Presentations for vertex-transitive graphs

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. The resulting notion of presentation allows us to represent every vertex-transitive graph.


Introduction
Every Cayley graph is vertex-transitive but the converse is not true, with the Petersen graph being a well-known example. 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. [4,7,17,19,20,25] and references therein. This paper offers a new algebraic way of defining graphs, which we will prove to have the power to present all vertex-transitive graphs.
The idea is to still define our graphs by means of generators and relators similarly to Cayley graphs defined via group presentations, but we now allow different vertices to obey different sets of relators. The fewer 'types' of vertices we have the closer our graph is to being a Cayley graph. This is perhaps best explained with an example: in Fig. 1 we have directed and labelled the Petersen graph with two letters r and b that make it look almost like a Cayley graph. But a closer look shows that if we start at any exterior vertex v and follow a sequence of edges labelled br brr then we return to v, while this is not true if v is one of the interior vertices. In the latter case, brr br is an example of a word that gives rise to a cycle.
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. Moreover, it entails Fig. 1 The Petersen graph, labelled by two letters r (for red) and b (for blue). The cycle obtained by reading the 'relation' rbrrb starting at the top square vertex is depicted in bold lines a set of generators, and for each generator s it prescribes the type of end-vertex of an edge labelled s for each type of starting vertex. The precise definition of partite presentations in the case where there are only two types of vertices, which we call 2partite presentations, is given in Sect. 3. The case with more classes is more involved, and it is given in Sect. 5.
We show how each partite presentation defines a graph, by imitating the standard definitions of a Cayley graph via a group presentation: either as a quotient of a free group by the normal subgroup generated by the relators (Definition 3.2), or as the 1-skeleton of the universal cover of the presentation complex (Definition 3.7). The resulting partite Cayley graph is always regular, with vertex-degree determined by the generating set, and it admits a group of automorphisms acting on its vertices semiregularly and with as many orbits as the number of types of vertices prescribed by its presentation (Proposition 5.14). In particular, 2-partite presentations always give rise to bi-Cayley graphs. We prove this, as well as a converse statement, in Sect. 4.
Our main result says that our formalism of partite Cayley graphs is general enough to describe all vertex-transitive graph: Theorem 1.1 Every countable, vertex-transitive, graph has a partite presentation.
In general, for the proof of this we allow for the vertex types to be in bijection with the vertex set of the graph in question. It would be interesting to study how much the number of vertex types can be reduced, see Sect. 7. As we remark there, there are vertextransitive graphs that require infinitely many vertex types in any partite presentation; most Diestel-Leader graphs [6] have this property. In the converse direction, we show, in Sect. 6, that every line graph of a Cayley graph admits a partite presentation with at most as many vertex types as the number of generators of .
The proof of Theorem 1.1 involves decomposing the edge-set into cycles. This decomposition is not obvious, and it is related to a conjecture of Leighton [17] disproved by Marušič [20]; see Sect. 5.2 for more.

Graphs and automorphisms
We work with the notion of graph as defined by Gersten [9]. A graph comprises a set of vertices V ( ), and a set of directed edges − → E ( ), endowed with a fix point free involution −1 : − → E ( ) → − → E ( ) and a terminus map τ : − → E ( ) → V ( ). Sometimes we will express the elements of − → E ( ) as directed pairs (v, w) with v, w ∈ V ( ), in which case we tacitly mean that τ ((v, w)) = w and (v, w) −1 = (w, v).
To a graph we can associate the set of undirected edges, E( ): Note that although we are talking about 'directed edges', we are not talking about 'directed graphs' in the sense of e.g. [5]. Our edges can be thought of as undirected pairs of vertices, but our formalism allows us to distinguish between two orientations for each of them. Moreover, our formalism allows for multiple edges between the same pair of vertices, and multiple loops at a single vertex. Thus the pair (V ( ), E( )) is a multigraph in the sense of [5].
A map of graphs φ : → is a pair of maps (φ V : For a graph , an endomorphism is a map from to itself, and it is called an automorphism if φ V and φ E are bijections. The sets of these maps are denoted End( ) and Aut( ), respectively.
We say that is vertex-transitive if Aut( ) acts transitively on V ( ), and edgetransitive if Aut( ) acts transitively on E( ). We say that is arc-transitive, or symmetric, if Aut( ) acts transitively on − → E ( ). We say is semi-symmetric if it is edge-transitive and regular but not vertex-transitive.
Given a set of undirected edges S ⊂ E( ) of a graph an orientation of S is a subsets O S ⊂ − → E ( ) such that O S / −1 = S, and O S ∩ O −1 S = ∅. A walk in is an alternating sequence v 0 e 1 v 1 . . . e n v n of vertices and directed edges such that τ (e i ) = v i and τ (e −1 i ) = v i−1 for every 1 ≤ i ≤ n.

Groups and Cayley graphs
Given a group G and a subset S ⊂ G, we define the (right) Cayley graph = Cay(G, S) to be the graph with vertex set V ( ) = G and directed edge set − → E ( ) = {(g, (gs)) | g ∈ G, s ∈ S}. Unless otherwise stated, we are not assuming that S generates G, so that Cayley graphs in this paper are not always connected. The group G acts on by automorphisms, by multiplication (on vertices) on the left.

Colourings
In this work a graph colouring will always refer to a colouring of the edges. A colouring of the undirected edges of is a map c : E( ) → X whereas a colouring of the directed edges is a map c : − → E ( ) → X , where X is an arbitrary set called the set of colours.

Covering spaces
A covering space (or cover) of a topological space X is a topological space C endowed with a continuous surjective map ψ : C → X such that for every x ∈ X , there exists an open neighbourhood U of x, such that ψ −1 (U ) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by ψ. Given a map of spaces φ : Y → X , and point y ∈ Y such that p(y) = x, we obtain an induced map between fundamental groups φ * : π 1 (Y , y) → π 1 (X , x) by composition. For a covering map φ we know that φ * is injective [13,Proposition 1.31]. If C is arc-connected, and π 1 (C, c) = 1, i.e. C is simply connected, we call C the universal cover of X , which is well-known to be unique when it exists.
Given a cover ψ : C → X and a map φ : Y → X (with Y path connected and locally path connected) we obtain a liftφ : Lastly we recall the classification of covering spaces: ) Let X be a path-connected, locally pathconnected, and semilocally simply-connected topological space. Then there is a bijection between the set of isomorphism classes of path-connected covering spaces ψ : C → X and the set of subgroups (up to conjugation) of π 1 (X ), obtained by associating the subgroup ψ * (π 1 (C)) to the covering space C.

