Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 1.7.5 If the line graph of a connected graph \( X \) is regular, then \( X \) is regular or bipartite and semiregular. | Proof. Suppose that \( L\left( X\right) \) is regular with valency \( k \) . If \( u \) and \( v \) are adjacent vertices in \( X \), then their valencies sum to \( k + 2 \) . Consequently, all neighbours of a vertex \( u \) have the same valency, and so if two vertices of \( X \) share a common neighbour, then they ha... | No |
Theorem 1.8.1 (Euler) If a connected plane graph has \( n \) vertices, e edges and \( f \) faces, then\n\n\[ n - e + f = 2. \] | A maximal planar graph is a planar graph \( X \) such that the graph formed by adding an edge between any two nonadjacent vertices of \( X \) is not planar. If an embedding of a planar graph has a face of length greater than three, then an edge can be added between two vertices of that face. Therefore, in any embedding... | No |
Lemma 2.2.1 Let \( G \) be a permutation group acting on \( V \) and let \( S \) be an orbit of \( G \) . If \( x \) and \( y \) are elements of \( S \), the set of permutations in \( G \) that map \( x \) to \( y \) is a right coset of \( {G}_{x} \) . Conversely, all elements in a right coset of \( {G}_{x} \) map \( x... | Proof. Since \( G \) is transitive on \( S \), it contains an element, \( g \) say, such that \( {x}^{g} = y \) . Now suppose that \( h \in G \) and \( {x}^{h} = y \) . Then \( {x}^{g} = {x}^{h} \), whence \( {x}^{h{g}^{-1}} = x \) . Therefore, \( h{g}^{-1} \in {G}_{x} \) and \( h \in {G}_{x}g \) . Consequently, all el... | Yes |
Lemma 2.2.2 (Orbit-stabilizer) Let \( G \) be a permutation group acting on \( V \) and let \( x \) be a point in \( V \) . Then\n\n\[ \left| {G}_{x}\right| \left| {x}^{G}\right| = \left| G\right| \] | Proof. By the previous lemma, the points of the orbit \( {x}^{G} \) correspond bijectively with the right cosets of \( {G}_{x} \) . Hence the elements of \( G \) can be partitioned into \( \left| {x}^{G}\right| \) cosets, each containing \( \left| {G}_{x}\right| \) elements of \( G \) . | Yes |
Lemma 2.2.3 Let \( G \) be a permutation group on the set \( V \) and let \( x \) be a point in \( V \) . If \( g \in G \), then \( {g}^{-1}{G}_{x}g = {G}_{{x}^{g}} \) . | Proof. Suppose that \( {x}^{g} = y \) . First we show that every element of \( {g}^{-1}{G}_{x}g \) fixes \( y \) . Let \( h \in {G}_{x} \) . Then\n\n\[ \n{y}^{{g}^{-1}{hg}} = {x}^{hg} = {x}^{g} = y \n\] \n\nand therefore \( {g}^{-1}{hg} \in {G}_{y} \) . On the other hand, if \( h \in {G}_{y} \), then \( {gh}{g}^{-1} \)... | Yes |
Lemma 2.3.1 The size of the isomorphism class containing \( X \) is | \[ \frac{n!}{\left| \operatorname{Aut}\left( X\right) \right| } \] Proof. This follows from the orbit-stabilizer lemma. We leave the details as an exercise. | No |
Corollary 2.3.3 Almost all graphs are asymmetric. | Proof. Suppose that the proportion of isomorphism classes of graphs on \( V \) that are asymmetric is \( \mu \) . Each isomorphism class of a graph that is not asymmetric contains at most \( n!/2 \) graphs, whence the average size of an isomorphism class is at most\n\n\[ n!\left( {\mu + \frac{\left( 1 - \mu \right) }{2... | Yes |
Lemma 2.4.1 Let \( G \) be a group acting transitively on the set \( V \), and let \( x \) be a point of \( V \). Then there is a one-to-one correspondence between the orbits of \( G \) on \( V \times V \) and the orbits of \( {G}_{x} \) on \( V \). | Proof. Let \( \Omega \) be an orbit of \( G \) on \( V \times V \), and let \( {Y}_{\Omega } \) denote the set \( \{ y : \left( {x, y}\right) \in \Omega \} \). We claim that the set \( {Y}_{\Omega } \) is an orbit of \( {G}_{x} \) acting on \( V \). If \( y \) and \( {y}^{\prime } \) belong to \( {Y}_{\Omega } \), then... | Yes |
Lemma 2.4.2 Let \( G \) be a transitive permutation group on \( V \) and let \( \Omega \) be an orbit of \( G \) on \( V \times V \) . Suppose \( \left( {x, y}\right) \in \Omega \) . Then \( \Omega \) is symmetric if and only if there is a permutation \( g \) in \( G \) such that \( {x}^{g} = y \) and \( {y}^{g} = x \)... | Proof. If \( \left( {x, y}\right) \) and \( \left( {y, x}\right) \) both lie in \( \Omega \), then there is a permutation \( g \in \) \( G \) such that \( {\left( x, y\right) }^{g} = \left( {{x}^{g},{y}^{g}}\right) = \left( {y, x}\right) \) . Conversely, suppose there is a permutation \( g \) swapping \( x \) and \( y ... | Yes |
Lemma 2.4.3 The automorphism group of \( J\left( {7,3,1}\right) \) contains a group isomorphic to \( \operatorname{Sym}\left( 8\right) \) . | Proof. There are 35 partitions of the set \( \{ 0,1,\ldots ,7\} \) into two sets of size four. Let \( X \) be the graph with these partitions as vertices, where two partitions are adjacent if and only if the intersection of a 4-set from one with a 4 -set from the other is a set of size two. Clearly, \( \operatorname{Au... | Yes |
Lemma 2.5.1 Let \( G \) be a transitive permutation group on \( V \) and let \( x \) be a point in \( V \) . Then \( G \) is primitive if and only if \( {G}_{x} \) is a maximal subgroup of \( G \) . | Proof. In fact, we shall be proving the contrapositive statement, namely that \( G \) has a nontrivial system of imprimitivity if and only if \( {G}_{x} \) is not a maximal subgroup of \( G \) . First some notation: We write \( H \leq G \) if \( H \) is a subgroup of \( G \), and \( H < G \) if \( H \) is a proper subg... | Yes |
Lemma 2.6.1 Let \( D \) be a directed graph such that the in-valency and out-valency of any vertex are equal. Then \( D \) is strongly connected if and only if it is weakly connected. | Proof. The difficulty is to show that if \( D \) is weakly connected, then it is strongly connected. Assume by way of contradiction that \( D \) is weakly, but not strongly, connected and let \( {D}_{1},\ldots ,{D}_{r} \) be the strong components of \( D \) . If there is an arc starting in \( {D}_{1} \) and ending in \... | Yes |
