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We would like to use Proposition3.10 to deduce $\mathcal {W}_{v_i}(\nu _{*}(\pi _1(\Gamma ))) = R_i$, and for this it now only remains to prove that the former is surjective onto $R_i$. Google Scholar, Higman, G.: Subgroups of finitely presented groups. Graduate Texts in Mathematics. Ann. For the three of them it is known wheter or not they are vertex-transitive, but it's never explained the reason. Then there is a bijection between the set of isomorphism classes of path-connected covering spaces $\psi : C \rightarrow X$ and the set of subgroups (up to conjugation) of $\pi _1(X)$, obtained by associating the subgroup $\psi _{*}(\pi _1(C))$ to the covering space C. We start by recalling one of the standard definitions of a Cayley graph, in order to then adapt it into the definition of a 2-partite Cayley graph. However, it is usually not easy to compute whether given vertex-transitive graphs are Cayley graphs. Suppose we have a covering map of graphs $\psi : \Delta \rightarrow \Gamma $ both of which have Cayley-like colourings $c_{\Delta }: \overrightarrow{E}(\Delta ) \rightarrow X$ and $c_{\Gamma }: \overrightarrow{E}(\Gamma ) \rightarrow X$ such that $c_{\Delta } = c_{\Gamma } \circ \psi $. For any vVi we have (v)=vi by (4). Let $\mathcal {R}'{:}{=} \{\mathcal {R}_a, a\in \mathcal {S}\}$. Moreover, our formalism allows for multiple edges between the same pair of vertices, and multiple loops at a single vertex. Discrete Mathematics. rev2023.6.2.43474. This representation isnt unique: if we choose different vertices for e0,e1 or a different action of G we potentially obtain different sets R, S and L. Note that BiCay(G,R,L,S) is regular if and only if |R|=|L|. Now pick $v \in V(\Delta )$ with $\epsilon (v) = v_i$. Graph X is symmetric. Define a map $\chi : S \times S \backslash \{(s,s^{-1}) \vert s \in \mathcal {S}\} \rightarrow \mathcal {S}'$ where. Eskin A, Fisher D, Whyte K. Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs. PCay(P)=PCayX|U|I||R to be the 1-skeleton of the universal cover of C(P). Where the same arguments apply directly the proofs will be omitted. If such an f exists, we say that (X,d) and $(Y,d')$ are quasi-isometric to each other. A colouring of the undirected edges of $\Gamma $ is a map $c: E(\Gamma ) \rightarrow X$ whereas a colouring of the directed edges is a map $c: \overrightarrow{E}(\Gamma ) \rightarrow X$, where X is an arbitrary set called the set of colours. Slider with three articles shown per slide. We say that a metric space (X,d) is quasi-isometric to a group G, if (X,d) is quasi-isometric to some, hence to every, finitely generated Cayley graph of G. Let P=X|U|I||R be a partite presentation with finite X. Let I:=S, and set R0:=W(1G)0(1(,(1G)0)). Note that $\mathcal {C}(P)$ is connected by condition (a) Finally. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Math. We say that a weak multicycle colouring c is partition-friendly, if c-1(x) is regular for all x. Define the $\vert \mathcal {S}\cup \mathcal {S}^{-1} \vert $-regular tree $T_{P}$ by. by using the fact that every cellular action on a CW-complex with finite stabilisers of cells is properly discontinuous [15, Theorem9,(2)=(10)]. Pick an orientation of OCE() of C. If is infinite then just choose an orientation with equal in and out degree, which can be constructed greedily. J. Combin. Graduate Texts in Mathematics, vol. We say a partite presentation P=X|U|I||R is uniform, if for every sS, all orbits of (s) have the same size. Math. Proc. We say that a locally finite (vertex-transitive) graph is finitely presented if it has a partite presentation with finitely many vertex classes and finitely many relators. To begin with, they have the same vertex set $V(C) = X = V(\Gamma )$. Math. . Vertices in a graph with the same number of closed walks. This follows by combining Proposition5.17 with the fact that these graphs are not quasi-isometric to any finitely generated group [7, Theorem1.4]. Ask Question Asked 8 years ago Modified 8 years ago Viewed 2k times 3 I have problems to prove wheter a regular graph is vertex-transitive or not. If instead we chose $(a^0)_1 {:}{=} y_1$ we would obtain $R = \{a,a^4\}$, $L = \{a^2,a^3\}$ and $S = \{a^4\}$. Thus $\Gamma $ is a Cayley graph if and only if it has Cayleyness 1. For this, pick $r \in R_i$, and note that as $R_i \subset K$ and $K \cong \mathcal {W}_{v_i}(\pi _1(C,v_i))$ by (2), there is a representative t of an element of $\pi _1(C,v_i)$ such that $\mathcal {W}_{v_i}(t) = r$. 71(3), 567591 (1983), Gardiner, A.: and Praeger, C.E. \end{aligned}$$, $r {:}{=} a_1 a_2 \ldots a_k \in \mathcal {R}$, $\chi (r) {:}{=} \chi (a_1,a_2)\chi (a_2,a_3) \ldots \chi (a_{k-1},a_k) \chi (a_k,a_1)$, $\mathcal {R}'{:}{=} \{\mathcal {R}_a, a\in \mathcal {S}\}$, $P {:}{=} \langle \mathcal {S}\vert \mathcal {U}\vert \mathcal {I}\vert \phi \vert \mathcal {R}' \rangle $, $$\begin{aligned}&V(L(\Gamma )) = \{ [(g,gs)] \vert g \in G, \ s \in \mathcal {S}\} \text{ and } \overrightarrow{E}(L(\Gamma ))\\&= \{ (g,s_1,s_2) \vert g \in G, \ s_1,s_2 \in \mathcal {S}\cup \mathcal {S}^{-1}, \ s_1 \not = s_2^{-1} \} \end{aligned}$$, $r \in \langle \langle \mathcal {R}\rangle \rangle _{MF_P}$, $\Phi : L(\Gamma ) \rightarrow \Gamma $, $p = \prod _{i=0}^{n-1} (g^i,s^i_1,s^i_2)$, $p= \prod _{j = 0}^{m-1} \left( \prod _{i \in I_j} (g^i,s^i_1,s^i_2) \right) $, $s_j \in \mathcal {S}\cup \mathcal {S}^{-1}$, $$\begin{aligned} p' {:}{=} \prod _{j = 0}^{m-1} \left( \left( \prod _{i \in I_j} (g^i,s^i_1,s^i_2) \right) (g_{j+1}, s_j^{-1}, s_{j-1}^{-1}) (g_{j-1}, s_{j-1}, s_{j}) \right) . To summarise, we can represent any bi-Cayley graph $\Gamma $ over G as $\text{ BiCay }(G,R,L,S)$ where $R, L, S \subset G$ with $R = R^{-1}$ and $L = L^{-1}$. Not every vertex-transitive graph is a Cayley graph. For a presentation $P = \langle \mathcal {S}\vert \mathcal {R}\rangle $ of a group one often alternatively defines the Cayley graph in the following more topological way. $\square $. Agelos Georgakopoulos. We thank Matthias Hamann for valuable discussions, and Derek Holt and Paul Martin for several corrections to an earlier draft of this work. Therefore, $\mathcal {C}(P)$ coincides with its own universal cover $\widehat{\mathcal {C}(P)}$. 