Algebraic definition
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. Let G be a group. A presentation S|R of G consists of a generating set S ⊂ G and a relator set R ⊂ F S such that F S / R = G, where F S denotes the free group with free generating set S, and R denotes its normal subgroup generated by R. For a group presentation S|R , we can construct the Cayley graph Cay S|R in the following manner. Let T S be the 2|S|-regular tree defined by by demanding e ∼ d whenever c(e) = c(d) and τ (e) ∼ τ (d) and τ (e −1 ) ∼ τ (d −1 ). Then Cay S|R can be defined as the quotient T S / ∼. The corresponding covering map is denoted by η : T S → Cay S|R . Note that as ∼ preserves c, we obtain a unique colouring c : This definition of the Cayley graph is standard. All Cayley graphs defined this way have even degrees: involutions in S give rise to pairs of 'parallel' edges with the same end-vertices. However, in certain contexts it is desirable to replace such pairs of parallel edges by single edges. To accommodate for this modification-which is important for us later as we want to capture odd-degree graphs such as the Petersen graph with our presentations-we now introduce modified presentations and Cayley graphs.
For a group presentation S|R , we define the modified presentation P = U, I|R where I:={s ∈ S : s 2 ∈ R } and U:=S\I. Define the corresponding modified free group M F P := S|{s 2 : s ∈ I} . (Thus M F P is a free product of infinite cyclic groups, one for each s ∈ U, and cyclic groups of order 2, one for each s ∈ I.) Let φ : F S → M F P be the unique homomorphism extending the identity on F S , as provided by the universal property of free groups, and let R: We proceed as above to define the colouring c and the relation ∼, and obtain the modified Cayley graph as the quotient T P / ∼.
We now modify the above construction of the Cayley graph, to obtain our partite Cayley graphs. The basic idea is to partition the vertex set into two (and later more than two) classes V 1 , V 2 , obeying different sets of relators R 0 , R 1 . This bipartition creates the need to distinguish our generators too into two classes S 1 , S 2 , the former corresponding to edges staying in the same partition class, and the latter corresponding to edges incident with both classes V 1 , V 2 .
We will formally define a 2-partite presentation as a 4-tuple P = S 1 , S 2 |R 0 , R 1 , and explain how this data is used to define a partite Cayley graph PCay(P), in analogy with the above definition of a Cayley graph Cay S|R corresponding to a group presentation P = S|R . The set S 1 is an arbitrary set of 'generators'. We partition S 2 into two sets, S 2 = {U, I}, so that S 1 , U, I are pairwise disjoint. Their union S:=S 1 ∪ U ∪ I will be our set of generators. The necessity of distinguishing S 2 into U, I is to allow for some involutions, namely the elements of I, to give rise to single edges in our graphs, just like in the above definition of modified Cayley graph.
As in our definition of modified Cayley graph, we let M F P := S|{s 2 : s ∈ I} . Let | · | S 2 be the unique homomorphism from M F P to Z/2Z extending We have that K :=K er(| · | S 2 ) is an index-two subgroup of M F P , and so its cosets Definition 3.1 For any two sets R 0 , R 1 ⊂ K , called relator sets, we call the tuple S 1 , S 2 |R 0 , R 1 a 2-partite presentation.
(The restriction R i ⊂ K does not have an analogue in the definition of Cayley graph; the intuition is that relators should start and finish at the same side of the bipartition V 1 , V 2 because they are supposed to yield cycles in the graph.) Given a 2-partite presentation P = S 1 , U, I|R 0 , R 1 , recall that M F P = S|{s 2 : s ∈ I} , and define the (|S ∪ S −1 |-regular) tree T P by Define the subgroups R 0 to be the normal closure of R 0 ∪ {srs −1 : r ∈ R 1 , s ∈ S 2 } in K , and Here R i ≤ K ≤ M F P is the analogue of the normal subgroup R of M F P in the definition of Cay(P), but now having two versions corresponding to our two classes of elements of M F P , namely {Ṽ 1 ,Ṽ 2 }:={K , S 2 K }. In analogy with the relation ∼ above, we now write v ∼ w whenever v −1 w ∈ R i for v, w ∈Ṽ i . We extend ∼ to the edges of T P via e ∼ d if c(e) = c(d), τ (e) ∼ τ (d), and τ (e −1 ) ∼ τ (d −1 ). Note that this is a generalisation of the modified Cayley graph. When I = ∅ we have a generalisation of the standard Cayley graph.
Borrowing terminology from groupoids, we define the vertex groups of our partite presentation to be G i :=K /R i for i ∈ Z/2Z.
The condition R i ⊂ K implies that if v ∼ w then v and w belong to the same coset V 0 orṼ 1 of K in M F P by the definitions. Thus factoring by ∼ projects the bipartition As in the case of Cayley graphs, relators in the presentation yield closed walks in , but now we need to start reading our relators at the correct side of the bipartition for this to be true: for every i ∈ Z/2Z and each r ∈ R i and v ∈ V i , if we start at v and follow the directed edges of with colours dictated by r one-by-one, we finish our walk at v.
We now explain how the Petersen graph can be obtained as a 2-partite Cayley graph: • M F P = a, b|b 2 , so that T P is the 3-regular tree; (There are many alternative ways to present R 0 , R 2 and R 1 = R 2 , and the above is just an example. Details as to why this presentation is correct can be found in the second author's PhD thesis [26].) The vertex groups G i = K /R i are generated by any generating set of K , in particular by {a, bab}. They abide by the relations that generate R i so in the case of R 0 these are a 5 , aba 2 b = a(bab) 2 and ba 5 b = (bab) 5 (when we write them in terms of the generators of K ). So we have G 0 = a, bab|a 5 , a(bab) 2 , (bab) 5 = bab|(bab) −10 , (bab) 5 as a = (bab) −2 =Z/5Z = bab and similarly G 1 = a, bab|(bab) 5 , (bab)a 2 , a 5 = a|a −10 , a 5 as (bab) = a −2 =Z/5Z = a .
The fact that G 0 is isomorphic to G 1 is not a coincidence as we remark at the end of this section. In Fig. 1, the vertices depicted as square correspond to V 0 =Ṽ 0 / ∼, and vertices depicted as circles correspond to V 1 =Ṽ 1 / ∼. Note that we have made S a subset of the group M F P , and so each s ∈ S has an inverse s −1 in M F P . With these inverses in mind we define S −1 :={s −1 : s ∈ S}. Note that s = s −1 exactly when s ∈ I. Moreover, as S 1 ⊂ K and G i = K /R i , we can think of S 1 as a subset of G i in the following proposition: Proposition 3.4 For every 2-partite presentation P = S 1 , U, I|R 0 , R 1 , the subgraph of :=PCay(P) with edges coloured by S 1 ∪ S −1 1 is isomorphic to the disjoint union of Cay(G 0 , S 1 ) and Cay(G 1 , S 1 ).
Proof Let T i be the subgraph of T P induced by the vertices ofṼ i , and i be the subgraph of induced by V i =Ṽ i / ∼. We will show that i is isomorphic to Cay(G i , S 1 ).
To begin with, recall thatṼ 0 = K and G 0 = K /R 0 , and so V 0 is canonically identified with G 0 . Thus to show that 0 is isomorphic to Cay(G 0 , S 1 ), we need to check that (v, w) is a directed edge of 0 coloured s whenever w = vs. The latter holds whenever v s ∈ η −1 (w) for every v ∈ η −1 (v), which is exactly when (v , v s) is a directed edge of T P coloured s. This in turn is equivalent to (v, w) being a directed edge of 0 coloured s because c = c • η.
This proves that 0 is isomorphic to Cay(G 0 , S 1 ). To prove that 1 is isomorphic to Cay(G 1 , S 1 ) we repeat the same argument multiplying on the left with a fixed element of S 2 throughout. Since V ( ) is the disjoint union of V 0 and V 1 , our statement follows. Proof By Proposition 3.4, the subgraph with edges coloured by S 1 ∪ S −1 1 is 2|S 1 |regular. It therefore suffices to prove that every vertex in has a unique outgoing edge coloured s for every s ∈ S 2 ∪ S −1 2 . Existence is easy by the definition of T P . To prove uniqueness, suppose in T P we have two edges which means that u 0 ∼ u 1 and hence (v 0 , u 0 ) ∼ (v 1 , u 1 ) proving our uniqueness statement.

