Corollary 1.5.4. If \( T \) is a tree and \( G \) is any graph with \( \delta \left( G\right) \geq \left| T\right| - 1 \) , then \( T \subseteq G \), i.e. \( G \) has a subgraph isomorphic to \( T \) . | Proof. Find a copy of \( T \) in \( G \) inductively along its vertex enumeration from Corollary 1.5.2. | No |
Lemma 1.5.5. Let \( T \) be a normal tree in \( G \) .\n\n(i) Any two vertices \( x, y \in T \) are separated in \( G \) by the set \( \lceil x\rceil \cap \lceil y\rceil \) .\n\n(ii) If \( S \subseteq V\left( T\right) = V\left( G\right) \) and \( S \) is down-closed, then the components of \( G - S \) are spanned by th... | Proof. (i) Let \( P \) be any \( x - y \) path in \( G \) . Since \( T \) is normal, the vertices of \( P \) in \( T \) form a sequence \( x = {t}_{1},\ldots ,{t}_{n} = y \) for which \( {t}_{i} \) and \( {t}_{i + 1} \) are always comparable in the tree oder of \( T \) . Consider a minimal such sequence of vertices in ... | Yes |
Proposition 1.5.6. Every connected graph contains a normal spanning tree, with any specified vertex as its root. | Proof. Let \( G \) be a connected graph and \( r \in G \) any specified vertex. Let \( T \) be a maximal normal tree with root \( r \) in \( G \) ; we show that \( V\left( T\right) = V\left( G\right) \) . \n\nSuppose not, and let \( C \) be a component of \( G - T \) . As \( T \) is normal, \( N\left( C\right) \) is a ... | Yes |
Proposition 1.6.1. A graph is bipartite if and only if it contains no odd cycle. | (1.5.1) Proof. Let \( G = \left( {V, E}\right) \) be a graph without odd cycles; we show that \( G \) is bipartite. Clearly a graph is bipartite if all its components are bipartite or trivial, so we may assume that \( G \) is connected. Let \( T \) be a spanning tree in \( G \), pick a root \( r \in T \), and denote th... | Yes |
Proposition 1.7.1. \( G \) is an \( {MX} \) if and only if \( X \) can be obtained from \( G \) by a series of edge contractions, i.e. if and only if there are graphs \( {G}_{0},\ldots ,{G}_{n} \) and edges \( {e}_{i} \in {G}_{i} \) such that \( {G}_{0} = G,{G}_{n} \simeq X \), and \( {G}_{i + 1} = {G}_{i}/{e}_{i} \) f... | Proof. Induction on \( \left| G\right| - \left| X\right| \) . | No |
A connected graph is Eulerian if and only if every vertex has even degree. | Proof. The degree condition is clearly necessary: a vertex appearing \( k \) times in an Euler tour (or \( k + 1 \) times, if it is the starting and finishing vertex and as such counted twice) must have degree \( {2k} \) .\n\nConversely, let \( G \) be a connected graph with all degrees even, and let\n\n\[ W = {v}_{0}{... | Yes |
Proposition 1.9.1. The induced cycles in \( G \) generate its entire cycle space. | Proof. By definition of \( \mathcal{C}\left( G\right) \) it suffices to show that the induced cycles in \( G \) generate every cycle \( C \subseteq G \) with a chord \( e \) . This follows at once by induction on \( \left| C\right| \) : the two cycles in \( C + e \) that have \( e \) but no other edge in common are sho... | Yes |
Proposition 1.9.2. The following assertions are equivalent for edge sets \( F \subseteq E \) :\n\n(i) \( F \in \mathcal{C}\left( G\right) \) ;\n\n(ii) \( F \) is a disjoint union of (edge sets of) cycles in \( G \) ;\n\n(iii) All vertex degrees of the graph \( \left( {V, F}\right) \) are even. | Proof. Since cycles have even degrees and taking symmetric differences preserves this,(i) \( \rightarrow \) (iii) follows by induction on the number of cycles used to generate \( F \) . The implication (iii) \( \rightarrow \) (ii) follows by induction on \( \left| F\right| \) : if \( F \neq \varnothing \) then \( \left... | Yes |