Lemma 2.6.2 Let \( G \) be a transitive permutation group on \( V \) . Then \( G \) is primitive if and only if each nondiagonal orbit is connected. | Proof. Suppose that \( G \) is imprimitive, and that \( {B}_{1},\ldots ,{B}_{r} \) is a system of imprimitivity. Assume further that \( x \) and \( y \) are distinct points in \( {B}_{1} \) and \( \Omega \) is the orbit of \( G \) (on \( V \times V \) ) that contains \( \left( {x, y}\right) \) . If \( g \in G \), then ... | Yes |
Lemma 3.1.1 The \( k \) -cube \( {Q}_{k} \) is vertex transitive. | Proof. If \( v \) is a fixed \( k \) -tuple, then the mapping\n\n\[ \n{\rho }_{v} : x \mapsto x + v \n\]\n\n(where addition is binary) is a permutation of the vertices of \( {Q}_{k} \) . This mapping is an automorphism because the \( k \) -tuples \( x \) and \( y \) differ in precisely one coordinate position if and on... | Yes |
Theorem 3.1.2 The Cayley graph \( X\left( {G, C}\right) \) is vertex transitive. | Proof. For each \( g \in G \) the mapping\n\n\[ \n{\rho }_{g} : x \mapsto {xg} \n\] \n\nis a permutation of the elements of \( G \) . This is an automorphism of \( X\left( {G, C}\right) \) because \n\n\[ \n\left( {yg}\right) {\left( xg\right) }^{-1} = {yg}{g}^{-1}{x}^{-1} = y{x}^{-1}, \n\] \n\nand so \( {xg} \sim {yg} ... | Yes |
Lemma 3.1.3 The Petersen graph is not a Cayley graph. | Proof. There are only two groups of order 10, the cyclic group \( {\mathbb{Z}}_{10} \) and the dihedral group \( {D}_{10} \) . You may verify that none of the cubic Cayley graphs on these groups are isomorphic to the Petersen graph (Exercise 2). | No |
Lemma 3.2.1 Let \( X \) be an edge-transitive graph with no isolated vertices. If \( X \) is not vertex transitive, then \( \operatorname{Aut}\left( X\right) \) has exactly two orbits, and these two orbits are a bipartition of \( X \) . | Proof. Suppose \( X \) is edge but not vertex transitive. Suppose that \( \{ x, y\} \in \) \( E\left( X\right) \) . If \( w \in V\left( X\right) \), then \( w \) lies on an edge and there is an element of \( \operatorname{Aut}\left( X\right) \) that maps this edge onto \( \{ x, y\} \) . Hence any vertex of \( X \) lies... | Yes |
Lemma 3.2.2 If the graph \( X \) is vertex and edge transitive, but not arc transitive, then its valency is even. | Proof. Let \( G = \operatorname{Aut}\left( X\right) \), and suppose that \( x \in V\left( X\right) \) . Let \( y \) be a vertex adjacent to \( x \) and \( \Omega \) be the orbit of \( G \) on \( V \times V \) that contains \( \left( {x, y}\right) \) . Since \( X \) is edge transitive, every arc in \( X \) can be mapped... | Yes |
Lemma 3.3.1 Let \( A \) and \( B \) be subsets of \( V\left( X\right) \), for some graph \( X \) . Then\n\n\[ \left| {\partial \left( {A \cup B}\right) }\right| + \left| {\partial \left( {A \cap B}\right) }\right| \leq \left| {\partial A}\right| + \left| {\partial B}\right| . | Proof. The details are left as an exercise. We simply note here that the difference between the two sides is twice the number of edges joining \( A \smallsetminus B \) to \( B \smallsetminus A \) . | No |
Corollary 3.3.2 Any two distinct edge atoms are vertex disjoint. | Proof. Assume \( \kappa = {\kappa }_{1}\left( X\right) \) and let \( A \) and \( B \) be two distinct edge atoms in \( X \) . If \( A \cup B = V\left( X\right) \), then, since neither \( A \) nor \( B \) contains more than half the vertices of \( X \), it follows that\n\n\[ \left| A\right| = \left| B\right| = \frac{1}{... | Yes |
Lemma 3.3.3 If \( X \) is a connected vertex-transitive graph, then its edge connectivity is equal to its valency. | Proof. Suppose that \( X \) has valency \( k \) . Let \( A \) be an edge atom of \( X \) . If \( A \) is a single vertex, then \( \left| {\partial A}\right| = k \) and we are finished. Suppose that \( \left| A\right| \geq 2 \) . If \( g \) is an automorphism of \( X \) and \( B = {A}^{g} \), then \( \left| B\right| = \... | Yes |
Theorem 3.4.1 (Menger) Let \( u \) and \( v \) be distinct vertices in the graph \( X \) . Then the maximum number of openly disjoint paths from \( u \) to \( v \) equals the minimum size of a set of vertices \( S \) such that \( u \) and \( v \) lie in distinct components of \( X \smallsetminus S \) . | We say that the subset \( S \) of the theorem separates \( u \) and \( v \) . Clearly, two vertices joined by \( m \) openly disjoint paths cannot be separated by any set of size less than \( m \) . The significance of this theorem is that it implies that two vertices that cannot be separated by fewer than \( m \) vert... | No |
Lemma 3.4.3 Let \( A \) and \( B \) be fragments in a graph \( X \) . Then\n\n(a) \( N\left( {A \cap B}\right) \subseteq \left( {A \cap N\left( B\right) }\right) \cup \left( {N\left( A\right) \cap B}\right) \cup \left( {N\left( A\right) \cap N\left( B\right) }\right) \) . | Proof. Suppose first that \( x \in N\left( {A \cap B}\right) \) . Since \( A \cap B \) and \( N\left( {A \cap B}\right) \) are disjoint, if \( x \in A \), then \( x \notin B \), and therefore it must lie in \( N\left( B\right) \) . Similarly, if \( x \in B \), then \( x \in N\left( A\right) \) . If \( x \) does not lie... | Yes |
Theorem 3.4.4 Let \( X \) be a graph on \( n \) vertices with connectivity \( \kappa \) . Suppose \( A \) and \( B \) are fragments of \( X \) and \( A \cap B \neq \varnothing \) . If \( \left| A\right| \leq \left| \bar{B}\right| \), then \( A \cap B \) is a fragment. | Proof. The intersections of \( A, N\left( A\right) \), and \( \bar{A} \) with the sets \( B, N\left( B\right) \), and \( \bar{B} \) partition \( V\left( X\right) \) into nine pieces, as shown in Figure 3.5. The cardinalities of five of these pieces are also defined in this figure. We present the proof as a number of st... | Yes |
Corollary 3.4.5 If \( A \) is an atom and \( B \) is a fragment of \( X \), then \( A \) is a subset of exactly one of \( B, N\left( B\right) \), and \( \bar{B} \) . | Proof. Since \( A \) is an atom, \( \left| A\right| \leq \left| B\right| \) and \( \left| A\right| \leq \left| \bar{B}\right| \) . Hence the intersection of \( A \) with \( B \) or \( \bar{B} \), if nonempty, would be a fragment. Since \( A \) is an atom, no proper subset of it can be a fragment. The result follows imm... | Yes |