2. In July 2022, did China have more nuclear weapons than Domino's Pizza locations? \end{aligned}$$, $$\begin{aligned} R_0&\text{ to } \text{ be } \text{ the } \text{ normal } \text{ closure } \text{ of } \mathcal {R}_0 \cup \{ s r s^{-1} : r \in \mathcal {R}_1, \ s \in \mathcal {S}_2\} \text{ in } K, \text{ and } \\ R_1&\text{ to } \text{ be } \text{ the } \text{ normal } \text{ closure } \text{ of } \mathcal {R}_1 \cup \{ s r s^{-1} : r \in \mathcal {R}_0, \ s \in \mathcal {S}_2\} \text{ in } K. \end{aligned}$$, $\{\tilde{V_1}, \tilde{V_2}\} {:}{=} \{K, \mathcal {S}_2K \}$, $\text{ PCay }\langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle = \text{ PCay }(P) {=}{:} \Gamma $, $\overrightarrow{E}(\Gamma ) = \overrightarrow{E}(T_P)/\sim $, $c: \overrightarrow{E}(T_P) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$, $c': \overrightarrow{E}(\Gamma ) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$, $\text{ PCay }(\langle \mathcal {S}_1 = \{a\}, \mathcal {U}= \emptyset , \mathcal {I}= \{b\} \vert \mathcal {R}_0 = \{a^5, aba^2b, b^2\}, \mathcal {R}_1 = \{a^5\}\rangle ) = \text{ PCay }\langle \{a\}, \emptyset , \{b\} \vert \{a^5, aba^2b\}, \{a^5\} \rangle $, $MF_P = \langle a, b \vert b^2 \rangle $, $K = \langle \langle a, bab \rangle \rangle \le MF_P$, $R_0 = \langle \langle a^5, aba^2b, ba^5b \rangle \rangle _{K}$, $R_1 = \langle \langle ba^5b, baba^2, a^5 \rangle \rangle _{K}$, $$\begin{aligned} G_0 =&\langle a, bab \vert a^5, a(bab)^2, (bab)^5 \rangle \\ =&\langle bab \vert (bab)^{-10}, (bab)^5 \rangle&\text{ as } a = (bab)^{-2}\\ =&{\mathbb Z}/5{\mathbb Z}= \langle bab \rangle \end{aligned}$$, $$\begin{aligned} G_1 =&\langle a, bab \vert (bab)^5, (bab)a^2, a^5 \rangle \\ =&\langle a \vert a^{-10}, a^5 \rangle&\text{ as } (bab) = a^{-2}\\ =&{\mathbb Z}/5{\mathbb Z}= \langle a \rangle . Recall that a Cayley graph can be naturally edge-coloured using the set of generators as colours. We recall that an action on a graph $\Gamma $ is semi-regular (or free) if $g \cdot x = h \cdot x$ implies $g = h$ for every $g,h \in G$ and $x\in V(\Gamma )$. 7. To see there is no multicycle colouring of L(P(5,2)), note that it has $\vert V(L(P(5,2))) \vert = \vert E(P(5,2)) \vert = 15$ vertices, so any multicycle will have to consist of triangles, pentagons, or 15-cycles. A multicycle colouring of a graph is a colouring c:E() such that the graph with vertex set V() and edge set c-1(x) is a multicycle for each x. To show this, pick any $r \in R_i$. Easily, $c'$ is a Cayley-like colouring. Then any Cayley-like colouring $c: \overrightarrow{E}(\Gamma ) \rightarrow X$ defines a map $\mathcal {W}_v: \mathcal {P}_v(\Gamma ) \rightarrow MF_X$ by $p = v e_1 v_1 \ldots e_n v_n \mapsto c(e_1) c(e_2) \ldots c(e_n)$. Praeger in (J London Math Soc 47(2):227-239, 1993) showed that a quasiprimitive action of a group G on a nonbipartite finite 2-arc transitive graph must be one of four of the eight O'Nan-Scott types. Proc. The forward direction is true: if has a multicycle colouring then it has a uniform partite presentation given in the proof of Theorem 5.7. We work with the notion of graph as defined by Gersten [9]. This information is encoded as a permutation (s) of the set of vertex classes. We define the following two subgroups of $Aut(\Gamma ) $: We remark that for any 2-partite presentation P, there is a subgroup of $Aut_c(\text{ PCay }(P))$ witnessing that $\text{ PCay }(P)$ is a bi-Cayley graph: For every 2-partite presentation $P = \langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle $ the vertex group $G_i$ is a subgroup of $\text {Aut}_c(\text{ PCay }(P))$. Our next proposition gives a sufficient condition for $\text{ PCay }(P)$ to be vertex-transitive in terms of the symmetry of $\mathcal {C}(P)$. Watkins, M.E. Details as to why this presentation is correct can be found in the second authors PhD thesis [26]. Math. Sci. Let $G$ be a group. Sci., pages 243256. Then the path p corresponds to a word Wx(p)MFP such that (Wx(p))(x)=y. Since its vertex degrees are odd, one of the colours in any weak multicycle colouring must induce a perfect matching. For a covering map $\phi $ we know that $\phi _{*}$ is injective [13, Proposition 1.31]. When it comes to vertex-transitive graphs the analogous question is still open and has been extensively studied, see [10, 22, 24] and references therein. To obtain M, note that if is finite, then |V(G)| is even since |E()|=n|V(G)|/2. For instance, the smallest transitive subgroup of automorphisms of the 10-node Petersen graph, shown in figure 4, has 20 elements. Phys. We know that $\text{ PCay }(P)$ is not always vertex-transitive, see e.g. This means that c is a partition-friendly weak multicycle colouring of as claimed. Each of the two orbits $O_i{:}{=} \{(g)_i : g \in G\}$ forms a (possibly disconnected) Cayley graph of $G$ with respect to the generating sets $R = R^{-1} = \{g \in G\vert e_0 (g)_0 \in E(\Gamma )\}$ and $L = L^{-1} = \{g \in G\vert e_1 (g)_1 \in E(\Gamma )\}$, respectively. When $\mathcal {I}=\emptyset $ we have a generalisation of the standard Cayley graph. 