Corollary 3.6
For a 2-partite presentation P = PCay S 1 , U, I|R 0 , R 1 the universal cover of :=PCay(P) is T P . Moreover, every edge with a colour in S 1 connects two vertices in V i for some i ∈ Z/2Z, and every edge with a colour in S 2 connects a vertex in V i to a vertex in V i+1 .
Proof Recall that ∼ defines a map of graphs η : T P → PCay(P), by η(x) = [x]. As both T P and T P / ∼ are |S ∪S −1 |-regular by Proposition 3.5, and η is locally injective, η is a cover. As the fundamental group of a tree is trivial we deduce that η is in fact the universal cover. By Proposition 3.4, edges labelled S 1 connect vertices in G i to vertices in G i , which are exactly the vertices in V i . Moreover, in T P edges labelled S 2 connect vertices inṼ i toṼ i+1 . Therefore, edges labelled S 2 in connect vertices inṼ

Topological definition
We now give an alternative definition of = PCay(P) following the standard topological approach of defining a Cayley graph.
Let X be a set. Define the rose Ro X to be a graph with a single vertex v and edge set E(Ro X ) = X , where each x ∈ X = E(Ro X ) signifies a loop at v. To be more precise, we let X −1 denote an abstract set disjoint from X and in bijection (denoted −1 ) with X , and let X ∪ X −1 be the set of directed edges of Ro X . The terminus map τ of Ro X maps all edges to v. We colour this rose by c : For a presentation P = S|R of a group one often alternatively defines the Cayley graph in the following more topological way. We start by constructing the presentation complex C(P) as follows. The 1-skeleton of C(P) is Ro S with vertex v. For each relator r ∈ R, we introduce a 2-cell D r and identify its boundary with the closed walk of Ro S dictated by r (see Definition 3.9). This completes the definition of C(P). The Cayley graph Cay S|R is the 1-skeleton of the universal cover of C(P).
We now generalise this construction to the context of our 2-partite presentations. We remark that it is not so easy to obtain the modified Cayley graphs using this construction because Ro S has even degree, so any cover will also have even degree. But treating I appropriately we will in fact be able to obtain graphs of odd degree. Definition 3.7 Let P = S 1 , U, I|R 0 , R 1 be a 2-partite presentation. We construct the presentation complex C(P) of P as follows. Start with two copies of Ro S 1 , with vertices v 0 and v 1 , respectively, and connect v 0 and v 1 with an edge for each element of S 2 ∪ S −1 2 ⊂ M F P . We will refer to this 1-complex C(P) as the presentation graph of P. We can extend the colouring of the two copies of Ro S 1 to a colouring c : To define the 2-cells of C(P), for each relator r = s 1 s 2 . . . s n ∈ R i , we start a walk p r at v i and extend this walk inductively with the edge labelled s i , i = 1, . . . , n. The path p r starts and ends at v i as R i ⊂ K . Attach a 2-cell along each such closed walk p r to obtain the presentation complex C(P) from C(P). Finally, we define the (topological) 2-partite Cayley graph PCay T S 1 , U, I|R 0 , R 1 to be the 1-skeleton of the universal cover of C(P).
Our next result, Theorem 3.11, says that this gives rise to the same graph as in Definition 3.2. To prove it, we will use the theory of covering spaces (Sect. 2.4). For this we need to turn our graphs into topological spaces, and we now recall the standard way to do so.
Given a graph with vertex set V , and any orientation on its edges O ⊂ − → E ( ), we define a topological space as follows. Associate a point to each vertex, and a closed interval I e = [0, 1] to each edge e ∈ O. Then define the quotient I e (0) ∼ τ (e −1 ) and T e (1) ∼ τ (e) to obtain the topological space It is not hard to see that when is connected this topological space is path-connected, locally path-connected and semilocally simply-connected. Moreover, different choices of O define homeomorphic topological spaces.
Next, we introduce a notion of edge-colouring that will be useful to establish that certain maps of graphs are covers. Recall that a Cayley graph can be naturally edgecoloured using the set of generators as colours. The Cayley-like colourings we now define imitate, and extend, this colouring.

Suppose we have two graphs and with Cayley-like colourings
Then any surjective map of graphs φ : → which respects these colourings, that is, satisfies c = c • φ, is a covering map of the associated topological spaces. Indeed, φ can't map any two edges that share an end vertex to the same edge, as this cannot respect the colourings.
Let P v ( ) be the set of walks in starting at a vertex v, and define the group M F X by the presentation X |{x x -1 : x ∈ X } . Then any Cayley-like colouring c :

