Growth rates of groups associated with face 2-coloured triangulations and directed Eulerian digraphs on the sphere

Let $\mathcal{G}$ be a properly face 2-coloured (say black and white) \break piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group. The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face two-coloured spherical triangulations with $t$ triangles of each colour. By relating properly face two-coloured spherical triangulations to directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to $\mathcal{C}$ we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.


Introduction
Let G be a graph. We will denote the vertex set of G by V (G) and the edge set of G by E(G). Suppose that there exists a face 2-coloured, black and white say, triangulation of the sphere, i.e. a spherical triangulation, G of G. Denote the set of white faces by W and the set of black faces by B. As the faces are properly face 2-coloured G is Eulerian and, by a well known result of Heawood [14], regardless of whether or not G is simple, G has a proper vertex 3-colouring. If G is simple, then the rotation at every vertex is a cycle, i.e. the triangulation is piecewise-linear. See Figure 1 for an illustration of a face 2-coloured spherical triangulation where the graph is simple. Define A W to be the abelian group with generating set V (G), subject to the relations {r + c + s = 0 : r, c, s are the vertices of a white face of G}. Define A B similarly but using the black faces. In [1] Blackburn and the current author proved that where C is a finite abelian group. In the same paper the question of the growth rate of the maximal order of C, in the terminology established in [13] the canonical group of the face 2-coloured spherical triangulation, was raised. More precisely: [1]). Let m t be the maximal order of the canonical group over all properly face two-coloured spherical triangulations of simple graphs with t faces of each colour. What is the value of lim sup t→∞ (m t ) 1/t ?
In order to establish these new bounds we will make use of a connection between canonical groups of face 2-coloured spherical triangulation and abelian sand-pile groups of directed Eulerian spherical digraphs. In Section 2 we will discuss the background for both of these groups as well as further motivation for addressing the above question.

Background and motivation 2.1 Spherical latin bitrades
Let G be a properly vertex 3-coloured simple graph with v vertices. If the edges of G can be partitioned into copies of K 3 , then such a partition is called a partial latin square. The set of triples of vertices of each of the copies of K 3 completely describes such a partial latin square and we will use the two descriptions interchangeably. We will refer to the graph G as the support graph of the partial latin square.
Let P be a partial latin square, then the three vertices in each triple of P are each from a different vertex colour classes, say R (the rows), C (the columns) and S (the symbols), in the support graph. Suppose that max{|R|, |C|, |S|} = n, then any triple of the partial latin square is of the form {r i , c j , s k }, where r i ∈ R, c j ∈ C and s k ∈ S, and such a triple can be thought of as the symbol k occurring in row i, column j of a n × n array.
Two partial latin squares are said to be isotopic if they are equal up to a relabelling of their sets of rows, columns and symbols. A partial latin square P is said to embed in an abelian group A if and only if it is isotopic to a partial latin subsquare contained in the Cayley table of A. An abelian group A is said to be a minimal abelian representation for the partial latin square P if P embeds in A and, for all embeddings of P in A, the isotopic copy of P in the Cayley table of A generates A.
Define A P to be the abelian group with generating set V (G), subject to the relations {r + c + s = 0 : {r, c, s} ∈ P }. The motivation for this definition is that if P embeds in an abelian group, then it embeds in A P and, in particular, any minimal abelian representation A of P is a quotient of the finite torsion subgroup of A P , see [1] and [11] for details.
A latin bitrade is an ordered pair (W, B) of non-empty partial latin squares such that for each triple {r i , c j , s k } ∈ W (respectively B) there exist unique That is, they are disjoint decompositions of the edge set of the same simple support graph. The arrays in Figure 2 correspond to a pair of partial latin squares which form a latin bitrade (W, B). Note that the two partial latin squares, W and B, are not isotopic. Suppose that G is a face 2-coloured spherical triangulation of a simple graph G with face colour classes W and B and a proper vertex 3-colouring given by R, C and S. Then the faces of W (respectively B) form a partial latin square. As W and B are decompositions of the same simple graph and, provided |W | > 1, no face occurs in both W and B, the pair (W, B) is a latin bitrade. For example, the face 2-coloured spherical triangulation illustrated in Figure 1 corresponds to the latin bitrade (W, B) in Figure 2, the white faces corresponding to the entries in W and the grey faces the entries in B.
In general the partial latin squares forming a bitrade do not necessarily embed in an abelian group, see [7]. However, the partial latin squares forming a bitrade (W, B) arising from a face 2-coloured spherical triangulation both embed in abelian groups, and hence W embeds in A W and B embeds in [7,10], answering a question from [9]. In [7] Cavenagh and Wanless conjectured that A W ∼ = A B ; this was proved in a more general setting in [1] as discussed in Section 1.