2.1 Examples Complete graph, K n In a complete graph K n of size n every node is symmetric to each other. Gordon and Breach (1970), Lauri, J., Scapellato, R.: Orbital Graphs and Strongly Regular Graphs. We want to refine c into a colouring $c'$ of the directed edges of $\Gamma $. Note that for any kK the path W(1G)0-1(k) connects (1G)0 to (g)0 for some gG because it uses an even number of edges e with c(e)S. In this section, we show that every line graph of a Cayley graph can be represented as a partite Cayley graph. Letting $\epsilon : \text{ PCay }(P) \rightarrow C(P)$ be the covering map, we can lift c to the edge-colouring ${\tilde{c}} =c \circ \epsilon $ of $\text{ PCay }(P)$. Letting :PCay(P)C(P) be the covering map, we can lift c to the edge-colouring c~=c of PCay(P). Recalling that each such relator was admitted as a relator (of the first kind) in $\mathcal {R}'$, we conclude that the word labelling p can be written as products of conjugates of words in $\mathcal {R}'$. Therefore, edges labelled $\mathcal {S}_2$ in $\Gamma $ connect vertices in ${\tilde{V}}_i/\sim \ = V_i$ to vertices in ${\tilde{V}}_{i+1}/ \sim \ = V_{i+1}$. [5]. Every Cayley graph is vertex-transitive but the converse is not true, with the Petersen graph being a well-known example. This colouring allows us to identify T with $T_P$. Sci., pages 243256. So for vV0 we have ^(v)=(v)=(v0)=v1 giving that ^(v)V1. If $n = 2$ we say that $\Gamma $ is bi-Cayley. Die Theorie der regulren Graphen. Since each vV() appears infinitely often as vi, we deduce that M0 is a spanning union of vertex-disjoint double-rays as desired. Moreover, as elements of $\text {Aut}({\widehat{\eta }}) \cong G_i$ preserve the cover, they preserve the colouring $c: \overrightarrow{E}(\Gamma ) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$ obtained by lifting our colouring of $\mathcal {C}(P)$ via ${\widehat{\eta }}$, and so we have realised $G_i$ as a subgroup of $\text {Aut}_c(\Gamma )$. Next, we want to associate each colour with a permutation Symn of the vertices of Kn. To define the desired partite presentation P, we start with, $\mathcal {U}= \{\omega \in \Omega \vert c^{-1}(\omega )$ is of degree 2$\}$, and. transitive, edge-transitive, or arc-transitive, if it is Aut X-vertex-transitive, AutX-edge-transitive, or Aut X-arc-transitive, respectively. Recall that Gx:=Wx,x/Rx is the right quotient of Wx,x by Rx, where Wx,z is the set of paths in C from x to z up to homotopy (in particular, Wx,x=1(C,x)), and, with Wz-1 the map from words in MFP to paths in C defined in Sect. so that the edge (g,s1,s2) connects [(g,gs1)] and [(gs1,gs1s2)]. Ways to find a safe route on flooded roads, Table generation error: ! A lot of research focuses on understanding how much larger the class of vertex-transitive graphs is or, what is essentially the same, on extending results from Cayley graphs to vertex-transitive graphs, see e.g. Fixing any base vertex class $b \in X$ leads to a partition of $V(T_P) = MF_P$, namely ${\tilde{V}}_{x} = W_{b,x}$. Springer-Verlag, Berlin (2005), Diestel, R., Leader, I.: A conjecture concerning a limit of non-Cayley graphs. Every connected vertex-transitive graph on an even number of vertices has a perfect matching, and each vertex in a connected vertex-transitive graph on an odd number of vertices is missed by a matching that covers all remaining vertices (Godsil and Royle 2001, p. 43; i.e., it has a near-perfect matching ). Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Note that we have made $\mathcal {S}$ a subset of the group $MF_P$, and so each $s\in \mathcal {S}$ has an inverse $s^{-1}$ in $MF_P$. First label. 1Leemann [16] states this for graphs of odd degree only, but the proof applies as is to the even degree case. The first examples of ^-transitive graphs were found by I. Therefore every K n is1-transitive graph. If n is even, then we can apply Theorem 5.11 recursively to decompose $E(\Gamma )$ into 2-regular spanning subgraphs, and attributing a distinct colour to the edges of each of those subgraphs yields a partition-friendly weak multicycle colouring. $\square $. The aim of this section is to show that every vertex-transitive graph has a partition-friendly weak multicycle colouring; hence it admits a partite presentation by Theorem5.7. Therefore, the action of $MF_P$ on $X = V(\Gamma )$ is transitive as required by (a) of Definition5.1. Note that there is a well-defined inverse $\mathcal {W}_v^{-1} : MF_X \rightarrow \mathcal {P}_v(\Gamma )$ as at every vertex $v' \in V(\Gamma )$ there is a unique edge $e \in \overrightarrow{E}(\Gamma )$ with colour c(e) and $\tau (e^{-1}) = v'$. Since its vertex degrees are odd, one of the colours in any weak multicycle colouring must induce a perfect matching. Soc. We remark that this sufficient condition is not necessary for $\text{ PCay }(P)$ to be vertex-transitive. A Math. In light of Leightons aforementioned conjecture, one can ask the following: Let be a vertex-transitive graph. But the backward direction is false, as shown by the following example. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erds on his 60th birthday), Vol. Does a locally finite vertex-transitive graph have finite Cayleyness if and only if is quasi-isometric to a Cayley graph? Soc. graph and its eigenvalues has also been investigated extensively, for example in [4, 26, 27]. If C is arc-connected, and $\pi _1(C,c) = 1$, i.e. This suggests. 3). Note that this is a generalisation of the modified Cayley graph. Springer, Berlin (2001), Book As $c_C$ is a Cayley-like colouring of C, we can consider $\mathcal {W}_{v_i}(p) \in MF_P$ by Definition3.8 and the discussion thereafter. I, volume 10, pages 91108.Vol. . For every 2-partite presentation $P = \langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle $, the graph $\Gamma {:}{=} \text{ PCay }(P)$ is regular, with vertex degree $\vert \mathcal {S}\cup \mathcal {S}^{-1} \vert = 2\vert \mathcal {S}_1 \vert + 2|\mathcal {U}| + |\mathcal {I}|$. Therefore, the action of MFP on X=V() is transitive as required by (a) of Definition5.1. Note that as $\Gamma $ is connected, for any two $x,y \in V(\Gamma )$ there is a path p connecting x and y. Groups Geom. That is true for Folkman graph - as the argument above shows, but it's not generally true for bipartite graphs. In particular, this leaves open thefollowing famous question. This implies that the word $s_0 \ldots s_{m-1}$ labelling this walk is a relation of $\Gamma $, and so it can be written as a product of conjugates of relators $\mathcal {R}$. In the latter case, brrbr is an example of a word that gives rise to a cycle. Georgakopoulos, A., Wendland, A. Moreover, for each I, let O={eE()|[e]c-1()}, and let () be the involution of V() exchanging the end-vertices of each edge in c-1(). We use f to define the colouring c:E()S1S1-1S as