Lemma 3.5.2 Let \( u \) and \( v \) be vertices in \( X \) such that no maximum matching misses both of them. Suppose that \( {M}_{u} \) and \( {M}_{v} \) are maximum matchings that miss \( u \) and \( v \), respectively. Then there is a path of even length in \( {M}_{u} \oplus {M}_{v} \) with \( u \) and \( v \) as it... | Proof. Our hypothesis implies that \( u \) and \( v \) are vertices of valency one in \( {M}_{u} \oplus {M}_{v} \) so, by our ruminations above, both vertices are end-vertices of paths in \( {M}_{u} \oplus {M}_{v} \) . As \( {M}_{u} \) and \( {M}_{v} \) have maximum size, these paths have even length. If they are end-v... | Yes |
Lemma 3.5.3 Let \( u \) and \( v \) be distinct vertices in \( X \) and let \( P \) be a path from \( u \) to \( v \) . If no vertex in \( V\left( P\right) \smallsetminus \{ u, v\} \) is critical, then no maximum matching misses both \( u \) and \( v \) . | Proof. The proof is by induction on the length of \( P \) . If \( u \sim v \), then no maximum matching misses both \( u \) and \( v \) ; hence we may assume that \( P \) has length at least two.\n\nLet \( x \) be a vertex on \( P \) distinct from \( u \) and \( v \) . Then \( u \) and \( x \) are joined in \( X \) by ... | Yes |
Lemma 3.6.2 Let \( G \) be a transitive permutation group on a set \( V \), let \( S \) be a subset of \( V \), and set \( c \) equal to the minimum value of \( \left| {S \cap {S}^{g}}\right| \) as \( g \) ranges over the elements of \( G \) . Then \( \left| S\right| \geq \sqrt{c\left| V\right| } \) . | Proof. We count the pairs \( \left( {g, x}\right) \) where \( g \in G \) and \( x \in S \cap {S}^{g} \) . For each element of \( G \) there are at least \( c \) such points in \( S \), and therefore there are at least \( c\left| G\right| \) such pairs. On the other hand, the elements of \( G \) that map \( x \) to \( y... | Yes |
Theorem 3.6.3 A connected vertex-transitive graph on \( n \) vertices contains a cycle of length at least \( \sqrt{3n} \) . | Proof. Let \( X \) be our graph and let \( G \) be its automorphism group. First we need to know that a connected vertex-transitive graph with valency at least three is at least 3-connected. This is a consequence of Theorem 3.4.2. Now we let \( C \) be a maximum-length cycle of \( X \) . Then by Exercise 19, \( \left| ... | No |
Lemma 3.7.2 If a group \( G \) acts regularly on the vertices of the graph \( X \) , then \( X \) is a Cayley graph for \( G \) relative to some inverse-closed subset of \( G \smallsetminus e \) . | Proof. Choose a fixed vertex \( u \) of \( X \) . Now, if \( v \) is any vertex of \( X \), then since \( G \) acts regularly on \( V\left( X\right) \), there is a unique element, \( {g}_{v} \) say, in \( G \) such that \( {u}^{{g}_{v}} = v \) . Define\n\n\[ C \mathrel{\text{:=}} \left\{ {{g}_{v} : v \sim u}\right\} \]... | Yes |
Lemma 3.7.3 If \( \theta \) is an automorphism of the group \( G \), then \( X\left( {G, C}\right) \) and \( X\left( {G,\theta \left( C\right) }\right) \) are isomorphic. | Proof. For any two vertices \( x \) and \( y \) of \( X\left( {G, C}\right) \) we have\n\n\[ \theta \left( y\right) \theta {\left( x\right) }^{-1} = \theta \left( {y{x}^{-1}}\right) \]\n\nand so \( \theta \left( y\right) \theta {\left( x\right) }^{-1} \in \theta \left( C\right) \) if and only if \( y{x}^{-1} \in C \) .... | Yes |
Theorem 3.8.1 Suppose that distinct group elements \( \alpha \) and \( \beta \) generate the finite group \( G \), and that \( X = X\left( {G,\{ \alpha ,\beta \} }\right) \) is the directed Cayley graph of \( G \) with connection set \( \{ \alpha ,\beta \} \) . Suppose further that \( \alpha \) and \( \beta \) have \( ... | Proof. Suppose that \( V\left( X\right) \) has a partition into \( r \) directed cycles. Define a permutation \( \pi \) of \( G \) by \( {x}^{\pi } = y \) if the arc \( \left( {x, y}\right) \) is in one of the directed cycles. If we define\n\n\[ P = \left\{ {x \in V\left( X\right) : {x}^{\pi } = {\alpha x}}\right\} ,\;... | Yes |
Theorem 3.9.1 Any connected vertex-transitive graph is a retract of a Cayley graph. | Proof. Suppose \( X \) is a connected vertex-transitive graph and let \( x \) be a vertex of \( X \) . Let \( C \) be the set\n\n\[ C \mathrel{\text{:=}} \left\{ {g \in G : x \sim {x}^{g}}\right\} . \]\n\nThen \( C \) is a union of right cosets of \( {G}_{x} \), and since \( x \) is not adjacent to itself \( C \cap {G}... | Yes |
Lemma 3.10.1 Let \( \mathcal{T} \) be a set of transpositions from \( \operatorname{Sym}\left( n\right) \) . Then \( \mathcal{T} \) is a generating set for \( \operatorname{Sym}\left( n\right) \) if and only if its graph is connected. | Proof. Let \( T \) be the graph of \( \mathcal{T} \), which has vertex set \( \{ 1,\ldots, n\} \) . Let \( G \) be the group generated by \( \mathcal{T} \) . If \( \left( {1i}\right) \) and \( \left( {ij}\right) \) are elements of \( \mathcal{T} \), then\n\n\[ \left( {1j}\right) = \left( {ij}\right) \left( {1i}\right) ... | Yes |
Lemma 3.10.2 Let \( \mathcal{T} \) be a set of transpositions from \( \operatorname{Sym}\left( n\right) \) . Then the following are equivalent:\n\n(a) \( \mathcal{T} \) is a minimal generating set for \( \operatorname{Sym}\left( n\right) \) .\n\n(b) The graph of \( \mathcal{T} \) is a tree.\n\n(c) The product of the el... | Proof. A connected graph on \( n \) vertices must have at least \( n - 1 \) edges, and has exactly \( n - 1 \) edges if and only if it is a tree. Thus (a) and (b) are equivalent. The equivalence of (b) and (c) is left as an exercise. | No |
Lemma 3.10.3 Let \( \mathcal{T} \) be a set of transpositions from \( \operatorname{Sym}\left( n\right) \) and let \( g \) and \( h \) be elements of \( \mathcal{T} \) . If the graph of \( \mathcal{T} \) contains no triangles, then \( g \) and \( h \) have exactly one common neighbour in \( X\left( {\operatorname{Sym}\... | Proof. The neighbours of a vertex \( g \) of \( X\left( {\operatorname{Sym}\left( n\right) ,\mathcal{T}}\right) \) have the form \( {xg} \) , where \( x \in \mathcal{T} \) . So suppose \( {xg} = {yh} \) is a common neighbour of \( g \) and \( h \) . Then \( {yx} = {hg} \), and any solution to this equation yields a com... | Yes |