Definition 3.9
For any g ∈ M F X , we say that W −1 v (g) is the walk (in ) dictated by the word g starting at v. This is a natural definition since we can express g as a word s 1 . . . s n with s i ∈ X ∪ X −1 , and obtain W −1 v (g) by starting at v and following the directed edges with colours c(s 1 ) . . . c(s n ); this is well-defined when c is Cayley-like.
It is straightforward to check that if p is homotopic to p , then W v ( p) = W v ( p ). Thus by restricting to the closed walks we can think of W v as a map from π 1 ( , v) to M F X , and so the above remarks imply that Suppose we have a covering map of graphs ψ : → both of which have Cayleylike colourings c : where with a slight abuse, we interpreted p as a walk in in the obvious way. We are now ready to prove that our two definitions of PCay(P) coincide:  Proof Our presentation graph C = C(P) is |S ∪S −1 |-regular by definition. Therefore, the universal cover of C is the |S ∪ S −1 |-regular tree T , and we can let θ : T → C be the corresponding covering map. Let c C : − → E (C) → S ∪ S −1 be the colouring of C as above. This lifts to a colouring c T : ). This colouring allows us to identify T with T P .
Let p ∈ π 1 (C, v i ). As c C is a Cayley-like colouring of C, we can consider W v i ( p) ∈ M F P by Definition 3.8 and the discussion thereafter. Any closed walk representing p must use an even number of edges coloured Recall that we can identify T with T P . If in doing so we identify the identity Let η : T P → be the covering map found in Corollary 3.6. Let c : Let : → C be the universal cover of C:=C(P). We know that and C are the 1-skeletons of and C, respectively, so we obtain the inclusion maps i : → and i : C → C. Furthermore, by restricting to the 1-skeleton we obtain a covering map : → C. As θ : T P → C is the universal cover of C, it can be lifted through : → C to a map : T P → so that • = θ by the definition of a universal cover. This gives us a map : T P → defined by :=i • . Note that all these maps respect the colourings of the edges as θ and do.
By Theorem 2.1, to show ∼ = it suffices to show that ν * (π 1 ( )) = * (π 1 ( )), or equivalently W v i (ν * (π 1 ( ))) = W v i ( * (π 1 ( ))) as W v i is a bijection. To do so, we will prove that the latter groups are both equal to R i , where R i is as defined after Definition 3.1.
To show that W v i (ν * (π 1 ( ))) = R i , let p be a closed walk representing some We would like to use Proposition 3.10 to deduce W v i (ν * (π 1 ( ))) = R i , and for this it now only remains to prove that the former is surjective onto R i . To show this, pick any r ∈ R i . As (1), with ∼ as in the definition of as a quotient of T P . This means that η(q(1)) = η(q(0)) = v, and so η •q is a loop representing an element of Proposition 1.26] that the inclusion of the one skeleton into a 2-simplex induces a surjection on the level of fundamental groups, and the kernel is exactly the normal closure of the words bounding the 2-cells. Thus i * : Finally, we claim that is contractable in , and so it represents some element of π 1 ( , v). Applying this to each factor of our above expression r = n j=1 w j r j w −1 j implies that t represents some element of Moreover, it is straightforward to check that as all the maps above respect the edge colourings, so does this isomorphisms of graphs.
From now on we just use the notation PCay(P) for the 2-partite Cayley graph obtained in either Definition 3.2 or 3.7.
As a corollary of the above proof, we deduce that the covers ν, are equal, and so and similarly V i = η(Ṽ i ) = (Ṽ i ), so V i is well defined for either the topological or graph definition, as in the notation of Fig. 2. From now on we will only use to denote this covering map.
The following corollary gathers some further facts that we obtained in the proof of Theorem 3.11 for future reference. Corollary 3.12 Let P = S 1 , U, I|R 0 , R 1 be a 2-partite presentation with partite Cayley graph :=PCay(P). We have where : → C(P) is the cover in Definition 3.7, and i : C(P) → C(P) the inclusion.
The generalised Petersen graph is denoted by P(n, k) and defined as follows. Let The classical example is the Petersen graph, P(5, 2), the smallest non-Cayley vertextransitive graph. The following statement, proved in the second author's PhD thesis [26], says that we can obtain every P(n, k) as a 2-partite Cayley graph. Note that from the definition of R i we have R 0 = s R 1 s −1 for any s ∈ S 2 . Therefore, we deduce that G i :=R i \K ∼ = R i+1 \K , where an isomorphism φ s,i : G i → G i+1 is given by conjugation by any s ∈ S 2 . This property isn't enough to guarantee vertex transitivity of , with a counter example given by P (4,2). This invites the following rather vague question.

Relationships to Bi-Cayley and Haar graphs
We recall that an action on a graph is semi-regular (or free) if g · x = h · x implies g = h for every g, h ∈ G and x ∈ V ( ). A vertex-transitive graph is said to be n Cayley over G if G is a semi-regular subgroup of Aut( ) with n orbits of vertices. If n = 2 we say that is bi-Cayley. Suppose is bi-Cayley over G. Pick two vertices e 0 , e 1 ∈ V ( ) from different orbits of G. As G has exactly two orbits in V ( ), and it acts regularly on each of them, for any x ∈ V ( ) there exists a unique i ∈ Z/2Z and g ∈ G such that g · e i = x, so we define x=:(g) i . Each of the two orbits O i :={(g) i : g ∈ G} forms a (possibly disconnected) Cayley graph of G with respect to the generating sets To capture the set E 01 of edges of the form (g) 0 (h) 1 ∈ E( ), we introduce the set S = {g ∈ G|(e 0 , (g) 1 ) ∈ − → E ( )} and note that S uniquely determines E 01 as any e ∈ E 01 coincides with h · e 0 (g) 1 for some g ∈ S and h ∈ G.
To summarise, we can represent any bi-Cayley graph over G as BiCay(G, R, L, S) where R, L, S ⊂ G with R = R −1 and L = L −1 . Then the set of directed edges of =:BiCay(G, R, L, S) is This representation isn't unique: if we choose different vertices for e 0 , e 1 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|.

Example 4.1
Consider again the Petersen graph = P(5, 2) as in Example 3.3 (Fig. 3). This has a natural action of G:=Z/5Z =< a > where a j : To represent this as a bi-Cayley graph with above notation, we could choose Recall that we have endowed :=PCay S 1 , U, I|R 0 , R 1 with a colouring c : − → E ( ) → S ∪S −1 . We want to talk about automorphisms that preserve this colouring.
The following definition distinguishes between preserving these colours globally or locally.

Definition 4.2 Let be a graph with a colouring c :
− → E ( ) → X . We define the following two subgroups of Aut( ): We remark that for any 2-partite presentation P, there is a subgroup of Aut c (PCay(P)) witnessing that PCay(P) is a bi-Cayley graph:  (PCay(P)). Moreover, G i acts regularly on V i (and on V i+1 ) for i ∈ Z/2Z, and so PCay(P) is bi-Cayley over G 0 ∼ = G 1 .
Proof Recall that for a covering map η : X → Y , the group of automorphisms f : X → X such that η • f = η is called the deck group of η and is denoted by Aut(η). It is known that if η is a universal cover Aut(η) = π 1 (Y ), and if X is connected and locally path connected then Aut(η) acts freely on η −1 (y) for any y ∈ Y [13].
Let :=PCay(P) and let η : → C be the universal cover of the presentation complex C(P) of P. Thus Aut( η) ∼ = π 1 (C(P)) ∼ = G i by the above remark and Corollary 3.12 1. As is the 1-skeleton of by Definition 3.7, we can think of Aut( η) ∼ = G i as a subgroup of Aut( ). Moreover, as elements of Aut( η) ∼ = G i preserve the cover, they preserve the colouring c : − → E ( ) → S ∪ S −1 obtained by lifting our colouring of C(P) via η, and so we have realised G i as a subgroup of Aut c ( ). As C(P) is a connected 2-complex it is locally path connected, therefore G i acts freely on η −1 (v i ) = V i by the above remarks.
A Haar graph is a bi-Cayley graph of the form BiCay(G, ∅, ∅, S). The following is an immediate consequence of the last two propositions.