follows: Note that this colouring is Cayley-like, as there is a unique edge of each colour incident with each vertex. Moreover, as $\mathcal {S}_1 \subset K$ and $G_i = K/R_i$, we can think of $\mathcal {S}_1$ as a subset of $G_i$ in the following proposition: For every 2-partite presentation $P = \langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle $, the subgraph of $\Gamma {:}{=} \text{ PCay }(P)$ with edges coloured by $\mathcal {S}_1 \cup \mathcal {S}_1^{-1}$ is isomorphic to the disjoint union of $\text{ Cay }(G_0, \mathcal {S}_1)$ and $\text{ Cay }(G_1, \mathcal {S}_1)$. We call a graph ^-transitive, if it is vertex-transitive, edge-transitive, but not arc-transitive. The fewer types of vertices we have the closer our graph is to being a Cayley graph. For each such that c-1() is 1-regular, we let be the permutation that exchanges the two end-vertices of each edge in c-1(). We can write p=i=0n-1(gi,s1i,s2i). For this we will use 1- and 2-factorisations of the complete graphs as a tool. As a corollary of the above proof, we deduce that the covers $\nu ,\epsilon $ are equal, and so. because ${\tilde{V}}_0 = K$ and ${\tilde{V}}_1 = \mathcal {S}_2 K$. Let $\Gamma $ be a graph with a colouring $c: \overrightarrow{E}(\Gamma ) \rightarrow X$. Therefore, $\Gamma $ has a perfect matching by Theorem5.8 as no matching can miss exactly 1 vertex in this case. Given two CW complexes Ci for iZ/2Z, recall that a simplicial map As we defined C(P) by glueing in a 2-cell along each closed walk dictated by an element of Rv,vV(C(P)), where we have chosen Rv=Wv(1(,v)), we have forced 1(C(P),v) to be trivial. We observe that the Diestel-Leader graph DL(m,n) for $m\ne n$ has infinite Cayleyness. Note that $\mathcal {W}^{-1}_{v}(w_jr_jw_j^{-1})$ is a loop of $\Delta $ as $\mathcal {W}^{-1}(r_j)$ is contractable in ${\widehat{\Delta }}$, and so it represents some element of $\pi _1(\Delta ,v)$. Given a set of undirected edges $S \subset E(\Gamma )$ of a graph $\Gamma $ an orientation of S is a subsets $O_S \subset \overrightarrow{E}(\Gamma )$ such that $O_S/^{-1}= S$, and $O_S \cap O_S^{-1} = \emptyset $. Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK, You can also search for this author in Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. If the presentation complex $\mathcal {C}(P)$ is vertex-transitive, then so is $\text{ PCay }(P)$. If there exists a simplicial map :CC such that (v0)=v1, then PCay(P) is vertex-transitive. Moreover, different choices of O define homeomorphic topological spaces. For Folkman graph an easy proof would be to see that if you look at the 4 immediate neighbours of a vertex v, in half the cases all of them are connected to another vertex distinct from v, and in half the cases this is not so. 10. PhD Thesis. Proc. $\Gamma $ is $\vert X \vert $-regular, for all $e, e' \in \overrightarrow{E}(\Gamma )$, if $c(e) = c(e')$ and $\tau (e) = \tau (e')$ then $e = e'$, and. Then we obtain R={a,a4}, L={a2,a3} and S={a0}. Algebraic Graph Theory. 52, 720 (2008), MathSciNet We remark that it is not so easy to obtain the modified Cayley graphs using this construction because $\text{ Ro}_{\mathcal {S}}$ has even degree, so any cover will also have even degree. As before, we have a natural colouring $c: \overrightarrow{E}(T_P) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$ defined by $c(w, ws) = s$, and as $\sim $ preserves c, the latter factors into $c': \overrightarrow{E}(\Gamma ) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$, i.e. For each $x\in X$ and each $r \in \mathcal {R}_x$, we introducing a 2-cell and glue its boundary along the walk of C(P) starting at x and dictated by r (as in Definition3.9). Can every vertex-transitive graph on at least 3 vertices be represented as a partite Cayley graph so that each vertex class contains at least two vertices? We have $W_{x,x} = \pi _1(C(P),x)$ and $\pi _1(\mathcal {C}(P),x) = R_x \backslash W_{x,x} {=}{:} G_x$, analogously to the 2-partite presentation case. We endow $T_{\mathcal {S}}$ with a colouring $c: \overrightarrow{E}(T_{\mathcal {S}}) \rightarrow \mathcal {S}\cup \mathcal {S}^{-1}$ defined by $c(w,ws) = s$ and $c(ws,w) = s^{-1}$. It therefore suffices to prove that every vertex in $\Gamma $ has a unique outgoing edge coloured s for every $s \in \mathcal {S}_2 \cup \mathcal {S}_2^{-1}$. : The groups of the generalized Petersen graphs. Consider the inclusion map i:CC from the presentation graph C:=C(P) to the presentation complex C:=C(P) of P. It is well-known [13, Proposition 1.26] that the inclusion of the one skeleton into a 2-complex induces a surjection on the level of fundamental groups, and the kernel is exactly the normal closure of the words bounding the 2-cells. 1 2 PRIMOZ POTO CNIK AND PABLO SPIGA Theorem 1.1. Recall that for a covering map $\eta : X \rightarrow Y$, the group of automorphisms $f: X \rightarrow X$ such that $\eta \circ f = \eta $ is called the deck group of $\eta $ and is denoted by $\text {Aut}(\eta )$. Thus every Cayley graph has a multicycle colouring, namely its natural colouring by the generators. Indeed, we can think of $\bigcup _{x,y \in X} W_{x,y}$ as the ground set, and define the groupoid operation $W_{x,y} \times W_{y,z} \rightarrow W_{x,z}$ by concatenation. Consider again the Petersen graph =P(5,2) as in Example3.3 (Fig. For this, we will use the following result of Godsil and Royle [11, Theorem 3.5.1]: (Godsil & Royle [11, Theorem 3.5.1]) Let be a connected, finite, vertex-transitive graph. We propose that this holds for all vertex-transitive graphs with a unique marked vertex. A graph $\Gamma $ is a weak multicycle, if it is a vertex-disjoint union of cycles, double-rays, and edges. 3, we can alternatively define PCay(P) as a graph quotient, following the lines of Definition3.2, as follows: As in Corollary3.6, it is not hard to see that TP is the universal cover of PCay(P). This motivates. For instance, consider the following examples: the generalized Petersen graphs $P_{2,7},\;P_{3,8}$ and the Folkman graph. 