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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 |
Example 4. Let \( Q = P \) be an \( n \) -polytope, and let \( M \) be a hyperplane. Then, \( {\Gamma }_{M}^{Q} \) is sharp if \( M \) is not perpendicular to the affine hull of an at least one-dimensional face of \( Q \) . | Null | No |
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 |
In \( {\mathbb{R}}^{3} \), let \( {I}_{j} \mathrel{\text{:=}} \left\lbrack {{a}_{j},{b}_{j}}\right\rbrack, j = 1,2,3 \), be line segments such that \( {b}_{1} - {a}_{1},{b}_{2} - {a}_{2},{b}_{3} - {a}_{3} \) are linearly independent. Then, for \( K \mathrel{\text{:=}} {I}_{1} + {I}_{3} \) , \( L \mathrel{\text{:=}} {I}... | Null | No |
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 |
If \( Z \) is a hexagonal prism with centrally symmetric basis, hence, \( {Z}^{ * } \) a hexagonal bipyramid, the projective line arrangement corresponding to it is of the type illustrated in Figure 22. | Null | No |
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 |
Example 2. Let \( {x}_{1},\ldots ,{x}_{q} \in \sigma \cap {\mathbb{Z}}^{n} \) . Then, \( S\left( {{x}_{1},\ldots ,{x}_{q}}\right) \cup \{ 0\} \), where\n\n\[ \nS\left( {{x}_{1},\ldots ,{x}_{q}}\right) \mathrel{\text{:=}} \left\lbrack {\left( {{x}_{1} + \sigma }\right) \cup \cdots \cup \left( {{x}_{q} + \sigma }\right) ... | Null | No |
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 |
Example 1. In \( {\mathbb{R}}^{2} \), let \( \sum \mathrel{\text{:=}} \left\{ {{\sigma }_{0},{\sigma }_{1},{\sigma }_{2}}\right\} \) where \( {\sigma }_{0} \mathrel{\text{:=}} \{ 0\} ,{\sigma }_{1} \mathrel{\text{:=}} {\mathbb{R}}_{ \geq 0}{e}_{1} \), and \[ {\sigma }_{2} \mathrel{\text{:=}} {\mathbb{R}}_{ \geq 0}\left... | Null | No |
Example 3. Let \( \sum \) in \( {\mathbb{R}}^{3} \) be spanned by the cube with vertices \( \pm {e}_{1} \pm {e}_{2} \pm {e}_{3} \) . We obtain \( L \) to be 4-dimensional and, hence, Pic \( \sum \cong \mathbb{Z} \) . | Null | No |
In Example 4, we replace the generator \( {e}_{1} + {e}_{2} + {e}_{3} \) of a 1-cone by \( e \mathrel{\text{:=}} 2{e}_{1} + 2{e}_{2} + 3{e}_{3} \), and change all faces containing \( {e}_{1} + {e}_{2} + {e}_{3} \) by taking \( e \) as the generating vector instead of \( {e}_{1} + {e}_{2} + {e}_{3} \) . (Figure 20). Now... | Null | No |
If \( \sum \) consists of an \( n \) -cone \( \sigma \) with \( k \) generators and all faces of \( \sigma \) , then \( \lambda = k - n \) and \( \mu \left( \sum \right) = 0 \) . | Null | No |
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. | Yes |
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 | \[ \text{Pic}\sum \cong {\mathbb{Z}}^{n}\text{.} \] | Yes |
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 |
Example 1. \( \mathfrak{a} \mathrel{\text{:=}} R \cdot \left( {{\xi }_{1}^{2} + {\xi }_{2}^{2} - 1}\right) \) is an ideal in \( R \mathrel{\text{:=}} \mathbb{C}\left\lbrack {{\xi }_{1},{\xi }_{2}}\right\rbrack \) . | Null | No |
Example 2. \( \mathfrak{a} \mathrel{\text{:=}} R \cdot \left( {{\xi }_{1} - 1}\right) \left( {{\xi }_{2} - 1}\right) \left( {{\xi }_{3} - 1}\right) \) is an ideal in \( R = \mathbb{C}\left\lbrack {{\xi }_{1},{\xi }_{2},{\xi }_{3}}\right\rbrack \) . | Null | No |
Example 4. \( \mathfrak{a} \mathrel{\text{:=}} R \cdot \left( {{\xi }_{1} - {\xi }_{2}}\right) + R \cdot \left( {{\xi }_{1} + {\xi }_{2}}\right) = R \cdot {\xi }_{1} + R \cdot {\xi }_{2} \) is an ideal in \( R = \mathbb{C}\left\lbrack {{\xi }_{1},{\xi }_{2}}\right\rbrack \) | Null | No |
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 |
Example 6. Let \( U \) be an affine variety and \( Z \subset U \) be a subvariety of \( U \) . Then, the inclusion \( Z \hookrightarrow U \) determines a morphism \( \varphi : Z \rightarrow U \) . One easily proves that \( {\varphi }^{ * } : {R}_{U} \rightarrow {R}_{Z} \) is a surjection and \( \ker {\varphi }^{ * } = ... | Null | 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... | No |