Corollary 4.5 Every Haar graph can be represented as a 2-partite Cayley graph, and every 2-partite Cayley graph PCay
Most of our motivation for introducing partite presentations came from studying vertex-transitive graphs. Our next proposition gives a sufficient condition for PCay(P) to be vertex-transitive in terms of the 'symmetry' of C(P). Given two CW complexes C i for i ∈ Z/2Z, recall that a simplicial map φ : C 0 → C 1 is a continuous map that maps each n-simplex to an n-simplex for every n. For a CW complex C, the group of bijective simplicial maps from C to itself is denoted by Aut(C).

Proposition 4.6
Let P be a 2-partite presentation. As above, the two vertices of the presentation complex C are denoted by v 0 and v 1 . If there exists a simplicial map

then PCay(P) is vertex-transitive.
Proof Set :=PCay(P). Lemma 4.3 says that G i acts transitively on V j for j ∈ Z/2Z. Thus it only remains to find an automorphism which maps a vertex in V 0 to a vertex in V 1 . We have a covering map : → C, where is the universal cover of C with 1-skeleton . By the lifting property φ • : → C lifts to an automorphism Thus when restricting φ to the 1-skeleton, , we obtain the required automorphism.
We remark that this sufficient condition is not necessary for PCay(P) to be vertex-transitive. For example, there is never such an automorphism for the partite presentations {a}, {}, {b}|{a n , aba k b}, {a n } of Theorem 3.13 unless k = ±1. However, we know that P(n, k) is vertex-transitive for many other choices of n and k (such as the case of the Petersen graph n = 5, k = 2), see [8].

Definition of partite presentations
In this section, we generalise our notion of partite presentation by allowing for more than two classes of vertices V i . This will allow us to describe vertex-transitive graphs such as the Coxeter graph which cannot be expressed as a bi-Cayley graph.
In Definition 3.1 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 R 0 , R 1 , and the definition of K implicitly distinguished our generators into those staying in the same vertex class, namely S 1 , from those swapping between the two vertex classes, namely S 2 . The two vertex classes V i were defined a-posteriori, and Corollary 3.6 confirms that the generators gave rise to edges of the partite Cayley graph behaving this way.
The following definition is a direct generalisation of Definition 3.1, although it is formulated a bit differently. We now make the vertex classes more explicit. The main complication arises from the fact that we have to specify, for each generator s, which vertex class any edge coloured by s will lead into if it starts at a given vertex class. This information is encoded as a permutation φ(s) of the set of vertex classes. As before, we distinguish our generators into two subsets U and I to allow for 'involutions' that make partite Cayley graphs with odd degrees possible.
We now give the formal definition: Definition 5.1 A partite presentation X |U|I|φ|R consists of the following data: 1. a set of vertex classes X ; 2. a generator set S, which is partitioned into two sets U and I; as before, we use S to define a group M F P := S|{s 2 |s ∈ I} (a free product of cyclic groups each of order 2 or ∞); 3. a map φ : S → Sym X from the generator set to the group Sym X of permutations of X ; We remark that any such map defines an action of M F P on X via s 1 . . . 4. a relator set R x ⊂ Stab(M F P , x) = for each x ∈ X , where Stab(M F P , x) denotes the stabiliser of x with respect to the aforementioned action of M F P . (This is a natural condition, as we want to return to our starting vertex when following a walk labelled by a relator, and in particular we want to return to the same vertex class.) The set {R x : x ∈ X } of these relator sets is denoted by R.
We now use such a presentation P = X |U|I|φ|R to define the partite Cayley graph PCay(P), in analogy with Definition 3.7. We start by defining the presentation graph C(P). This has vertex set X , and directed edge set by c(x, φ(s)x):=s, and note that this is a Cayley-like colouring as in Definition 3.8.
The partite presentation complex C(P) is the 2-complex obtained from C(P) as follows. For each x ∈ X and each r ∈ 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 Definition 3.9). It is straightforward to check that this is a closed walk using (4).
Note that C(P) is connected by condition (a) Finally, Definition 5. 2 We define the partite Cayley graph PCay(P) = PCay X |U|I|φ|R to be the 1-skeleton of the universal cover of C(P).
Letting : PCay(P) → C(P) be the covering map, we can lift c to the edgecolouringc = c • of PCay(P).
Note that if X is a singleton, then we recover the usual group presentations and Cayley graphs by the above definitions. Our 2-partite presentations S 1 , U , I |R 0 , R 1 of Sect. 3 are tantamount to partite presentations as in Definition 5.1 with X = {0, 1}, where φ(s 1 ) = (0)(1) for s 1 ∈ S 1 and φ(s 2 ) = (0, 1) for s 2 ∈ S 2 :=U ∪ I , with U = S 1 ∪ U and I = I . As in Sect. 3, we can alternatively define PCay(P) as a graph quotient, following the lines of Definition 3.2, as follows: 1. Let S:=U ∪ I and define the group M F P by the presentation S|{s 2 : s ∈ I} ; this is a free product of infinite cyclic groups, one for each s ∈ U, and cyclic groups of order 2, one for each s ∈ I. Define the tree T P by This is a (2|U| + |I|)-regular tree, and it comes with a colouring c : − → E (T P ) → S ∪ S −1 by c(w, ws) = s. 2. We can extend the map φ of (3) from S to an action of M F P by composition: we let x · s 1 . . . s n :=φ(s n ) • . . . • φ(s 1 )(x) for all x ∈ X and s i ∈ S. Let W x,y = {w ∈ M F P |φ(w)(x) = y} for x, y ∈ X . Fixing any 'base' vertex class b ∈ X leads to a partition of V (T P ) = M F P , namelyṼ x = W b,x . Note that two vertices in We define PCay(P) to be the corresponding quotient T P / ∼. As in Corollary 3.6, it is not hard to see that T P is the universal cover of PCay(P). Define V x , x ∈ X as the image ofṼ x under the quotient of ∼. We have W x,x = π 1 (C(P), x) and π 1 (C(P), x) = R x \W x,x =:G x , analogously to the 2-partite presentation case. We call G x , x ∈ X the vertex groups.
We remark that the vertex set of PCay(P) can be given the structure of a groupoid G PCay(P) . Indeed, we can think of x,y∈X W x,y as the ground set, and define the groupoid operation W x,y × W y,z → W x,z by concatenation. Another way to think of this groupoid is G PCay(P) ∼ = π 1 (C(P), X ), the universal groupoid of the presentation complex C(P), with paths starting and ending in V (C).
The main result of this section is that every vertex-transitive graph is isomorphic to PCay(P) for some partite presentation P. For the proof of this we will need to decompose the edges of into cycles. The next section discusses such decompositions.