173, 3rd edn. For any $v \in V_{i}$ we have $\epsilon (v) = v_i$ by (4). The best answers are voted up and rise to the top, Not the answer you're looking for? A vertex-transitive graph is symmetric if and only if each vertex-stabilizer Gv acts transitively on the set of vertices adjacent to v. For example, there are just two distinct 3-regular graphs with 6 vertices; one is K3,3 and the other is the ladder L3. In other words, c-1(x) is either a disjoint union of cycles or a perfect matching for all x. Cambridge Philos. 7 to argue that there are partite Cayley graphs that cannot be represented by a partite presentation with finite X. The following result will be used later to show that every vertex-transitive graph admits a partite presentation. In Cycles and rays (Montreal, PQ, 1987), volume 301 of NATO Adv. Recall that for a covering map :XY, the group of automorphisms f:XX such that f= is called the deck group of and is denoted by Aut(). The necessity of distinguishing $\mathcal {S}_2$ into $\mathcal {U},\mathcal {I}$ is to allow for some involutions, namely the elements of $\mathcal {I}$, to give rise to single edges in our graphs, just like in the above definition of modified Cayley graph. The labelling of the Petersen graph used in Example4.1. Thus $i_{*} : \pi _1(C,v_i) \rightarrow \pi _1(\mathcal {C},v_i)$ is a surjection. 1 we have directed and labelled the Petersen graph with two letters r and b that make it look almost like a Cayley graph. How can I shave a sheet of plywood into a wedge shim? Izv. In the finite case Leighton called this a multicycle. Graph Theory 40 (2020), 533-557] to determine the pairs (n,d) for which a vertex-transitive nut graph of order n and degree d exists . I think doing it repeatedly in random places would yield fairly non-symmetric things. Moreover, for each $\omega \in \mathcal {I}$, let $O_{\omega } = \{e \in \overrightarrow{E}(\Gamma ) \vert [e] \in c^{-1}(\omega )\}$, and let $\phi (\omega )$ be the involution of $V(\Gamma )$ exchanging the end-vertices of each edge in $c^{-1}(\omega )$. Csoka E, Lippner G. Invariant random perfect matchings in Cayley graphs. A map of graphs $\phi : \Gamma \rightarrow \Delta $ is a pair of maps $(\phi _V: V(\Gamma ) \rightarrow V(\Delta ), \phi _E: \overrightarrow{E}(\Gamma ) \rightarrow \overrightarrow{E}(\Delta ))$ where $\phi _E$ commutes with $^{-1}$ and $\phi _V \circ \tau = \tau \circ \phi _E$. A Haar graph is a bi-Cayley graph of the form $\text{ BiCay }(G,\emptyset ,\emptyset ,S)$. Bethesda, MD 20894, Web Policies Note that if X is a singleton, then we recover the usual group presentations and Cayley graphs by the above definitions. Universit de Genve, (2016). The proof of Theorem5.9 in the locally finite case can be found in [3] or [16, Proposition 3.2.17]1 If is not locally finite, then it is easy to construct a perfect matching greedily. Let $\mathcal {I}{:}{=} S$, and set $\mathcal {R}_0 {:}{=} \mathcal {W}_{(1_{G})_0}(\pi _1(\Gamma ,(1_{G})_0))$. Akad. Define the corresponding modified free group $MF_P {:}{=} \langle \mathcal {S}\vert \{s^2 : s \in \mathcal {I}\} \rangle $. To do this, for each U we choose an orientation OE() of c-1()}E() (recall this means that (OO-1)/-1=c-1() and OO-1=). 4. For a path $p: [0,1] \rightarrow \Gamma $ with $p(0), p(1) \in V(\Gamma )$ and a lift ${\tilde{p}}: [0,1] \rightarrow \Delta $ of p by $\psi $, it is straightforward to check that. Both these graphs are vertex-transitive, and K3,3 is symmetric, but L3 is not because there . A graph is said to be a multicycle, if either every component of is a cycle of a fixed length, or every component of is a double-ray, or every component of is an edge. Still, it is possible to express L(P(5,2)) as a partite Cayley graph: PCaya(12)(3),b(1)(23)|{b5,a10,a2b},{a-2b4},{a5,b10,b2a}, The line graph L(P(5,2)) of the Petersen graph. We have a covering map :^C, where ^ is the universal cover of C with 1-skeleton . Our claim now easily follows, e.g. Allnodesymmetric (vertex-transitive) graphs are 1-transitive. We now make the vertex classes more explicit. Recall that a graph is defined using a directed edge set E(), but we can also consider the undirected edge set E()=E()/-1, so that an undirected edge is a pair {e,d} such that e-1=d and d-1=e. $\square $. Let $R {:}{=} \langle \langle \mathcal {R}\rangle \rangle $ be the normal closure of $\mathcal {R}$ in $F_{\mathcal {S}}$. So a complete graph would be vertex transitive, but also edge transitive? Is there a reason beyond protection from potential corruption to restrict a minister's ability to personally relieve and appoint civil servants? We say $\Gamma $ is semi-symmetric if it is edge-transitive and regular but not vertex-transitive. \end{aligned}$$, $$\begin{aligned} \frac{1}{A} d(x,z) - B \le d'(f(x),f(z) \le A d(x,z) + B \end{aligned}$$, $i_{*} : \pi _1(C,x) \rightarrow \pi _1(\mathcal {C},x)$, $\pi _1(\mathcal {C},x) = \pi _1(C,x)/R_x = W_{x,x}/R_x = G_x$, ${\widehat{\eta }} : {\widehat{\Gamma }} \rightarrow \mathcal {C}$, $O_{\omega } \cup O_{\omega }^{-1} = c^{-1}(\omega )$, $\Gamma =\text{ Cay }\langle \mathcal {S}\vert \mathcal {R}\rangle $, $\mathcal {U}= \{ e, e^{-1} \} \cup \{m_{i,j} \vert m \in M, i,j \in \{-1,1\} \text{ where } m^2 \not = 1\}$, $\mathcal {I}= \{m_{i,j} \vert m \in M, i,j \in \{-1,1\} \text{ where } m^2 = 1\}$, $\mathcal {S}' = \mathcal {U}\cup \mathcal {I}$, $$\begin{aligned} \phi : \begin{array}{c} m_i\\ e,e^{-1} \end{array} \mapsto \begin{array}{c} m\\ 1_\mathcal {S}\end{array}. A note on curvature and fundamental group. Since the Cayleyness of a vertex-transitive graph $\Gamma $ divides $\vert V(\Gamma ) \vert $, a potential approach to answering this question is to enquire if for every prime $p \in {\mathbb N}$, there is a vertex-transitive graph on $p^k$ vertices for some $k\in {\mathbb N}$ that is not a Cayley graph. The 2-partite Cayley graph $\text{ PCay }\langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle = \text{ PCay }(P) {=}{:} \Gamma $ is the quotient $T_P/\sim $. J. Res. As $G$ has exactly two orbits in $V(\Gamma )$, and it acts regularly on each of