Proposition 1.9.3. Together with \( \varnothing \), the cuts in \( G \) form a subspace \( {\mathcal{C}}^{ * } \) of \( \mathcal{E}\left( G\right) \) . This space is generated by cuts of the form \( E\left( v\right) \) . | Proof. Let \( {\mathcal{C}}^{ * } \) denote the set of all cuts in \( G \), together with \( \varnothing \) . To prove that \( {\mathcal{C}}^{ * } \) is a subspace, we show that for all \( D,{D}^{\prime } \in {\mathcal{C}}^{ * } \) also \( D + {D}^{\prime } \) \( \left( { = D - {D}^{\prime }}\right) \) lies in \( {\mat... | Yes |
Lemma 1.9.4. Every cut is a disjoint union of bonds. | Proof. Consider first a connected graph \( H = \left( {V, E}\right) \), a connected subgraph \( C \subseteq H \), and a component \( D \) of \( H - C \) . Then \( H - D \), too, is connected (Fig. 1.9.2), so the edges between \( D \) and \( H - D \) form a minimal cut. By the choice of \( D \), this cut is precisely th... | Yes |
Theorem 1.9.5. The cycle space \( \mathcal{C} \) and the cut space \( {\mathcal{C}}^{ * } \) of any graph satisfy\n\n\[ \mathcal{C} = {\mathcal{C}}^{* \bot }\;\text{ and }\;{\mathcal{C}}^{ * } = {\mathcal{C}}^{ \bot }.\] | Proof. (See also Exercise 30.) Let us consider a graph \( G = \left( {V, E}\right) \) . Clearly, any cycle in \( G \) has an even number of edges in each cut. This implies \( \mathcal{C} \subseteq {\mathcal{C}}^{* \bot } \) .\n\nConversely, recall from Proposition 1.9.2 that for every edge set \( F \notin \mathcal{C} \... | No |
Theorem 1.9.6. Let \( G \) be a connected graph and \( T \subseteq G \) a spanning tree. Then the corresponding fundamental cycles and cuts form a basis of \( \mathcal{C}\left( G\right) \) and of \( {\mathcal{C}}^{ * }\left( G\right) \), respectively. If \( G \) has \( n \) vertices and \( m \) edges, then\n\n\[ \dim \... | Proof. Since an edge \( e \in T \) lies in \( {D}_{e} \) but not in \( {D}_{{e}^{\prime }} \) for any \( {e}^{\prime } \neq e \), the cut\n\n\( \left( {1.5.3}\right) \)\n\n\( {D}_{e} \) cannot be generated by other fundamental cuts. The fundamental cuts therefore form a linearly independent subset of \( {\mathcal{C}}^{... | Yes |
Corollary 2.1.3. If \( G \) is \( k \) -regular with \( k \geq 1 \), then \( G \) has a 1 -factor. | Proof. If \( G \) is \( k \) -regular, then clearly \( \left| A\right| = \left| B\right| \) ; it thus suffices to show by Theorem 2.1.2 that \( G \) contains a matching of \( A \) . Now every set \( S \subseteq A \) is joined to \( N\left( S\right) \) by a total of \( k\left| S\right| \) edges, and these are among the ... | Yes |
For every set of preferences, \( G \) has a stable matching. | Proof. Call a matching \( M \) in \( G \) better than a matching \( {M}^{\prime } \neq M \) if \( M \) makes the vertices in \( B \) happier than \( {M}^{\prime } \) does, that is, if every vertex \( b \) in an edge \( {f}^{\prime } \in {M}^{\prime } \) is incident also with some \( f \in M \) such that \( {f}^{\prime ... | Yes |
A graph \( G \) has a 1-factor if and only if \( q\left( {G - S}\right) \leq \left| S\right| \) for all \( S \subseteq V\left( G\right) \) . | Proof. Let \( G = \left( {V, E}\right) \) be a graph without a 1-factor. Our task is to\nbad set find a bad set \( S \subseteq V \), one that violates Tutte’s condition.\n\nWe may assume that \( G \) is edge-maximal without a 1-factor. Indeed, if \( {G}^{\prime } \) is obtained from \( G \) by adding edges and \( S \su... | Yes |
Every bridgeless cubic graph has a 1-factor. | Proof. We show that any bridgeless cubic graph \( G \) satisfies Tutte’s condition. Let \( S \subseteq V\left( G\right) \) be given, and consider an odd component \( C \) of \( G - S \) . Since \( G \) is cubic, the degrees (in \( G \) ) of the vertices in \( C \) sum to an odd number, but only an even part of this sum... | Yes |