Multicycle colourings
Leighton [17] asked whether finite vertex-transitive graphs have similar colouring structures to Cayley graphs of groups. For a Cayley graph = Cay(G, S), the generators canonically induce a colouring c : E( ) → S as above, so that c −1 (s) is a disjoint union of cycles of the same length for every s ∈ S. In the finite case Leighton called this a multicycle. A double-ray is a 2-way infinite path.

Definition 5.3
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. 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 ∈ .
Thus every Cayley graph has a multicycle colouring, namely its natural colouring by the generators. Leighton [17] conjectured that every finite vertex-transitive graph has a multicycle colouring [17], but this was shown to be false by Marušič [20], a counter-example being the line graph of the Petersen graph:  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. Any 15-cycle in L(P (5, 2)) would yield a Hamiltonian cycle in P(5, 2), which we know does not exist. Moreover, the only triangles in L(P (5,2)) are formed by edges incident with a single vertex of P(5, 2). As P(5, 2) is not bipartite, there is no way to Fig. 4 The line graph L(P (5, 2)) of the Petersen graph partition the triangles into disjoint sets that pass through all vertices. So we can only use sets of five cycles, which correspond to sets of edge disjoint pentagons in P(5, 2). As P(5, 2) is cubic, there is no set of pentagons that visits every edge exactly once.
Still, it is possible to express L(P(5, 2)) as a partite Cayley graph: Our aim now is to weaken the notion of a multicycle colouring enough that every vertex-transitive graph will admit one, so that the weakened notion will allow us to find partite presentations. This is the essence of Theorem 5.7.

Definition 5.5
A graph is a weak multicycle, if it is a vertex-disjoint union of cycles, double-rays, and edges. 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 ∈ .
We say that a weak multicycle colouring c is partition-friendly, if c −1 (x) is regular for all x ∈ . In other words, c −1 (x) is either a disjoint union of cycles or a perfect matching for all x.
As we will see in the following section, every vertex-transitive graph has a partitionfriendly weak multicycle colouring. The condition of vertex transitivity here cannot be relaxed to just regularity. Indeed, let be the 3-regular graph in Fig. 5. Since its vertex degrees are odd, one of the colours in any weak multicycle colouring must induce a perfect matching. But does not have a perfect matching M, because removing v and the vertex matched to v by M results in at least one component with an odd number of vertices.

Multicycle colourings and partite presentations
We say a partite presentation P = X |U|I|φ|R is uniform, if for every s ∈ S, all orbits of φ(s) have the same size. In other words, if c is a multicycle colouring on C(P). In light of Leighton's aforementioned conjecture, one can ask the following: 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. But the backward direction is false, as shown by the following example. Consider the 2-partite presentation P = {a}, {b}, ∅|{a}, {a 2 } . This is trivially uniform, like every 2-partite presentation. However, PCay(P), shown in Fig. 6, does not have a multicycle colouring.

Question 5.6 Let be a vertex-transitive graph. Does have a multi-cycle colouring if and only if it is the partite Cayley graph of a uniform partite presentation?
The following result will be used later to show that every vertex-transitive graph admits a partite presentation.

Theorem 5.7 A connected graph has a partition-friendly weak multicycle colouring if and only if it admits a partite presentation.
Proof 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. In the following proof we have to transition between colourings of the directed edges and colourings of the undirected edges. Apart from this, the proof boils down to a straightforward checking of the conditions of the corresponding definitions.
For the forward direction, suppose is connected and it has a partition-friendly weak multicycle colouring c : E( ) → . To define the desired partite presentation P, we start with Since c is partition-friendly, we have U ∪ I = . We want to refine c into a colouring c of the directed edges of . To do this, for each ω ∈ U we choose an orientation is a multicycle, we can choose O ω so that each of its cycles is oriented, that is, for each vertex v ∈ V ( ) there is exactly one e ∈ O ω with τ (e) = x. Thus O ω defines a permutation φ(ω) of X = V ( ), by letting φ(ω)(x) be the unique y ∈ X such that (x, y) ∈ O ω . Moreover, for each ω ∈ I, let ∈ c −1 (ω)}, and let φ(ω) be the involution of V ( ) exchanging the end-vertices of each edge in c −1 (ω). Thus φ satisfies (b) of Definition 5.1 by construction (we will check (a) below).
We now define c by Easily, c is a Cayley-like colouring. This allows us to define W v on as described after Definition 3.8. Note that as is connected, for any two x, y ∈ V ( ) there is a path p connecting x and y. Then the path p corresponds to a word W x ( p) ∈ M F P such that φ(W x ( p))(x) = y. Therefore, the action of M F P on X = V ( ) is transitive as required by (a) of Definition 5.1.
To complete the definition of our partite presentation P, we choose the relators We claim that coincides with the presentation graph C(P). To begin with, they have the same vertex set V (C) = X = V ( ). Moreover, and so our claim is proved.
As we defined C(P) by glueing in a 2-cell along each closed walk dictated by an element of R v , v ∈ V (C(P)), where we have chosen R v = W v (π 1 ( , v)), we have forced π 1 (C(P), v) to be trivial. Therefore, C(P) coincides with its own universal cover C(P). Thus PCay(P), defined as the 1-skeleton of C(P), is C(P) = . Therefore, P is a partite presentation for .
For the converse direction, let = PCay(P) for some partite presentation P. Let : → C(P) be the covering map, and c C : − → E (C) → U ∪ U −1 ∪ I the colouring induced by the generators of P, as in the definition of PCay(P). We collapse c C into a colouring c C of the undirected edges of C defined by We can collapse c : − → E ( ) → U ∪ U −1 ∪ I similarly to obtain an undirected colouring c : E( ) → U ∪ I. Note that c C is a partition-friendly weak multicycle colouring, with c −1 (i) being of degree 1 for i ∈ I and c −1 (u) being of degree 2 for u ∈ U, by the definitions. As c C • = c , it is easy to verify that c C • = c . This implies that c −1 (x) has the same degree as c −1 C (x), and that every vertex has at least one incident edge coloured s for each s ∈ I ∪ U. This means that c is a partition-friendly weak multicycle colouring of as claimed.