them, for any $x \in V(\Gamma )$ there exists a unique $i \in {\mathbb Z}/2{\mathbb Z}$ and $g \in G$ such that $g \cdot e_i = x$, so we define $x {=}{:} (g)_i$. Introduction. census of vertex-transitive graphs up to 31 vertices [22], or the census of small 4-valent half-arc-transitive graphs and arc-transitive digraphs of valency 2 [19]. Gardiner, A.: and Praeger, C.E. As $\mathcal {C}(P)$ is a connected 2-complex it is locally path connected, therefore $G_i$ acts freely on ${\widehat{\eta }}^{-1}(v_i) = V_i$ by the above remarks. Soc., (1975). . We colour it by c:E(C(P))SS-1 defined by c(x,(s)x):=s, and note that this is a Cayley-like colouring as in Definition3.8. Our main result says that our formalism of partite Cayley graphs is general enough to describe all vertex-transitive graph: Every countable, vertex-transitive, graph has a partite presentation. Thus satisfies (b) of Definition5.1 by construction (we will check (a) below). However, PCay(P), shown in Fig. Next, we introduce a notion of edge-colouring that will be useful to establish that certain maps of graphs are covers. 11(1), 211243 (2017), Article As 1(C,x)=Gx we have an action of Gx on ^ and its 1-skeleton PCay(P)=: by deck transformations. There are two essentially different types of such cubic graphs. Next, we prove that $\text{ PCay }(P)$ is isomorphic to $L(\Gamma )$. This definition of the Cayley graph is standard. Soc. We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. Next, we want to associate each colour $\omega \in \Omega $ with a permutation $\pi _\omega \in Sym_n$ of the vertices of $K_n$. (Lovasz [18, Problem 11]) Let $\Gamma $ be a finite cubic vertex-transitive graph. It is a straightforward consequence of the varcMilnor lemma [21] that if $P = \langle X \vert \mathcal {U}\vert \mathcal {I}\vert \phi \vert \mathcal {R}\rangle $ is a partite presentation with finite X, then $\Gamma {:}{=} \text{ PCay }(P)$ is quasi-isometric to (any Cayley graph of) $G_x$. Choose S1R such that S1S1-1= and yet S1S1-1=R. Therefore, P is a partite presentation for . We start by constructing the presentation complex $\mathcal {C}(P)$ as follows. This is perhaps best explained with an example: in Fig. }. It is a straightforward consequence of the varcMilnor lemma [21] that if P=X|U|I||R is a partite presentation with finite X, then :=PCay(P) is quasi-isometric to (any Cayley graph of) Gx. This property isnt enough to guarantee vertex transitivity of $\Gamma $, with a counter example given by P(4,2). As both $T_P$ and $T_P/\sim $ are $\vert \mathcal {S}\cup \mathcal {S}^{-1} \vert $-regular by Proposition3.5, and $\eta $ is locally injective, $\eta $ is a cover. The following definition is a direct generalisation of Definition3.1, although it is formulated a bit differently. For a vertex-transitive graph with partite presentation P does Autc-loc(PCayX|U|I||R) act vertex-transitively on =PCay(P) where c is the colouring coming from P? J. Algebraic Combin. This motivates. Recall that $G_x {:}{=} W_{x,x}/ R_x$ is the right quotient of $W_{x,x}$ by $R_x$, where $W_{x,z}$ is the set of paths in C from x to z up to homotopy (in particular, $W_{x,x} = \pi _1(C,x)$), and, with $\mathcal {W}^{-1}_z$ the map from words in $MF_P$ to paths in C defined in Sect. 7 to argue that there are partite Cayley graphs that cannot be represented by a partite presentation with finite X. - 173.208.223.114. One can show that the relations in $\mathcal {R}_x$ hold in $L(\Gamma )$ for all $x \in \mathcal {S}$. Thus the pair $(V(\Gamma ), E(\Gamma ))$ is a multigraph in the sense of [5]. As M0i-1 is finite, and every vertex has infinite degree, this is always possible. Indeed, we can think of x,yXWx,y as the ground set, and define the groupoid operation Wx,yWy,zWx,z by concatenation. We know that $\Delta $ and C are the 1-skeletons of ${\widehat{\Delta }}$ and $\mathcal {C}$, respectively, so we obtain the inclusion maps $i: \Delta \rightarrow {\widehat{\Delta }}$ and $i: C \rightarrow \mathcal {C}$. The vertex groups are still isomorphic due to the fact that $\pi _1$ does not depend on the choice of a base point: For every partite presentation $P = \langle X \vert \mathcal {U}\vert \mathcal {I}\vert \phi \vert \mathcal {R}\rangle $, and every $x, y \in X$, the vertex groups $G_{x}, G_{y}$ are isomorphic. We say that $\Gamma $ is k-regular if $d(v)=k$ for every $v\in V(\Gamma )$. As $\Gamma $ is the 1-skeleton of ${\widehat{\Gamma }}$ by Definition3.7, we can think of $\text {Aut}({\widehat{\eta }}) \cong G_i$ as a subgroup of $ \text {Aut}(\Gamma )$. Cambridge University Press, Cambridge (2002), MATH This example motivates our definition of a partite presentation, which prescribes a number of types of vertices, and a set of relators for each type. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. This suggests. Presentations for vertex-transitive graphs, $^{-1}: \overrightarrow{E}(\Gamma ) \rightarrow \overrightarrow{E}(\Gamma )$, $\tau : \overrightarrow{E}(\Gamma ) \rightarrow V(\Gamma )$, $E(\Gamma ) {:}{=} \overrightarrow{E}(\Gamma )/^{-1}$, $|\overrightarrow{E}(\Gamma )|= 2|E(\Gamma )|$, $(\phi _V: V(\Gamma ) \rightarrow V(\Delta ), \phi _E: \overrightarrow{E}(\Gamma ) \rightarrow \overrightarrow{E}(\Delta ))$, $\phi _V \circ \tau = \tau \circ \phi _E$, $O_S \subset \overrightarrow{E}(\Gamma )$, $\overrightarrow{E}(\Gamma )= \{ (g,(gs)) \mid g\in G, s\in S\}$, $c: \overrightarrow{E}(\Gamma ) \rightarrow X$, $\phi _{*}: \pi _1(Y,y) \rightarrow \pi _1(X,x)$, $\phi _{*}(\pi _1(Y,y)) \subset \psi _{*}(\pi _1(C,c))$, $\langle \mathcal {S}\vert \mathcal {R}\rangle $, $F_{\mathcal {S}} / \langle \langle \mathcal {R}\rangle \rangle = G$, $\langle \langle \mathcal {R}\rangle \rangle $, $\text{ Cay }\langle \mathcal {S}\vert \mathcal {R}\rangle $, $$\begin{aligned} V(T_{\mathcal {S}})&{:}{=} F_{\mathcal {S}}, \text{ and } \\ \overrightarrow{E}(T_\mathcal {S})&{:}{=} \{(w, ws) \vert w \in F_{\mathcal {S}}, s \in \mathcal {S}\cup \mathcal {S}^{-1}\}. 70, 211218 (1971), Gersten, S.M. (3), 71(3):524546, 1995. open archive 1. 