Theorem 3.10.4 Let \( \mathcal{T} \) be a minimal generating set of transpositions for \( \operatorname{Sym}\left( n\right) \) . If the graph of \( \mathcal{T} \) is asymmetric, then\n\n\[ \operatorname{Aut}\left( {X\left( {\operatorname{Sym}\left( n\right) ,\mathcal{T}}\right) }\right) \cong \operatorname{Sym}\left( n... | Proof. Let \( T \) be the graph of \( \mathcal{T} \) . Since \( \mathcal{T} \) is a minimal generating set, \( T \) is a tree and hence contains no cycles. Then by Lemma 3.10.3 we can determine from \( X\left( {\operatorname{Sym}\left( n\right) ,\mathcal{T}}\right) \) which pairs of transpositions in \( \mathcal{T} \) ... | No |
Lemma 4.1.1 The graphs \( J\left( {v, k, i}\right) \) are at least arc transitive. | Proof. Consider the vertex \( \{ 1,\ldots, k\} \) . The stabilizer of this vertex contains \( \operatorname{Sym}\left( k\right) \times \operatorname{Sym}\left( {v - k}\right) \) . Clearly, any two \( k \) -sets meeting this initial vertex in an \( i \) -set can be mapped to each other by this group. | Yes |
Lemma 4.1.3 (Tutte) If \( X \) is an s-arc transitive graph with valency at least three and girth \( g \), then \( g \geq {2s} - 2 \) . | Proof. We may assume that \( s \geq 3 \), since the condition on the girth is otherwise meaningless. It is easy to see that \( X \) contains a cycle of length \( g \) and a path of length \( g \) whose end-vertices are not adjacent. Therefore \( X \) contains a \( g \) -arc with adjacent end-vertices and a \( g \) -arc... | Yes |
Lemma 4.1.4 (Tutte) If \( X \) is an \( s \) -arc transitive graph with girth \( {2s} - 2 \) , it is bipartite and has diameter \( s - 1 \) . | Proof. We first observe that if \( X \) has girth \( {2s} - 2 \), then any \( s \) -arc lies in at most one cycle of length \( {2s} - 2 \), and so if \( X \) is \( s \) -arc transitive, it follows that every \( s \) -arc lies in a unique cycle of length \( {2s} - 2 \) . Clearly, \( X \) has diameter at least \( s - 1 \... | Yes |
Lemma 4.2.1 Let \( X \) and \( Y \) be directed graphs and let \( f \) be a homomorphism from \( X \) onto \( Y \) such that every edge in \( Y \) is the image of an edge in \( X \) . Suppose \( {y}_{0},\ldots ,{y}_{r} \) is a path in \( Y \) . Then for each vertex \( {x}_{0} \) in \( X \) such that \( f\left( {x}_{0}\... | ## Proof. Exercise. | No |
Lemma 4.3.1 Let \( X \) be a strongly connected directed graph, let \( G \) be a transitive subgroup of its automorphism group, and, if \( u \in V\left( X\right) \), let \( N\left( u\right) \) be the set of vertices \( v \) in \( V\left( X\right) \) such that \( \left( {u, v}\right) \) is an arc of \( X \) . If there i... | Proof. Suppose \( u \in V\left( X\right) \) and \( {G}_{u} \upharpoonright N\left( u\right) \) is the identity group. By Lemma 2.2.3, if \( v \in V\left( X\right) \), then \( {G}_{v} \) is conjugate in \( G \) to \( {G}_{u} \) . Hence \( {G}_{v}|N\left( v\right) \) must be the identity for all vertices \( v \) of \( X ... | Yes |
Lemma 4.3.2 Let \( X \) be a connected cubic graph that is \( s \) -arc transitive, but not \( \left( {s + 1}\right) \) -arc transitive. Then \( X \) is \( s \) -arc regular. | Proof. We note that if \( X \) is cubic, then \( {X}^{\left( s\right) } \) has out-valency two. Now let \( G \) be the automorphism group of \( X \), let \( \alpha \) be an \( s \) -arc in \( X \), and let \( H \) be the subgroup of \( G \) fixing each vertex in \( \alpha \) . Then \( G \) acts vertex transitively on \... | No |
Theorem 4.4.1 The Petersen graph cannot be 3-edge coloured. | Proof. Let \( P \) denote the Petersen graph, and suppose for a contradiction that it can be 3-edge coloured. Since \( P \) is cubic, each colour class is a 1- factor of \( P \) . A simple case argument shows that each edge lies in precisely two 1-factors (Figure 4.4 shows the two 1-factors containing the vertical\n\n\ | No |
Lemma 4.5.1 The graph \( J\left( {v, k, k - 1}\right) \) is distance transitive. | Proof. The key is to prove that two vertices \( u \) and \( v \) have distance \( i \) in \( J\left( {v, k, k - 1}\right) \) if and only if \( \left| {u \cap v}\right| = k - i \) (viewing \( u \) and \( v \) as \( k \) -sets). We leave the details as an exercise. | No |
Lemma 4.5.3 A connected s-arc transitive graph with girth \( {2s} - 2 \) is distance-transitive with diameter \( s - 1 \) . | Proof. Let \( X \) satisfy the hypotheses of the lemma and let \( \left( {u,{u}^{\prime }}\right) \) and \( \left( {v,{v}^{\prime }}\right) \) be pairs of vertices at distance \( i \) . Since \( X \) has diameter \( s - 1 \) by Lemma 4.1.4, we see that \( i \leq s - 1 \) . The two pairs of vertices are joined by paths ... | Yes |
Lemma 4.6.1 The diameter of \( J\left( {7,3,0}\right) \) is three and its girth is six. | Proof. If we denote \( J\left( {7,3,0}\right) \) by \( Y \), then \( {Y}_{1}\left( u\right) \) consists of the triples disjoint from \( u \), while \( {Y}_{2}\left( u\right) \) consists of the triples that meet \( u \) in two points, and \( {Y}_{3}\left( u\right) \) consists of the triples that meet \( u \) in one poin... | No |
Lemma 4.6.2 There is a one-to-one correspondence between six-cycles in \( J\left( {7,3,0}\right) \) and partitions of \( \Omega \) of the form \( \{ {abc},{de},{fg}\} \) . | Proof. The partition \( \{ {abc},{de},{fg}\} \) corresponds to the six-cycle \( {ade},{bfg} \) , cde, afg, bde, cfg. To show that every six-cycle has this form, it suffices to consider six-cycles through 123. Without loss of generality we can assume that the neighbours of 123 in the six-cycle are 456 and 457 . The vert... | Yes |
Lemma 4.6.3 Every heptad meets every six-cycle in \( J\left( {7,3,0}\right) \) . | Proof. The seven triples of a heptad contain 21 pairs of points; since two distinct triples have only one point in common, these pairs must be distinct. Hence each pair of points from \( \{ 1,\ldots ,7\} \) lies in exactly one triple from the heptad. If the point \( i \) lies in \( r \) triples, then it lies in \( {2r}... | Yes |