Lemma 2.3.1. Let \( k \in \mathbb{N} \), and let \( H \) be a cubic multigraph. If \( \left| H\right| \geq {s}_{k} \) , then \( H \) contains \( k \) disjoint cycles. | Proof. We apply induction on \( k \) . For \( k \leq 1 \) the assertion is trivial, so let \( k \geq 2 \) be given for the induction step. Let \( C \) be a shortest cycle in \( H \) .\n\nWe first show that \( H - C \) contains a subdivision of a cubic multigraph \( {H}^{\prime } \) with \( \left| {H}^{\prime }\right| \... | Yes |
There is a function \( f : \mathbb{N} \rightarrow \mathbb{R} \) such that, given any \( k \in \mathbb{N} \), every graph contains either \( k \) disjoint cycles or a set of at most \( f\left( k\right) \) vertices meeting all its cycles. | We show the result for \( f\left( k\right) \mathrel{\text{:=}} {s}_{k} + k - 1 \) . Let \( k \) be given, and let \( G \) be any graph. We may assume that \( G \) contains a cycle, and so it has a maximal subgraph \( H \) in which every vertex has degree 2 or 3 . Let \( U \) be its set of degree 3 vertices.\n\nLet \( \... | Yes |
A multigraph contains \( k \) edge-disjoint spanning trees if and only if for every partition \( P \) of its vertex set it has at least \( k\left( {\left| P\right| - 1}\right) \) cross-edges. | For the proof of Theorem 2.4.1, let a multigraph \( G = \left( {V, E}\right) \) and \( G = \left( {V, E}\right) \) \( k \in \mathbb{N} \) be given. Let \( \mathcal{F} \) be the set of all \( k \) -tuples \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \) of edge-disjoint spanning forests in \( G \) with the maximum tota... | Yes |
Corollary 2.4.2. Every \( {2k} \) -edge-connected multigraph \( G \) has \( k \) edge-disjoint spanning trees. | Proof. Every set in a vertex partition of \( G \) is joined to other partition sets by at least \( {2k} \) edges. Hence, for any partition into \( r \) sets, \( G \) has at least \( \frac{1}{2}\mathop{\sum }\limits_{{i = 1}}^{r}{2k} = {kr} \) cross-edges. The assertion thus follows from Theorem 2.4.1. | Yes |
Lemma 2.4.3. For every \( {e}^{0} \in E \smallsetminus E\left\lbrack {F}^{0}\right\rbrack \) there exists a set \( U \subseteq V \) that is connected in every \( {F}_{i}^{0}\left( {i = 1,\ldots, k}\right) \) 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 \( U \mathrel{\text{:=}} V\left( {C}^{0}\right) \) . \n\nLet \( i \in \{ 1,\ldots, k\} \) be given; we have to ... | Yes |
Theorem 2.4.4. (Nash-Williams 1964)\n\n\( A \) multigraph \( G = \left( {V, E}\right) \) can be partitioned into at most \( k \) forests if and only if \( \parallel G\left\lbrack U\right\rbrack \parallel \leq k\left( {\left| U\right| - 1}\right) \) for every non-empty set \( U \subseteq V \) . | Proof. The forward implication was shown above. Conversely, we show\n\n\( \left( {1.5.3}\right) \)\n\nthat every \( k \) -tuple \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \in \mathcal{F} \) partitions \( G \), i.e. that \( E\left\lbrack F\right\rbrack = \) \( E \) . If not, let \( e \in E \smallsetminus E\left\lbra... | Yes |
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 \( {v}_{P} \in P \) for every \( P \in \mathcal{P} \) . | We prove by induction on \( \left| G\right| \) that for every path cover \( \mathcal{P} = \) \( {P}_{i} \) \( \left\{ {{P}_{1},\ldots ,{P}_{m}}\right\} \) of \( G \) with \( \operatorname{ter}\left( \mathcal{P}\right) \) minimal there is a set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) as \( {v}_{i} \) clai... | Yes |
In every finite partially ordered set \( \left( {P, \leq }\right) \), the minimum number of chains with union \( P \) is equal to the maximum cardinality of an antichain in \( P \) . | If \( A \) is an antichain in \( P \) of maximum cardinality, then clearly \( P \) cannot be covered by fewer than \( \left| A\right| \) chains. The fact that \( \left| A\right| \) chains will suffice follows from Theorem 2.5.1 applied to the directed graph on \( P \) with the edge set \( \{ \left( {x, y}\right) \mid x... | No |