2. To define the desired partite presentation P, we start with. From the above discussion we obtain a multicycle colouring MSymS of KS where each colour is identified with a permutation of S. The generating set of our partite presentation P comprises the formal symbols, U={e,e-1}{mi,j|mM,i,j{-1,1}wherem21} and, I={mi,j|mM,i,j{-1,1}wherem2=1}. Moreover, it is straightforward to check that as all the maps above respect the edge colourings, so does this isomorphisms of graphs. (Easily, this is an equivalence relation.) Proceedings of the Calgary International Conference on Combinatorial Structures and their Applications held at the University of Calgary, Calgary, Alberta, Canada, vol. Every regular connected bi-Cayley graph BiCay(G,R,L,S) where RR-1=LL-1= and |R|=|L| can be constructed as a 2-partite Cayley graph. If true, it would imply that all finite vertex-transitive cubic graphs have a uniform partite presentation. Now define a map $\nu : \Gamma \rightarrow C$ by letting $\nu (v) = v_i$ whenever $v \in V_i = \eta ({\tilde{V}}_i)$. This gives us a map ${\widehat{\Phi }}: T_P \rightarrow {\widehat{\Delta }}$ defined by ${\widehat{\Phi }}{:}{=} i \circ \Phi $. Easily, $\Phi (p)$ is homotopic to $\Phi (p')$. This will allow us to describe vertex-transitive graphs such as the Coxeter graph which cannot be expressed as a bi-Cayley graph. For this, let $\{e_i\}_{i\in {\mathbb {N}}}$ be an enumeration of the edges of $\Gamma $, and let $\{v_i\}_{i\in {\mathbb {N}}}$ be a sequence of vertices of $\Gamma $ in which each $v\in V(\Gamma )$ appears infinitely often. $\mathcal {W}_v$ is a group isomorphism from $\pi _1(\Gamma ,v)$ to a subgroup of $MF_X$. $\square $. Then $\text{ Cay }\langle \mathcal {S}\vert \mathcal {R}\rangle $ can be defined as the quotient $T_{\mathcal {S}}/\sim $. For this, let {ei}iN be an enumeration of the edges of , and let {vi}iN be a sequence of vertices of in which each vV() appears infinitely often. A weak multicycle colouring of a graph is a colouring c:E() such that the graph with vertex set V() and edge set c-1(x) is a weak multicycle for each x. Moreover, in $T_P$ edges labelled $\mathcal {S}_2$ connect vertices in ${\tilde{V}}_i$ to ${\tilde{V}}_{i+1}$. In other words, if c is a multicycle colouring on C(P). Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids ). For example, there is never such an automorphism for the partite presentations {a},{},{b}|{an,abakb},{an} of Theorem3.13 unless k=1. Now we want to group the edges of p by the stars of vertices of $\Gamma $ they lie in. 10. It is straightforward to check that this is a closed walk using (4). Indeed, let be the 3-regular graph in Fig. Moreover, $G_x$ acts regularly on $V_x$, and so $\text{ PCay }(P)$ is $\vert X \vert $-Cayley. Define the Cayleyness of a (vertex-transitive) graph $\Gamma $ as the minimum number of vertex classes in any partite presentation of $\Gamma $. Every Haar graph can be represented as a 2-partite Cayley graph, and every 2-partite Cayley graph $\text{ PCay }(\langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle )$ with $\mathcal {S}_1 = \mathcal {U}= \emptyset $ is a Haar graph. Moreover, $G_i$ acts regularly on $V_i$ (and on $V_{i+1}$) for $i \in {\mathbb Z}/2{\mathbb Z}$, and so $\text{ PCay }(P)$ is bi-Cayley over $G_0 \cong G_1$. A Haar graph is a bi-Cayley graph of the form BiCay(G,,,S). We greedily construct an M0 as above containing e0 as follows. Cambridge University Press, 2 edition, pp. 1, the vertices depicted as square correspond to $V_0= {\tilde{V}}_0/\sim $, and vertices depicted as circles correspond to $V_1= {\tilde{V}}_1/\sim $. If such an f exists, we say that (X,d) and (Y,d) are quasi-isometric to each other. We now give an alternative definition of $\Gamma = \text{ PCay }(P)$ following the standard topological approach of defining a Cayley graph. Thus $O_{\omega }$ defines a permutation $\phi (\omega )$ of $X = V(\Gamma )$, by letting $\phi (\omega )(x)$ be the unique $y\in X$ such that $(x,y)\in O_{\omega }$. Thus it is an isomorphism proving our claim. Set :=PCay(P). Frucht R, Graver JE, Watkins ME. Don't have to recite korbanot at mincha? 2). We remark that this sufficient condition is not necessary for PCay(P) to be vertex-transitive. Then construct an auxiliary graph $\Delta $ with, By definition, $\Delta $ is k-regular and bipartite, with bipartition $V^+ = \{v^+ \vert v \in V(\Gamma )\}$ and $V^- = \{v^- \vert v \in V(\Gamma )\}$. Thenis arc-transitive and one of the following holds: (i): |V| 70andis one of the six exceptions1,.,6, dened in Section1.2; To see there is no multicycle colouring of L(P(5,2)), note that it has |V(L(P(5,2)))|=|E(P(5,2))|=15 vertices, so any multicycle will have to consist of triangles, pentagons, or 15-cycles. We will show that $\Gamma _i$ is isomorphic to $\text{ Cay }(G_i, \mathcal {S}_1)$. The first kind is just obtained by rewriting the elements of $\mathcal {R}$ in terms of the new generators. Leighton [17] asked whether finite vertex-transitive graphs have similar colouring structures to Cayley graphs of groups. We call $G_x, x \in X$ the vertex groups. As $\pi _1(\mathcal {C},x) = G_x$ we have an action of $G_x$ on ${\widehat{\Gamma }}$ and its 1-skeleton $\text{ PCay }(P) {=}{:} \Gamma $ by deck transformations. Inst. To represent this as a bi-Cayley graph with above notation, we could choose $(a^0)_0 {:}{=} x_0$ and $(a^0)_1 {:}{=} y_0$. Math. M consists of either one cycle, (and is Hamiltonian), or of two disjoint cycles of the same length. Since each $v\in V(\Gamma )$ appears infinitely often as $v_i$, we deduce that $M_0$ is a spanning union of vertex-disjoint double-rays as desired. It would be interesting to generalise results about finitely presented groups such as [14] to finitely presented graphs in our sense. We are now ready to prove that our two definitions of $\text{ PCay }(P)$ coincide: For every 2-partite presentation $P = \langle \mathcal {S}_1, \mathcal {U}, \mathcal {I}\vert \mathcal {R}_0, \mathcal {R}_1 \rangle $, the 2-partite Cayley graphs $\Gamma = \text{ PCay }(P)$ and $\Delta = \text{ PCay}_T (P)$ are isomorphic. Every regular connected bi-Cayley graph $\text{ BiCay }(G,R,L,S)$ where $R \cap R^{-1} = L \cap L^{-1} = \emptyset $ and $\vert R \vert = \vert L \vert $ can be constructed as a 2-partite Cayley graph. Accessibility In the finite case Leighton called this a multicycle. Letbe a nite connected edge- and vertex-transitive4-valent graph admitting a non-identity automorphism xing more than1/3of the vertices. Lemma4.3 says that Gi acts transitively on Vj for jZ/2Z. As P(5,2) is cubic, there is no set of pentagons that visits every edge exactly once. $MF_P = \langle a, b \vert b^2 \rangle $, so that $T_P$ is the 3-regular tree; $K = \langle \langle a, bab \rangle \rangle \le MF_P$; $R_0 = \langle \langle a^5, aba^2b, ba^5b \rangle \rangle _{K}$, and. In Definition3.1 of a 2-partite presentation we did not explicitly talk about the two vertex classes, but they were implicit in that definition: we had two sets of relators $\mathcal {R}_0, \mathcal {R}_1$, and the definition of K implicitly distinguished our generators into those staying in the same vertex class, namely $\mathcal {S}_1$, from those swapping between the two vertex classes, namely $\mathcal {S}_2$. If in doing so we identify the identity $1_{MF_P} \in V(T_P)$ of $MF_P$ with some vertex in $\theta ^{-1}(v_0)$ (which we easily can) then (2) implies. Since c-1() is a multicycle, we can choose O so that each of its cycles is oriented, that is, for each vertex vV() there is exactly one eO with (e)=x. The two vertex classes Vi were defined a-posteriori, and Corollary3.6 confirms that the generators gave rise to edges of the partite Cayley graph behaving this way. - Dima Pasechnik Sep 8, 2012 at 4:11 oops, sorry, forgot that this actually is disproved by examples of "quadratic forms" graphs. $\square $. Historical motivation In 1969, Lovsz [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying together two seemingly unrelated concepts: traversability and symmetry of graphs. The following is an immediate consequence of the last two propositions. In other words, $c^{-1}(x)$ is either a disjoint union of cycles or a perfect matching for all x. Inclusion in an NLM database does not imply endorsement of, or agreement with, We greedily construct an $M_0$ as above containing $e_0$ as follows. Math. To do this, for each $\omega \in \mathcal {U}$ we choose an orientation $O_{\omega }\subset \overrightarrow{E}(\Gamma ) $ of $c^{-1}(\omega )\} \subset {E}(\Gamma )$ (recall this means that $(O_{\omega } \cup O_{\omega }^{-1})/\,^{-1} = c^{-1}(\omega )$ and $O_{\omega } \cap O_{\omega }^{-1} = \emptyset $). For a vertex-transitive graph $\Gamma $ with partite presentation P does $\text {Aut}_{c-loc}(\text{ PCay }\langle X \vert \mathcal {U}\vert \mathcal {I}\vert \phi \vert \mathcal {R}\rangle )$ act vertex-transitively on $\Gamma = \text{ PCay }(P)$ where c is the colouring coming from P? Set $\mathcal {S}' = \mathcal {U}\cup \mathcal {I}$, the generators of P. We need to associate a permutation $\phi (s)$ of the vertex classes with each $s\in \mathcal {S}'$, and we do so by, Let $\theta : \overrightarrow{E}(K_{\mathcal {S}}) \rightarrow M \cup M^{-1}$ be the colouring of $K_{\mathcal {S}}$ by $M \cup M^{-1}$. If n is even, then a special case of Baranyais theorem [2] gives us a 1-factorisation of Kn, i.e. Ser. We claim that $\Gamma $ coincides with the presentation graph C(P). a partition of the edges into perfect matchings. One can show that the relations in Rx hold in L() for all xS. Therefore, Wv0((1(PCay(P),v)))=Wv0((1(,(1G)0))), and so by Theorem 2.1 we have PCay(P). be locally (G;s)-distance transitive with automorphism group Gif, for each vertex x, the stabiliser G x is transitive on the set of vertices at distance ifrom x, for i= 1;:::;s. If the design Dis a ne, then the graph is locally ( G;4)-distance transitive if and only if Dis G-pairwise transitive, [8, Proposition 2.7(ii)]. This has vertex set X, and directed edge set {(x,(s)(x)| for all xX and sU(U)-1I} where (s-1)=(s)-1. : On subgroups and Schreier graphs of finitely generated groups. $\mathcal {R}_v = \mathcal {W}_v(\pi _1(\Gamma ,v)) \subset MF_P$. 10, 2630 (1935), Article This invites the following rather vague question. 3 Since we need to pay attention to the directions of the edges of $\Gamma $, each such pair s,t will give rise to four generators of P, indexed by the elements of $\{-1,1\}^2$. , Combining this with Theorem5.8 and Theorem5.9, we now obtain. Let ^ be the universal cover of C, with covering map ^:^C. As we will see in the following section, every vertex-transitive graph has a partition-friendly weak multicycle colouring. In passing, let us mention the following still open conjecture. Define the following function. Have you any examples that is vertex but not edge transitive. As $M_0^{i-1}$ is finite, and every vertex has infinite degree, this is always possible. Gustav Beier, Christian Garcia, Thasa Tamusiunas, Journal of Algebraic Combinatorics 207. Let P=X|U|I||R be a partite presentation. : A geometrical approach to imprimitive graphs. For this presentation we have. Now $\prod _{j = 0}^{m-1} (g_{j-1}, s_{j-1}, s_{j})$ is a closed walk in $L(\Gamma )$ no three consecutive edges of which are contained in the star of a vertex of $\Gamma $ because of the way we chose the $I_j$. : Intersections of finitely generated subgroups of free groups and resolutions of graphs. Proceedings of the Calgary International Conference on Combinatorial Structures and their Applications held at the University of Calgary, Calgary, Alberta, Canada, vol. A walk in $\Gamma $ is an alternating sequence $v_0 e_1 v_1 \ldots e_n v_n$ of vertices and directed edges such that $\tau (e_i) = v_i$ and $\tau (e_i^{-1}) = v_{i-1}$ for every $1\le i \le n$. Thus by restricting to the closed walks we can think of $\mathcal {W}_v$ as a map from $\pi _1(\Gamma ,v)$ to $MF_X$, and so the above remarks imply that. Having constructed $M_0$, we inductively construct the $M_i, i\ge 1$ so that $M_i$ contains $e_i$ unless $e_i$ is already in \(\bigcup _{j