Lemma 4.7.1 Let \( F \) be a 1 -factor of \( {K}_{6} \) and let e be an edge of \( {K}_{6} \) that is not contained in \( F \) . Then there is a unique 1 -factor on \( e \) that contains an edge of \( F \) . | Proof. Two edges of \( {K}_{6} \) lie in a 1-factor if and only if they are disjoint, and two disjoint edges lie in a unique 1-factor. Since \( e \notin F \), it meets two distinct edges of \( F \), and hence is disjoint from precisely one edge of \( F \), with which it lies in a unique 1-factor. | Yes |
Lemma 5.1.1 The incidence graph \( X \) of a partial linear space has girth at least six. | Proof. If \( X \) contains a four-cycle \( p, L, q, M \), then \( p \) and \( q \) are incident to two lines. Since the girth of \( X \) is even and not four, it is at least six. | No |
Theorem 5.2.1 Let \( \mathcal{I} \) be a partial linear space that contains a triangle. Then \( \mathcal{I} \) is a (possibly degenerate) projective plane if and only if its incidence graph \( X\left( \mathcal{I}\right) \) has diameter three and girth six. | Proof. Let \( \mathcal{I} \) be a projective plane that contains a triangle. Any two points lie at distance two in \( X\left( \mathcal{I}\right) \), similarly for any two lines. Now, consider a line \( L \) and a point \( p \) not on \( L \) . Any line \( M \) through \( p \) must meet \( L \) in a point \( {p}^{\prime... | Yes |
Theorem 5.4.1 Let \( \mathcal{I} \) be a partial linear space that contains noncollinear points and nonconcurrent lines. Then \( \mathcal{I} \) is a generalized quadrangle if and only if its incidence graph \( X\left( \mathcal{I}\right) \) has diameter four and girth eight. | Proof. Let \( \mathcal{I} \) be a generalized quadrangle, and consider the distances in \( X\left( \mathcal{I}\right) \) from a point \( p \) . A line is distance one from \( p \) if it contains \( p \), and at distance three otherwise (by the condition defining a generalized quadrangle). A point is at distance two fro... | Yes |
Lemma 5.5.1 Let \( W\left( q\right) \) be the point/line incidence structure whose points and lines are the totally isotropic points and totally isotropic lines of \( {PG}\left( {3, q}\right) \) . Then \( W\left( q\right) \) is a generalized quadrangle. | Proof. We need to prove that given a point \( p \) and a line \( L \) not containing \( p \), there is a unique point on \( L \) that is collinear with \( p \) . Suppose that the point \( p \) is spanned by the vector \( u \) . Any point collinear with \( p \) is spanned by a vector in \( {u}^{ \bot } \) . The 3-dimens... | Yes |
Lemma 5.6.2 If \( d\left( {v, w}\right) = d \), then \( v \) and \( w \) have the same valency. | Proof. Since \( X \) is bipartite of diameter \( d \), any neighbour \( {v}^{\prime } \) of \( v \) has distance \( d - 1 \) from \( w \) . Therefore, there is a unique path of length \( d - 1 \) from \( {v}^{\prime } \) to \( w \) that contains precisely one neighbour of \( w \) . Each such path contains a different n... | Yes |
Lemma 5.6.2 If \( d\left( {v, w}\right) = d \), then \( v \) and \( w \) have the same valency. | Proof. Since \( X \) is bipartite of diameter \( d \), any neighbour \( {v}^{\prime } \) of \( v \) has distance \( d - 1 \) from \( w \) . Therefore, there is a unique path of length \( d - 1 \) from \( {v}^{\prime } \) to \( w \) that contains precisely one neighbour of \( w \) . Each such path contains a different n... | Yes |
Lemma 5.6.3 Every vertex in \( X \) has valency at least two. | Proof. Let \( C \) be a cycle of length \( {2d} \) in \( X \) . Clearly, the vertices of \( C \) have valency at least two. Let \( x \) be a vertex not on \( C \), let \( P \) be the shortest path joining \( x \) to \( C \), and denote the length of \( P \) by \( i \) . Travelling around the cycle for \( d - i \) steps... | Yes |
Lemma 5.6.4 Any two vertices lie in a cycle of length \( {2d} \) . | Proof. Let \( v \) and \( w \) be any two vertices of \( X \) . Let \( P \) be the shortest path between them. By repeatedly choosing any neighbour of an endpoint of \( P \) not already in \( P \), we can extend \( P \) to a geodesic path of length \( d \) with endpoints \( x \) and \( y \) . Then \( x \) has a neighbo... | Yes |
Lemma 5.6.5 Let \( C \) be a cycle of length \( {2d} \) . Then any two vertices at the same distance in \( C \) from a thick vertex in \( C \) have the same valency. | Proof. Let \( v \) be a thick vertex contained in \( C \) and let \( w \) be its antipode in \( C \) (that is, the unique vertex in \( C \) at distance \( d \) from \( v \) ). Now, because \( v \) is thick, it has at least one further neighbour \( {v}^{\prime } \), and hence there is a path \( P \) from \( {v}^{\prime ... | Yes |
Lemma 5.6.6 The minimum distance \( k \) between any pair of thick vertices in \( X \) is a divisor of \( d \) . If \( d/k \) is odd, then all the thick vertices have the same valency; if it is even, then the thick vertices share at most two valencies. Moreover, any vertex at distance \( k \) from a thick vertex is its... | Proof. Let \( v \) and \( w \) be two thick vertices of \( X \) such that \( d\left( {v, w}\right) = k \), and let \( x \) be any other thick vertex of \( X \) . By extending the path from \( x \) to the closer of \( v \) and \( w \), we can form a cycle \( C \) of length \( {2d} \) containing \( v, w \) and \( x \) . ... | Yes |
Theorem 5.6.7 A generalized polygon \( X \) that is not thick is either a cycle, the \( k \) -fold subdivision of a multiple edge, or the \( k \) -fold subdivision of a thick generalized polygon. | Proof. If \( X \) has no thick vertices at all, then it is a cycle. Otherwise, the previous lemma shows that any path between two thick vertices of \( X \) has length a multiple of \( k \) with every \( k \) th vertex being thick and the remainder thin. Therefore, we can define a graph \( {X}^{\prime } \) whose vertice... | Yes |
Lemma 5.7.1 If \( X \) is a generalized hexagon of order \( \left( {2,2}\right) \), then the graph \( {X}_{5}\left( u\right) \cup {X}_{6}\left( u\right) \) is the subdivision \( S\left( Y\right) \) of a cubic graph \( Y \) on 32 vertices. | Proof. It is straightforward to confirm that the 48 vertices of \( {X}_{5}\left( u\right) \) are each adjacent to two vertices of \( {X}_{6}\left( u\right) \), and each vertex of \( {X}_{6}\left( u\right) \) is adjacent to three from \( {X}_{5}\left( u\right) \) . | Yes |