Weak multicycle colourings of vertex-transitive graphs
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 Theorem 5.7.
For this, we will use the following result of Godsil  This result generalises to infinite vertex-transitive graphs as follows: 3,16]) Let be a countably infinite, connected, vertex-transitive graph. Then has a perfect matching.
The proof of Theorem 5.9 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.
In passing, let us mention the following still open conjecture. If true, it would imply that all finite vertex-transitive cubic graphs have a uniform partite presentation. The following old theorem of Petersen is a rather straightforward application of Hall's Marriage theorem [12]. Although it is well-known, we include a proof for convenience.
Theorem 5.11 (J. Petersen [23]) Every regular graph of positive (finite and) even degree has a spanning 2-regular subgraph.
Proof Let be a 2k-regular graph. If is finite then it contains an Euler tour C (i.e. a closed walk that uses each edge exactly once) by Euler's theorem [5]. Pick If is infinite then just choose an orientation with equal in and out degree, which can be constructed greedily. Then construct an auxiliary graph with By definition, is k-regular and bipartite, with bipartition V + = {v + |v ∈ V ( )} and V − = {v − |v ∈ V ( )}. For any finite A ⊂ V + , as is k-regular, the neighbourhood has size at least k × |A|/k = |A|. So by Hall's Marriage theorem [12], contains a perfect matching M ⊂ E( ). Then the spanning subgraph S ⊂ given by Combining this with Theorem 5.8 and Theorem 5.9, we now obtain Lemma 5.12 Every countable, vertex-transitive, graph has a partition-friendly weak multicycle colouring.
Proof We first consider the case where is (finite or) locally finite. As is vertextransitive it is n-regular for some n ∈ N. If n is even, then we can apply Theorem 5.11 recursively to decompose E( ) 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.
If n is odd, then we first find a perfect matching M, colour its edges with the same colour, and treat \M as above to obtain a partition-friendly weak multicycle colouring. To obtain M, note that if is finite, then |V (G)| is even since |E( )| = n|V (G)|/2. Therefore, has a perfect matching by Theorem 5.8 as no matching can miss exactly 1 vertex in this case. If is infinite, then Theorem 5.9 provides a perfect matching.
If is not locally finite, then each vertex has countably infinite degree. We will decompose E( ) into an edge-disjoint union of multicycles {M i } i∈N , where each M i is a spanning subgraph consisting of pairwise vertex-disjoint double-rays. For this, let {e i } i∈N be an enumeration of the edges of , and let {v i } i∈N be a sequence of vertices of in which each v ∈ V ( ) appears infinitely often. We greedily construct an M 0 as above containing e 0 as follows. We start with M 0 0 = e 0 , and for i = 1, 2, . . ., we extend the (possibly trivial) path in M i−1 0 containing v i into a longer path by adding an edge of − M i−1 0 at each of its end-vertices. As M i−1 0 is finite, and every vertex has infinite degree, this is always possible. Finally, we let M 0 := i∈N M i 0 . Since each v ∈ V ( ) appears infinitely often as v i , we deduce that M 0 is a spanning union of vertex-disjoint double-rays as desired.
Having constructed M 0 , we inductively construct the M i , i ≥ 1 so that M i contains e i unless e i is already in j<i E(M j ), by noticing that − j<i E(M j ) is a regular graph with countably infinite degree itself, and repeating the above procedure. Then {M i } i∈N is the desired partition-friendly weak multicycle colouring of . This combined with Theorem 5.7 yields our main result: Theorem 1.1 Every countable, vertex-transitive, graph has a partite presentation.
We conclude this section with the following question.

Question 5.13
For a vertex-transitive graph with partite presentation P does Aut c−loc (PCay X |U|I|φ|R ) act vertex-transitively on = PCay(P) where c is the colouring coming from P?

Generalised results
Here we extend some of our earlier results from 2-partite to general partite presentations. Where the same arguments apply directly the proofs will be omitted. First we generalise Lemma 4.3: Proposition 5.14 For a partite presentation P = X |U|I|φ|R there is a natural inclusion of the vertex group G x ≤ Aut c (PCay(P)) for each x ∈ X . Moreover, G x acts regularly on V x , and so PCay(P) is |X |-Cayley.
The vertex groups are still isomorphic due to the fact that π 1 does not depend on the choice of a base point: 15 For every partite presentation P = X |U|I|φ|R , and every x, y ∈ X , the vertex groups G x , G y are isomorphic.
the non-expert reader. This will be used in Sect. 7 to argue that there are partite Cayley graphs that cannot be represented by a partite presentation with finite X .
A quasi-isometry between metric spaces (X , d) and (Y , d ) is a (not necessarily continuous) function f : X → Y satisfying the following two statements for some constants A, B ∈ R + : for every x, z ∈ X , and for every y If such an f exists, we say that (X , d) and (Y , d ) are quasi-isometric to each other. (Easily, this is an equivalence relation.) It is well-known, and easy to check, that any two finitely generated Cayley graphs of the same group are quasi-isometric to each other. 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.

Proposition 5.17
Let P = X |U|I|φ|R be a partite presentation with finite X . Then :=PCay(P) is quasi-isometric to G x for every x ∈ X.
Proof Consider the inclusion map i : C → C 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. Thus i * : Let be the universal cover of C, with covering map η : → C. As π 1 (C, x) = G x we have an action of G x on -and it's 1-skeleton PCay(P)=: -by deck transformations. We know the quotient of a universal cover by the group of deck transformations is the space itself [13, p 70]. Thus the quotient of by G x is C, and so the quotient of by G x is C. Since C is finite when X is, we deduce that the action of G x on is co-compact.
Lastly, we claim that the action of G x by deck transformations on is properly discontinuous. Any compact subset K ⊂ is bounded in the graph metric. By Proposition 5.14, G x acts regularly on V x , and in particular the stabiliser of each vertex is trivial. Our claim now easily follows, e.g. by using the fact that every cellular action on a CW-complex with finite stabilisers of cells is properly discontinuous [15, Theorem 9, (2)=(10)].
To summarise, the action of G x on is properly discontinuous and co-compact. The Švarc-Milnor lemma [21] says exactly that G x is finitely generated, and quasiisometric to for any such action.