Example 9. \( \dim {\mathbb{C}}^{n} = n \) . | Null | No |
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... | Yes |
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} \... | Yes |
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 |
Proposition 1.9.8. \( B{B}^{t} = A + D \) | Null | No |
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... | Yes |
Proposition 3.1.2. The block graph of a connected graph is a tree. | Null | 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 |
Theorem 3.4.1. (Mader 1978)\n\nGiven a graph \( G \) with an induced subgraph \( H \), there are always \( {M}_{G}\left( H\right) \) independent \( H \) -paths in \( G \) . | Null | No |
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 |
Theorem 4.1.1. (Jordan Curve Theorem for Polygons)\n\nFor 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. | Null | No |
Lemma 4.1.2. Let \( {P}_{1},{P}_{2},{P}_{3} \) be three arcs, between the same two endpoint but otherwise disjoint.\n\n(i) \( {\mathbb{R}}^{2} \smallsetminus \left( {{P}_{1} \cup {P}_{2} \cup {P}_{3}}\right) \) has exactly three regions, with frontiers \( {P}_{1} \cup {P}_{2},{P}_{2} \cup {P}_{3} \) and \( {P}_{1} \cup... | Null | No |
Lemma 4.1.3. Let \( {X}_{1},{X}_{2} \subseteq {\mathbb{R}}^{2} \) be disjoint sets, each the union of finitely many points and arcs, and let \( P \) be an arc between a point in \( {X}_{1} \) and one in \( {X}_{2} \) whose interior \( \overset{ \circ }{P} \) lies in a region \( O \) of \( {\mathbb{R}}^{2} \smallsetminu... | Null | No |
Theorem 4.1.4. Let \( \varphi : {C}_{1} \rightarrow {C}_{2} \) be a homeomorphism between two circles on \( {S}^{2} \), let \( {O}_{1} \) be a region of \( {C}_{1} \), and let \( {O}_{2} \) be a region of \( {C}_{2} \). Then \( \varphi \) can be extended to a homeomorphism \( {C}_{1} \cup {O}_{1} \rightarrow {C}_{2} \c... | Null | No |
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 ... | Yes |
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 |
Proposition 4.2.7. The face boundaries in a 3-connected plane graph are precisely its non-separating induced cycles. | Proof. Let \( G \) be a 3-connected plane graph, and let \( C \subseteq G \) . If \( C \) is a (4.1.1) non-separating induced cycle, then by the Jordan curve theorem its two\n\n(4.1.2)\n\nfaces cannot both contain points of \( G \smallsetminus C \) . Therefore it bounds a face of \( G \) .\n\nConversely, suppose that \... | Yes |
Theorem 4.2.9. (Euler's Formula)\n\nLet \( G \) be a connected plane graph with \( n \) vertices, \( m \) edges, and \( \ell \) faces. Then\n\n\[ n - m + \ell = 2. \] | Proof. We fix \( n \) and apply induction on \( m \) . For \( m \leq n - 1, G \) is a tree\n\n(1.5.1)\n\n\( \left( {1.5.3}\right) \) and \( m = n - 1 \) (why?), so the assertion follows from Proposition 4.2.4.\n\nNow let \( m \geq n \) . Then \( G \) has an edge \( e \) that lies on a cycle; let \( {G}^{\prime } \mathr... | Yes |
Corollary 4.2.10. A plane graph with \( n \geq 3 \) vertices has at most \( {3n} - 6 \) edges. Every plane triangulation with \( n \) vertices has \( {3n} - 6 \) edges. | Proof. By Proposition 4.2.8 it suffices to prove the second assertion. In a plane triangulation \( G \), every face boundary contains exactly three edges, and every edge lies on the boundary of exactly two faces (Lemma 4.2.2). The bipartite graph on \( E\left( G\right) \cup F\left( G\right) \) with edge set \( \{ {ef} ... | Yes |
Any two planar embeddings of a 3-connected graph are equivalent. | Proof. Let \( G \) be a 3-connected graph with planar embeddings \( \rho : G \rightarrow H \) and \( {\rho }^{\prime } : G \rightarrow {H}^{\prime } \) . By Theorem 4.3.1 it suffices to show that \( {\rho }^{\prime } \circ {\rho }^{-1} \) is a graph-theoretical isomorphism, i.e. that \( \rho \left( C\right) \) bounds a... | Yes |