Lemma 5.7.2 For \( c \in \{ r, g, b\} \), the 16 edges of \( Y \) subdivided by the 16 vertices of \( {X}_{5}\left( u\right) \) with first entry \( c \) form a one-factor of \( Y \) . | Proof. The distance from \( c \in \{ r, g, b\} \) to any vertex of \( {X}_{5}\left( u\right) \) with first entry \( c \) is four, and so there is a path of length at most eight between any two such vertices. Since \( X \) has no cycles of length 10, two such vertices cannot subdivide incident edges of \( Y \) . | Yes |
Theorem 5.7.3 Let \( Y \) be the graph of Figure 5.3 and let \( R \) be the set of edges in one of the colour classes. Then for every edge \( e \in R \), there is a unique edge \( {e}^{\prime } \in R \) at distance 10 from e. Moreover,\n\n(a) There is a unique partition of the eight pairs \( \left\{ {e,{e}^{\prime }}\r... | Proof. A few minutes with a photocopy of \( Y \) and a pencil will be far more convincing than any written proof, so this is left as an exercise. | No |
Lemma 5.8.1 Let \( X \) be a graph with diameter \( d \) and girth \( {2d} + 1 \) . Then \( X \) is regular. | Proof. First we shall show that any two vertices at distance \( d \) have the same valency, and then we shall show that this implies that all vertices have the same valency. Let \( v \) and \( w \) be two vertices of \( X \) such that \( d\left( {v, w}\right) = d \) . Let \( P \) be the path of length \( d \) joining t... | Yes |
Lemma 5.9.1 An independent set \( C \) in a Moore graph of diameter two and valency seven contains at most 15 vertices. If \( \left| C\right| = {15} \), then every vertex not in \( C \) has exactly three neighbours in \( C \) . | Proof. Let \( X \) be a Moore graph of diameter two and valency seven. Suppose that \( C \) is an independent set in \( X \) with \( c \) vertices in it. Without loss of generality we may assume that the vertices are labelled so that the vertices \( \{ 1,\ldots ,{50} - c\} \) are the ones not in \( C \) . If \( i \) is... | Yes |
Lemma 5.10.1 In a 2-design with \( k < v \) we have \( b \geq v \) . | Proof. Putting \( t = 2 \) and \( s = 1 \) into (5.2) we get that \( r\left( {k - 1}\right) = \left( {v - 1}\right) {\lambda }_{2} \) , and so \( k < v \) implies that \( r - {\lambda }_{2} > 0 \) . By the remark at the end of Section 8.6, it follows that \( N{N}^{T} \) is invertible. It follows that the rows of \( N \... | Yes |
Lemma 5.10.2 The dual \( {\mathcal{D}}^{ * } \) of a symmetric design \( \mathcal{D} \) is a symmetric design with the same parameters. | Proof. If \( N \) is the incidence matrix of \( \mathcal{D} \), then \( {N}^{T} \) is the incidence matrix of \( {\mathcal{D}}^{ * } \) . Since \( \mathcal{D} \) is a 2-design, we have \( N{N}^{T} = \left( {r - {\lambda }_{2}}\right) I + {\lambda }_{2}J \), and thus \( {N}^{T} = {N}^{-1}\left( {\left( {r - {\lambda }_{... | Yes |
Theorem 5.10.3 A bipartite graph is the incidence graph of a symmetric 2-design if and only if it is distance regular with diameter three. | Proof. Let \( \mathcal{D} \) be a symmetric \( 2 - \left( {v, k,{\lambda }_{2}}\right) \) design with incidence graph \( X \) . Any two points lie at distance two in \( X \), and similarly for blocks. Therefore, a block lies at distance three from a point not on the block, and this is the diameter of \( X \) . Now, con... | Yes |
Theorem 5.10.4 The block intersection graph of a Steiner triple system with \( v > 7 \) is distance regular with diameter two. | Proof. Let \( \mathcal{D} \) be a \( 2 - \left( {v,3,1}\right) \) design, and let \( X \) be the block intersection graph of \( \mathcal{D} \) . Every point lies in \( \left( {v - 1}\right) /2 \) blocks, and so \( X \) is regular with valency \( 3\left( {v - 3}\right) /2 \) . If we consider two blocks that intersect, t... | Yes |
Lemma 6.2.1 Let \( X \) and \( Y \) be cores. Then \( X \) and \( Y \) are homomorphically equivalent if and only if they are isomorphic. | Proof. Suppose \( X \) and \( Y \) are homomorphically equivalent and that \( f \) : \( X \rightarrow Y \) and \( g : Y \rightarrow X \) are the homomorphisms between them. Then because both \( f \circ g \) and \( g \circ f \) must be surjective, we see that both \( f \) and \( g \) are surjective, so \( X \) and \( Y ... | Yes |
Lemma 6.2.2 Every graph \( X \) has a core, which is an induced subgraph and is unique up to isomorphism. | Proof. Since \( X \) is finite and the identity mapping is a homomorphism, the family of subgraphs of \( X \) to which \( X \) has a homomorphism is finite and nonempty and hence has a minimal element with respect to inclusion. Since a core is a retract, it is clearly an induced subgraph. Now, suppose that \( {Y}_{1} \... | Yes |
Lemma 6.2.3 Two graphs \( X \) and \( Y \) are homomorphically equivalent if and only if their cores are isomorphic. | Proof. If there is a homomorphism \( f : X \rightarrow Y \), then we have a sequence of homomorphisms\n\n\[ \n{X}^{ \bullet } \rightarrow X \rightarrow Y \rightarrow {Y}^{ \bullet } \n\]\n\nwhich composes to give a homomorphism from \( {X}^{ \bullet } \) to \( {Y}^{ \bullet } \) . Hence, if \( X \) and \( Y \) are homo... | Yes |
Theorem 6.3.1 Let \( X, Y \), and \( Z \) be graphs. If \( f : Z \rightarrow X \) and \( g : Z \rightarrow \) \( Y \), then there is a unique homomorphism \( \phi \) from \( Z \) to \( X \times Y \) such that \( f = {p}_{X} \circ \phi \) and \( g = {p}_{Y} \circ \phi \) . | Proof. Assume that we are given homomorphisms \( f : Z \rightarrow X \) and \( g \) : \( Z \rightarrow Y \) . The map\n\n\[ \phi : z \mapsto \left( {f\left( z\right), g\left( z\right) }\right) \]\n\n is readily seen to be a homomorphism from \( Z \) to \( X \times Y \) . Clearly, \( {p}_{X} \circ \phi = f \) and \( {p}... | Yes |
Theorem 6.4.1 If \( F \) is a graph and \( \psi \) is a homomorphism from \( X \) to \( Y \) , then the adjoint of \( \psi \) is a homomorphism from \( {F}^{Y} \) to \( {F}^{X} \) . | Proof. Suppose that \( f \) and \( g \) are adjacent vertices of \( {F}^{Y} \) and that \( {x}_{1} \) and \( {x}_{2} \) are adjacent vertices in \( X \) . Then \( \psi \left( {x}_{1}\right) \sim \psi \left( {x}_{2}\right) \), and therefore \( f\left( {\psi \left( {x}_{1}\right) }\right) \sim g\left( {\psi \left( {x}_{2... | Yes |