Line graphs of Cayley graphs admit partite presentations
In this section, we show that every line graph of a Cayley graph can be represented as a partite Cayley graph. For this we will use 1-and 2-factorisations of the complete graphs as a tool. Let K n be the complete graph on n-vertices. If n is odd then K n has a Hamiltonian decomposition, a partition of the edges into spanning cycles [1]. If n is even, then a special case of Baranyai's theorem [2] gives us a 1-factorisation of K n , i.e. a partition of the edges into perfect matchings.
Thus in either case, we have found a partition-friendly multicycle colouring c : E(K n ) → of K n . Next, we want to associate each colour ω ∈ with a permutation π ω ∈ Sym n of the vertices of K n . To do so, for each ω ∈ such that c −1 (ω) is 2-regular, we pick an orientation , and let π ω be the corresponding permutation (sending each vertex to its successor in O ω . For each ω ∈ such that c −1 (ω) is 1-regular, we let π ω be the permutation that exchanges the two end-vertices of each edge in c −1 (ω). Proposition 6.1 Let = Cay S|R be a Cayley graph. Then the line graph L( ) can be represented as PCay(P) for a partite presentation P with at most |S| vertex classes.
Proof The partite presentation P we will construct will have one vertex class for each generator in S. Since the edges of L( ) are precisely the pairs of incident edges of , we will identify the generators of P with pairs of generators s, t ∈ S. Since we need to pay attention to the directions of the edges of , each such pair s, t will give rise to four generators of P, indexed by the elements of {−1, 1} 2 . Similarly, each s ∈ S will give rise to two generators of P, since there are pairs of incident edges of labelled by s, and there are two choices for their directions. The relators of P will be of two kinds. The first kind is just obtained by rewriting the elements of R in terms of the new generators. The second kind will correspond to closed walks in L( ) contained in the star of a vertex of .
We proceed with the formal definition of P. The vertex classes of P will be identified with the generating set S of . Let K S denote the complete graph with V (K S ) = S.
Here we make the identification that (m i, j ) −1 = (m −1 ) − j,−i . We now define the sets of relators R a , a ∈ S of P. For each relator r :=a 1 a 2 . . . a k ∈ R we add χ(r ) :=χ(a 1 , a 2 )χ (a 2 , a 3 ) . . . χ(a k−1 , a k )χ (a k , a 1 ) to R a ±1 1 . (These are the relators of the first kind as explained at the beginning of the proof.) Lastly, we add relations (of the second kind) corresponding to the star of each vertex of as follows. Let a 1 . . . a k ∈ W S be any word equalling the identity in M F P , and add χ (a 1 , a 2 ) . . . χ(a k , a 1 ) to R a ±1 1 , where χ(s, s −1 ) is the empty word. Let R :={R a , a ∈ S}. We have now constructed our presentation P:= S|U|I|φ|R .
Next, we prove that PCay(P) is isomorphic to L( ). First label so that the edge (g, s 1 , s 2 ) connects [(g, gs 1 )] and [(gs 1 , gs 1 s 2 )]. Let C:=C(P) be the presentation graph of P. Then we can define a map : L( ) → C by letting ([(g, gs)]) = s and letting ((g, s 1 , s 2 )) be the edge of colour χ(s 1 , s 2 ) coming from s ±1 1 ∈ S. One can show that the relations in R x hold in L( ) for all x ∈ S. It remains to show that these relations suffice.
Intuitively we are going to argue that any closed walk p in L( ) is labelled by some r ∈ R M F P interwoven with relations coming from the stars at the vertices. One can observe this by just projecting p to a closed walk in , where after some cancelations happening within the stars of vertices, we are left with a closed walk labelled by a word r than can be expressed in terms of the relators in R. We proceed with this formally.
Define a topological map : L( ) → by mapping [(g, gs)] ∈ V (L( )) to the midpoint of the edge (g, gs), and (g, s 1 , s 2 ) ∈ E(L( )) to the arc in the star of gs 1 connecting the midpoints of [(g, gs 1 )] and [(gs 1 , gs 1 s 2 )]. Consider a closed walk p in L( ). We can write p = n−1 i=0 (g i , s i 1 , s i 2 ). As ( p) is a closed walk in we know it can be contracted to a path given by g 0 g 1 . . . g m−1 for g i ∈ G. Now we want to group the edges of p by the stars of vertices of they lie in. For this, we subdivide the interval {0, . . . , n − 1} into disjoint subintervals {I j } m−1 j=0 such that (g i , s i 1 , s i 2 ) lies in the star of g j for all i ∈ I j and 0 ≤ j ≤ m − 1 (we can assume without loss of generality that no I j has to be the union of an initial and a final subinterval of {0, . . . , n − 1} by rotating p appropriately). Thus p = m−1 j=0 i∈I j (g i , s i 1 , s i 2 ) . To each j we can also associate s j ∈ S ∪ S −1 so that g j s j = g j+1 ; these are the generators that p uses in order to move from one star to the next.
We modify p into a closed walk p by inserting pairs of edges that have the same end-vertices and opposite directions each time that p moves from one star to the next. More formally, we define  Notice that by contracting these pairs of opposite edges (g j+1 , s −1 j , s −1 j−1 )(g j−1 , s j−1 , s j ) we obtain p. Moreover, the sub-walk of p stays within the star of g j by definition, and it is a closed walk starting and ending at [(g j−1 , g j )]. Therefore, it is labelled by one of our relators of the second kind. Easily, ( p) is homotopic to ( p ). Moreover, since a closed walk contained in a star is 0-homotopic. Now m−1 j=0 (g j−1 , s j−1 , s j ) is a closed walk in L( ) no three consecutive edges of which are contained in the star of a vertex of because of the way we chose the I j . This implies that the word s 0 . . . s m−1 labelling this walk is a relation of , and so it can be written as a product of conjugates of relators R. Recalling that each such relator was admitted as a relator (of the first kind) in R , we conclude that the word labelling p can be written as products of conjugates of words in R .
We explicate an example of this below.

Conclusion
In this paper, we showed that every vertex-transitive graph admits a partite presentation, but we were not able to limit the number of vertex classes required. This suggests

Problem 7.1 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?
Define the Cayleyness of a (vertex-transitive) graph as the minimum number of vertex classes in any partite presentation of . Thus is a Cayley graph if and only if it has Cayleyness 1.

Problem 7.2 Is there a vertex-transitive graph of Cayleyness (at least) n for every n ∈ N?
Since the Cayleyness of a vertex-transitive graph divides |V ( )|, a potential approach to answering this question is to enquire if for every prime p ∈ N, there is a vertex-transitive graph on p k vertices for some k ∈ N that is not a Cayley graph.
We observe that the Diestel-Leader graph DL(m, n) for m = n has infinite Cayleyness. This follows by combining Proposition 5.17 with the fact that these graphs are not quasi-isometric to any finitely generated group [7,Theorem 1.4]. This motivates

Problem 7.3 Does a locally finite vertex-transitive graph have finite Cayleyness if and only if is quasi-isometric to a Cayley graph?
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. Is this equivalent to π 1 ( ) being generated by walks of bounded length? It would be interesting to generalise results about finitely presented groups such as [14] to finitely presented graphs in our sense.
It is not hard to show, using group presentations, that there are finitely many finite extensions of any finitely presented group. 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. We hope that partite presentations will be useful in developing an analogous proof.