Proposition 3.1.3. A graph is 2-connected if and only if it can be constructed from a cycle by successively adding \( H \) -paths to graphs \( H \) already constructed (Fig. 3.1.2). | Proof. Clearly, every graph constructed as described is 2-connected. Conversely, let a 2-connected graph \( G \) be given. Then \( G \) contains a cycle, and hence has a maximal subgraph \( H \) constructible as above. Since any edge \( {xy} \in E\left( G\right) \smallsetminus E\left( H\right) \) with \( x, y \in H \) ... | Yes |
A graph \( G \) is 3-connected if and only if there exists a sequence \( {G}_{0},\ldots ,{G}_{n} \) of graphs with the following properties:\n\n(i) \( {G}_{0} = {K}^{4} \) and \( {G}_{n} = G \) ;\n\n(ii) \( {G}_{i + 1} \) has an edge \( {xy} \) with \( d\left( x\right), d\left( y\right) \geq 3 \) and \( {G}_{i} = {G}_{... | If \( G \) is 3-connected, a sequence as in the theorem exists by Lemma 3.2.1. Note that all the graphs in this sequence are 3-connected.\n\nConversely, let \( {G}_{0},\ldots ,{G}_{n} \) be a sequence of graphs as stated; we show that if \( {G}_{i} = {G}_{i + 1}/{xy} \) is 3-connected then so is \( {G}_{i + 1} \), for ... | Yes |
Let \( G = \left( {V, E}\right) \) be a graph and \( A, B \subseteq V \) . Then the minimum number of vertices separating \( A \) from \( B \) in \( G \) is equal to the maximum number of disjoint \( A - B \) paths in \( G \) . | We apply induction on \( \parallel G\parallel \) . If \( G \) has no edge, then \( \left| {A \cap B}\right| = k \) and we have \( k \) trivial \( A - B \) paths. So we assume that \( G \) has an edge \( e = {xy} \) . If \( G \) has no \( k \) disjoint \( A - B \) paths, then neither does \( G/e \) ; here, we count the ... | Yes |
Lemma 3.3.2. If an alternating walk \( W \) as above ends in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) , then \( G \) contains a set of disjoint \( A - B \) paths exceeding \( \mathcal{P} \) . | Proof. We may assume that \( W \) has only its first vertex in \( A \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) and only its last vertex in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) . Let \( H \) be the graph on \( V\left( G\right) \) whose edge set is the symmetric difference of \( E\l... | Yes |
Corollary 3.3.5. Let \( a \) and \( b \) be two distinct vertices of \( G \) .\n\n(i) If \( {ab} \notin E \), then the minimum number of vertices \( \neq a, b \) separating \( a \) from \( b \) in \( G \) is equal to the maximum number of independent \( a - b \) paths in \( G \) .\n\n(ii) The minimum number of edges se... | Proof. (i) Apply Theorem 3.3.1 with \( A \mathrel{\text{:=}} N\left( a\right) \) and \( B \mathrel{\text{:=}} N\left( b\right) \) .\n\n(ii) Apply Theorem 3.3.1 to the line graph of \( G \), with \( A \mathrel{\text{:=}} E\left( a\right) \) and \( B \mathrel{\text{:=}} E\left( b\right) \) . | Yes |
Theorem 3.3.6. (Global Version of Menger's Theorem) \( \\left\\lbrack \\begin{array}{l} {4.2.7} \\ {6.6.1} \\ {9.4.2} \\end{array}\\right\\rbrack \) (i) A graph is \( k \) -connected if and only if it contains \( k \) independent paths between any two vertices.\n\n(ii) A graph is \( k \) -edge-connected if and only if ... | Proof. (i) If a graph \( G \) contains \( k \) independent paths between any two vertices, then \( \\left| G\\right| > k \) and \( G \) cannot be separated by fewer than \( k \) vertices; thus, \( G \) is \( k \) -connected.\n\nConversely, suppose that \( G \) is \( k \) -connected (and, in particular, has more than \(... | Yes |