Theorem 6.4.2 For any graphs \( F, X \), and \( Y \), we have \( {F}^{X \times Y} \cong {\left( {F}^{X}\right) }^{Y} \) . | Proof. It is immediate that \( {F}^{X \times Y} \) and \( {\left( {F}^{X}\right) }^{Y} \) have the same number of vertices. We start by defining the natural bijection between these sets, and then we will show that it is an isomorphism.\n\nSuppose that \( g \) is a map from \( V\left( {X \times Y}\right) \) to \( F \) .... | Yes |
Corollary 6.4.3 For any graphs \( F, X \), and \( Y \), we have\n\n\[ \left| {\operatorname{Hom}\left( {X \times Y, F}\right) }\right| = \left| {\operatorname{Hom}\left( {Y,{F}^{X}}\right) }\right| . \] | Proof. We have just seen that \( {F}^{X \times Y} \cong {\left( {F}^{X}\right) }^{Y} \), and so they have the same number of loops, which are precisely the homomorphisms. | Yes |
Lemma 6.5.1 Let \( X \) and \( Y \) be fixed graphs. Suppose that for all graphs \( Z \) we have\n\n\[ \left| {\operatorname{Hom}\left( {Z, X}\right) }\right| = \left| {\operatorname{Hom}\left( {Z, Y}\right) }\right| \]\n\nThen \( X \) and \( Y \) are isomorphic. | Proof. Let \( \operatorname{Inj}\left( {A, B}\right) \) denote the set of injective homomorphisms from a graph \( A \) to a graph \( B \) . We aim to show that for all \( Z \) we have \( \left| {\operatorname{Inj}\left( {Z, X}\right) }\right| = \) \( \left| {\operatorname{Inj}\left( {Z, Y}\right) }\right| \) . By takin... | Yes |
Lemma 6.5.2 For any graphs \( F, X \), and \( Y \) we have\n\n\[ \n{F}^{X \cup Y} \cong {F}^{X} \times {F}^{Y} \n\] | Proof. For any graph \( Z \), we have\n\n\[ \n\left| {\operatorname{Hom}\left( {Z,{F}^{X \cup Y}}\right) }\right| = \left| {\operatorname{Hom}\left( {Z \times \left( {X \cup Y}\right), F}\right) }\right| \n\]\n\n\[ \n= \left| {\operatorname{Hom}\left( {\left( {Z \times X}\right) \cup \left( {Z \times Y}\right), F}\righ... | Yes |
Theorem 6.6.1 Suppose \( \chi \left( X\right) > n \) . Then \( {K}_{n}^{X} \) is \( n \) -colourable if and only if \( \chi \left( {X \times Y}\right) > n \) for all graphs \( Y \) such that \( \chi \left( Y\right) > n \) . | Proof. By Corollary 6.4.3,\n\n\[ \left| {\operatorname{Hom}\left( {X \times {K}_{n}^{X},{K}_{n}}\right) }\right| = \left| {\operatorname{Hom}\left( {{K}_{n}^{X},{K}_{n}^{X}}\right) }\right| > 0, \]\n\nand therefore \( X \times {K}_{n}^{X} \) is \( n \) -colourable. Consequently, if \( \chi \left( X\right) > n \) and \(... | Yes |
Theorem 6.6.2 The map graph \( {K}_{n}^{{K}_{n + 1}} \) is \( n \) -colourable. | Proof. We construct a proper \( n \) -colouring \( \phi \) of \( {K}_{n}^{{K}_{n + 1}} \) . For any \( f \in {K}_{n}^{{K}_{n + 1}} \) , there are two distinct vertices \( i \) and \( j \) such that \( f\left( i\right) = f\left( j\right) \) . Define \( \phi \left( f\right) \) to be the least value in the range of \( f \... | Yes |
Corollary 6.6.3 Suppose that the graph \( X \) contains a clique of size \( n + 1 \) . Then \( {K}_{n}^{X} \) is \( n \) -colourable. | Proof. Since \( {K}_{n + 1} \rightarrow X \), by Theorem 6.4.1\n\n\[ \n{K}_{n}^{X} \rightarrow {K}_{n}^{{K}_{n + 1}} \n\]\n\nBy the theorem, \( {K}_{n}^{{K}_{n + 1}} \) is \( n \) -colourable, and so \( {K}_{n}^{X} \) is \( n \) -colourable. | Yes |
Theorem 6.6.4 All loops in \( {K}_{n}^{{K}_{n}} \) are isolated vertices. The subgraph of \( {K}_{n}^{{K}_{n}} \) induced by the vertices without loops is \( n \) -colourable. | Proof. Suppose \( f \in {K}_{n}^{{K}_{n}} \) and \( f \) is a proper \( n \) -colouring of \( {K}_{n} \) . If \( g \) is adjacent to \( f \), then \( g\left( i\right) \neq f\left( j\right) \) for \( j \) in \( V\left( {K}_{n}\right) \smallsetminus i \) . This implies that \( g\left( i\right) = \) \( f\left( i\right) \)... | Yes |
Theorem 6.6.5 If \( X \) is connected and not \( n \)-colourable, then \( {K}_{n}^{X} \) contains a unique n-clique, namely the constant functions. | Proof. By Lemma 6.4.4, the subgraph of \( {K}_{n}^{X} \) induced by the constant functions is an \( n \)-clique. We need to prove this is the only \( n \)-clique.\n\nIf \( \chi \left( X\right) > n \) and \( f \) is a homomorphism from \( X \) to \( {K}_{n}^{{K}_{n}} \), then, by the previous theorem, \( f \) must map e... | Yes |
Theorem 6.6.6 Suppose \( n \geq 2 \) and let \( X \) and \( Y \) be connected graphs, each containing an \( n \) -clique. If \( X \) and \( Y \) are not \( n \) -colourable, neither is \( X \times Y \) . | Proof. Let \( {x}_{1},\ldots ,{x}_{n} \) and \( {y}_{1},\ldots ,{y}_{n} \) be the respective \( n \) -cliques in \( X \) and \( Y \) and suppose, by way of contradiction, that there is a homomorphism \( f \) from \( X \times Y \) into \( {K}_{n} \) . Consider the induced homomorphism from \( Y \) into \( {K}_{n}^{X} \)... | Yes |
Corollary 6.6.7 Let \( X \) be a graph such that every vertex lies in an \( n \) - clique and \( \chi \left( X\right) > n \) . If \( Y \) is a connected graph with \( \chi \left( Y\right) > n \), then \( \chi \left( {X \times Y}\right) > n \) . | Proof. Suppose by way of contradiction that there is a homomorphism \( f \) from \( X \times Y \) into \( {K}_{n} \) . Then consider the induced mapping \( {\Phi }_{f} \) from \( X \) into \( {K}_{n}^{Y} \) . Because \( {K}_{n}^{Y} \) has no loops, every \( n \) -clique in \( X \) is mapped injectively onto the unique ... | Yes |
Lemma 6.7.2 If \( X \) is uniquely \( n \) -colourable, then each proper \( n \) -colouring of \( X \) is an isolated vertex in \( {K}_{n}^{X} \) . | Proof. Let \( f \) be a proper \( n \) -colouring of \( X \) and let \( x \) be a vertex in \( X \) . Since \( X \) is uniquely \( n \) -colourable, each of the \( n - 1 \) colours other than \( f\left( x\right) \) must occur as the colour of a vertex in the neighbourhood of \( x \) . It follows that if \( g \sim f \),... | Yes |
Lemma 6.7.3 If \( \chi \left( X\right) > n \), then there is a homomorphism from \( {K}_{n}^{X} \) to the subgraph of \( {K}_{n}^{X \times {K}_{n}} \) induced by the loopless vertices. | Proof. Let \( {p}_{X} \) be the projection homomorphism from \( X \times {K}_{n} \) to \( X \), and let \( \varphi \) be the induced mapping from \( {K}_{n}^{X} \) to \( {K}_{n}^{X \times {K}_{n}} \) . (See Theorem 6.4.1, where this was introduced.) If \( g \in {K}_{n}^{X} \), then \( g \) is not a proper colouring of ... | Yes |