Lemma 4.4.2. A graph contains \( {K}^{5} \) or \( {K}_{3,3} \) as a minor if and only if it contains \( {K}^{5} \) or \( {K}_{3,3} \) as a topological minor. | Proof. By Proposition 1.7.2 it suffices to show that every graph \( G \) with a \( {K}^{5} \) minor contains either \( {K}^{5} \) as a topological minor or \( {K}_{3,3} \) as a minor. So suppose that \( G \succcurlyeq {K}^{5} \), and let \( K \subseteq G \) be minimal such that \( K = M{K}^{5} \). Then every branch set... | Yes |
Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \... | Proof. Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality ... | Yes |
Lemma 4.4.5. If \( \left| G\right| \geq 4 \) and \( G \) is edge-maximal with \( T{K}^{5}, T{K}_{3,3} \nsubseteq G \) , then \( G \) is 3-connected. | Proof. We apply induction on \( \left| G\right| \) . For \( \left| G\right| = 4 \), we have \( G = {K}^{4} \) and the assertion holds. Now let \( \left| G\right| > 4 \), and let \( G \) be edge-maximal without a \( T{K}^{5} \) or \( T{K}_{3,3} \) . Suppose \( \kappa \left( G\right) \leq 2 \), and choose \( {G}_{1} \) a... | Yes |
Theorem 4.4.6. (Kuratowski 1930; Wagner 1937) \( \left\lbrack \begin{array}{l} {4.5.1} \\ {12.4.3} \end{array}\right\rbrack \)\n\nThe following assertions are equivalent for graphs \( G \) :\n\n(i) \( G \) is planar;\n\n(ii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a minor;\n\n(iii) \( G \) contain... | Proof. Combine Corollary 4.2.11 with Lemmas 4.4.2, 4.4.3 and 4.4.5. | Yes |
Corollary 4.4.7. Every maximal planar graph with at least four vertices is 3-connected. | Proof. Apply Lemma 4.4.5 and Theorem 4.4.6. | No |
A 3-connected graph is planar if and only if every edge lies on at most (equivalently: exactly) two non-separating induced cycles. | The forward implication follows from Propositions 4.2.7 and (4.2.6) (4.2.7) 4.2.2 (and Proposition 4.2.6 for the 'exactly two' version); the backward implication follows from Theorems 3.2.3 and 4.5.1. | Yes |
Proposition 4.6.1. For any connected plane multigraph \( G \), an edge set \( E \subseteq E\left( G\right) \) is the edge set of a cycle in \( G \) if and only if \( {E}^{ * } \mathrel{\text{:=}} \left\{ {{e}^{ * } \mid e \in E}\right\} \) is a minimal cut in \( {G}^{ * } \) . | Proof. By conditions (i) and (ii) in the definition of \( {G}^{ * } \), two vertices \( {v}^{ * }\left( {f}_{1}\right) \) and \( {v}^{ * }\left( {f}_{2}\right) \) of \( {G}^{ * } \) lie in the same component of \( {G}^{ * } - {E}^{ * } \) if and only if \( {f}_{1} \) and \( {f}_{2} \) lie in the same region of \( {\mat... | Yes |
Proposition 4.6.2. If \( {G}^{ * } \) is an abstract dual of \( G \), then the cut space of \( {G}^{ * } \) is the cycle space of \( G \), i.e., \[ {\mathcal{C}}^{ * }\left( {G}^{ * }\right) = \mathcal{C}\left( G\right) \] | Proof. Since the cycles of \( G \) are precisely the bonds of \( {G}^{ * } \), the subspace \( \mathcal{C}\left( G\right) \) they generate in \( \mathcal{E}\left( G\right) = \mathcal{E}\left( {G}^{ * }\right) \) is the same as the subspace generated by the bonds in \( {G}^{ * } \) . By Lemma 1.9.4, \( {}^{5} \) this is... | Yes |
A graph is planar if and only if it has an abstract dual. | Proof. Let \( G \) be a planar graph, and consider any drawing. Every \( {\text{component}}^{6}C \) of this drawing has a plane dual \( {C}^{ * } \) . Consider these \( {C}^{ * } \) as abstract multigraphs, and let \( {G}^{ * } \) be their disjoint union. Then the bonds of \( {G}^{ * } \) are precisely the minimal cuts... | Yes |
Theorem 5.1.1. (Four Colour Theorem)\n\nEvery planar graph is 4-colourable. | Null | No |
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