Corollary 3.4.2. Given a graph \( G \) with an induced subgraph \( H \), there are at least \( \frac{1}{2}{\kappa }_{G}\left( H\right) \) independent \( H \) -paths and at least \( \frac{1}{2}{\lambda }_{G}\left( H\right) \) edge-disjoint \( H \) -paths in \( G \) . | Proof. To prove the first assertion, let \( k \) be the maximum number of independent \( H \) -paths in \( G \) . By Theorem 3.4.1, there are sets \( X \subseteq V\left( {G - H}\right) \) and \( F \subseteq E\left( {G - H - X}\right) \) with\n\n\[ k = \left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\... | Yes |
Lemma 3.5.1. There is a function \( h : \mathbb{N} \rightarrow \mathbb{N} \) such that every graph of average degree at least \( h\left( r\right) \) contains \( {K}^{r} \) as a topological minor, for every \( r \in \mathbb{N} \) . | Proof. For \( r \leq 2 \), the assertion holds with \( h\left( r\right) = 1 \) ; we now assume that \( r \geq 3 \) . We show by induction on \( m = r,\ldots ,\left( \begin{array}{l} r \\ 2 \end{array}\right) \) that every graph \( G \) with average degree \( d\left( G\right) \geq {2}^{m} \) has a topological minor \( X... | Yes |
There is a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that every \( f\left( k\right) \) -connected graph is \( k \) -linked, for all \( k \in \mathbb{N} \) . | 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 \( K \) \( d\left( G\right) \geq \delta \left( G\right) \geq \kappa \left( G\right) \geq h\left( {3k}\right) \) ; ... | Yes |
Lemma 4.2.1. Let \( G \) be a plane graph, \( f \in F\left( G\right) \) a face, and \( H \subseteq G \) a subgraph.\n\n(i) \( H \) has a face \( {f}^{\prime } \) containing \( f \) .\n\n(ii) If the frontier of \( f \) lies in \( H \), then \( {f}^{\prime } = f \) . | Proof. (i) Clearly, the points in \( f \) are equivalent also in \( {\mathbb{R}}^{2} \smallsetminus H \) ; let \( {f}^{\prime } \) be the equivalence class of \( {\mathbb{R}}^{2} \smallsetminus H \) containing them.\n\n(ii) Recall from Section 4.1 that any arc between \( f \) and \( {f}^{\prime } \smallsetminus f \) me... | Yes |
Lemma 4.2.2. Let \( G \) be a plane graph and \( e \) an edge of \( G \). \( \begin{array}{r} \left\lbrack {4.5.1}\right\rbrack \\ \left\lbrack {4.5.2}\right\rbrack \\ \left\lbrack {12.5.4}\right\rbrack \end{array} \) (i) If \( X \) is the frontier of a face of \( G \), then either \( e \subseteq X \) or \( X \cap \ove... | Proof. We prove all three assertions together. Let us start by considering one point \( {x}_{0} \in e \). We show that \( {x}_{0} \) lies on the frontier of either exactly two faces or exactly one, according as \( e \) lies on a cycle in \( G \) or not. We then show that every other point in \( e \) lies on the frontie... | Yes |
Corollary 4.2.3. The frontier of a face is always the point set of a subgraph. | The subgraph of \( G \) whose point set is the frontier of a face \( f \) is said to bound \( f \) and is called its boundary; we denote it by \( G\left\lbrack f\right\rbrack \) . A face is said to be incident with the vertices and edges of its boundary. By Lemma 4.2.1 (ii), every face of \( G \) is also a face of its ... | No |
Proposition 4.2.4. A plane forest has exactly one face. | Proof. Use induction on the number of edges and Lemma 4.1.3. | No |
Lemma 4.2.5. If a plane graph has different faces with the same boundary, then the graph is a cycle. | Proof. Let \( G \) be a plane graph, and let \( H \subseteq G \) be the boundary of\n\n(4.1.1)\n\ndistinct faces \( {f}_{1},{f}_{2} \) of \( G \) . Since \( {f}_{1} \) and \( {f}_{2} \) are also faces of \( H \), Proposition 4.2.4 implies that \( H \) contains a cycle \( C \) . By Lemma 4.2.2 (ii), \( {f}_{1} \) and \(... | Yes |
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