Lemma 6.8.1 If \( f \) is a retraction from a connected graph \( X \) to a proper subgraph \( Y \), then it is a folding. | Proof. We proceed by induction on the number of vertices in \( X \) . Suppose \( f \) is a retraction from \( X \) to \( Y \), that is, \( f \) is a homomorphism from \( X \) to \( Y \) and \( f \mid Y \) is the identity. If \( X = Y \), we have nothing to prove. Otherwise, since \( X \) is connected, there is a vertex... | Yes |
Lemma 6.8.2 Every homomorphism \( h \) from \( X \) to \( Y \) can be expressed as the composition \( f \circ g \), where \( g \) is a folding and \( f \) a local injection. | Proof. Let \( \pi \) be the kernel of \( h \) . If \( u \) and \( v \) are vertices of \( X \), write \( u \approx v \) if \( u \) and \( v \) lie in the same cell of \( \pi \) and are equal or at distance two in \( X \) . This is a symmetric and reflexive relation on the vertices of \( X \) . Hence its transitive clos... | Yes |
Lemma 6.8.3 If \( X \) covers \( Y \) and \( Y \) is a tree, then \( X \) is the disjoint union of copies of \( Y \) . | Proof. Suppose \( f \) is a covering map from \( X \) to \( Y \) . Since \( f \) is a local isomorphism, if \( x \in V\left( X\right) \), then the valency of \( f\left( x\right) \) in \( Y \) equals the valency of \( x \) in \( X \) . This implies that the image of any cycle in \( X \) is a cycle in \( Y \) , and hence... | Yes |
Lemma 6.9.1 Let \( X \) be a connected nonbipartite graph. If every 2-arc lies in a shortest odd cycle of \( X \), then \( X \) is a core. | Proof. Let \( f \) be a homomorphism from \( X \) to \( X \) . This necessarily maps a shortest odd cycle of \( X \) onto an odd cycle of the same length, so any two vertices in the cycle have different images under \( f \) . Since every 2 -arc lies in a shortest odd cycle, this shows that \( f \) is a local injection,... | Yes |
Lemma 6.9.2 Let \( X \) be a reduced triangle-free graph with diameter two. For any pair of distinct nonadjacent vertices \( u \) and \( v \), there is a vertex adjacent to \( u \) but not to \( v \) . | Proof. Suppose for a contradiction that \( N\left( u\right) \subseteq N\left( v\right) \) . Since \( X \) is reduced, there is some vertex \( w \) adjacent to \( v \) but not to \( u \) . Since \( X \) has no triangles, \( w \) is not adjacent to any neighbour of \( u \), which implies that the distance between \( u \)... | Yes |
Lemma 6.9.3 Let \( X \) be a triangle-free graph with diameter two. Then \( X \) is a core if and only if it is reduced. | Proof. Our comments above establish that a graph that is not reduced is not a core. So we assume that \( X \) is reduced and show that each 2-arc in \( X \) lies in a 5-cycle, whence the result follows from Lemma 6.9.1.\n\nAssume that \( \left( {u, v, w}\right) \) is a 2-arc. Then \( w \) is at distance two from \( u \... | Yes |
Lemma 6.10.1 For \( k \geq 2 \), the Cayley graph \( \operatorname{And}\left( k\right) \) is a reduced triangle-free graph with diameter two. | Proof. First we show that \( \operatorname{And}\left( k\right) \) is reduced. If \( \operatorname{And}\left( k\right) \) has two distinct vertices with the same neighbours, then there must be an element \( g \) in \( G \) such that \( g \neq 0 \) and \( g + C = C \) . It follows that both \( g + 1 \) and \( g - 1 \) li... | Yes |
Lemma 6.10.2 Each independent set of vertices of \( \operatorname{And}\left( k\right) \) is contained in the neighbourhood of a vertex. | Proof. Consider the \( k - 1 \) pairs of adjacent vertices of the form \( \{ {3i},{3i} - 1\} \) for \( 1 \leq i < k \) . Now, any vertex \( g \in C \) has the form \( {3q} + 1 \) . If \( q \geq i \), then \( g \) is adjacent to \( {3i} \), but not to \( {3i} - 1 \) . If \( q < i \), then \( g \) is adjacent to \( {3i} ... | Yes |
Lemma 6.11.1 If \( k \geq 2 \), then the number of 3-colourings of \( \operatorname{And}\left( k\right) \) is \( 6\left( {{3k} - 1}\right) \), and they are all equivalent under its automorphism group. | Proof. In any 3-colouring of \( \operatorname{And}\left( k\right) \), the average size of a colour class is \( k - \frac{1}{3} \) . Since the maximum size of a colour class is \( k \), two colour classes have size \( k \), and the third has size \( k - 1 \) . By Lemma 6.10.2, the two big colour classes are neighbourhoo... | Yes |
Lemma 6.11.2 Let \( X \) be a triangle-free regular graph with valency \( k \geq 2 \) , and suppose that \( P \) is a path of length two in \( X \) . If \( X \smallsetminus P \cong \operatorname{And}\left( {k - 1}\right) \) , then \( X \cong \operatorname{And}\left( k\right) \) . | Proof. Let \( P \) be the path \( \left( {u, v, w}\right) \) and let \( Y \) denote \( X \smallsetminus P \) . Since \( X \) is triangle-free and regular, the neighbours of \( u, v \), and \( w \) that are in \( Y \) form independent sets of size \( k - 1, k - 2 \), and \( k - 1 \) respectively. Since \( Y \) is regula... | Yes |
Theorem 6.13.1 If \( X \) is a vertex-transitive graph, then its core \( {X}^{ \bullet } \) is vertex transitive. | Proof. Let \( x \) and \( y \) be two distinct vertices of \( {X}^{ \bullet } \) . Then there is an automorphism of \( X \) that maps \( x \) to \( y \) . The composition of this automorphism with a retraction from \( X \) to \( {X}^{ \bullet } \) is a homomorphism \( f \) from \( X \) to \( {X}^{ \bullet } \) . The re... | Yes |
Theorem 6.13.2 If \( X \) is a vertex-transitive graph, then \( \left| {V\left( {X}^{ \bullet }\right) }\right| \) divides \( \left| {V\left( X\right) }\right| \) . | Proof. We show that the fibres of any homomorphism from \( X \) to \( {X}^{ \bullet } \) have the same size. Let \( f \) be a homomorphism from \( X \) to \( X \) whose image \( Y \) is a core of \( X \) . For any element \( g \) of \( \operatorname{Aut}\left( X\right) \), the translate \( {Y}^{g} \) is mapped onto \( ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.