Directed Eulerian spherical digraphs and abelian sand-pile groups
Let G be a graph; we will denote the degree of a vertex v ∈ V (G) by deg G (v) and the maximum degree over all vertices of G by ∆(G). Let G be an embedding of G in a sphere. We arbitrarily fix an orientation for the vertices, and denote the rotation at a vertex v ∈ V (G) by ρ(v). Suppose ρ(v) = (u 1 , u 2 , . . . , u deg G (v) ) for some v ∈ V (G); if G is a triangulation and G is a simple graph, then the set of vertices {u 0 , u 1 , . . . , u deg G (v)−1 } induces a cycle in G where, interpreting u deg G (v) as u 0 , the edges are between u i and u i+1 . In a slight abuse of notation we will denote this cycle as ρ(v). Let D be a (not necessarily simple) digraph. Label the vertices of D as v 1 , v 2 , . . . , v n . The adjacency matrix A = [a ij ] of D is the n × n matrix where the entry a ij equals the number of arcs from vertex v i to vertex v j . The asymmetric Laplacian of D is the n × n matrix L(D) = B − A where B is the diagonal matrix whose entry b ii is the out-degree of v i . A sink in a digraph is a vertex with 0 out-degree.
A digraph D is Eulerian if the out-degree at each vertex of D equals the in-degree of each vertex of D. In this case, for each v ∈ V (D) we will refer to out-degree and in-degree of v simply as the degree of A recent and comprehensive survey of results on abelian sand-pile groups of digraphs is given by [15].
In [17] Ribó Mor uses a probabilistic argument via Suen's Inequality, [19], to establish an upper bound on the order of the abelian sand-pile group in an undirected planar graph in terms of the number of vertices. In the same thesis Ribó Mor establishes a tighter bound using non-probabilistic techniques. This bound has subsequently been improved on in [4].
Consider an embedding of an Eulerian digraph. If each face of the embedding is a directed cycle, equivalently the arc rotation at each vertex alternates between incoming and outgoing arcs, the embedding is called a directed Eulerian digraph embedding, see [2] and [3]. If the embedding is on the sphere we call it a directed Eulerian spherical digraph. Eulerian digraph embeddings in surfaces of arbitrary genus have been studied in [2] and [8], and in [3] Bonnington et al. provide Kuratowski type theorems for directed Eulerian spherical digraphs.
In the following Subsection we will discuss a connection between the canonical groups of face 2-coloured spherical triangulations and the abelian sand-pile groups of directed Eulerian spherical digraphs.

Canonical groups and abelian sand-pile groups
Let G be a face 2-coloured spherical triangulation with a proper vertex 3colouring where the vertex colour classes are R, C and S. Let I ∈ {R, C, S}; we will construct a directed Eulerian spherical digraph D I (G) (or simply D I ) with vertex set I. The digraph will potentially have, for any pair of vertices u and v, multiple arcs from u to v. Let {I 0 , I 1 , I 2 } = {R, C, S}. Consider a vertex i ∈ I 0 , then the rotation at i is , where, without loss of generality, u j ∈ I 1 and v j ∈ I 2 for all 1 ≤ j ≤ 1 2 deg G (i) and the edge e j between u j and v j in the rotation is contained in a black face. Then in D I there are 1 2 deg G (i) outgoing arcs a j with initial vertex i, one for each black face, and the terminal vertex for arc a j is the vertex in I contained in the white face containing edge e j . Clearly, the graph D I inherits a spherical embedding from G in which the arc rotation at each vertex alternates between incoming and outgoing arcs. Hence G is a directed Eulerian spherical embedding. Figure 3 illustrates the graph D R (the arcs of which are shown as dashed) obtained from a face 2-coloured spherical triangulation.  Proof. In short, we reverse the construction above.
Denote the faces of D as Consider an arc of D, a say that has x as its initial vertex and y as its terminal vertex. Then on one side of a there is a new vertex u and on the other a new vertex w. Replace a with two triangular faces; a black face with vertex set {x, u, w} and a white face with vertex set {y, u, w}. As D is strongly connected this results in a triangulation of the sphere and as D is a directed Eulerian digraph the resulting triangulation is properly face 2-coloured.
We now list some observations on Lemma 1 and the above construction.  The following lemma is implicit in [1].
Lemma 2. Let G be a face 2-coloured spherical triangulation with a proper vertex 3-colouring where the vertex colour classes are R, C and S. Let I ∈ {R, C, S}, then D I is strongly connected and S( In the following sections we will focus on bounding the number of spanning arborecsences in the directed graph D I , where I ∈ {R, C, S}, obtained from a face 2-coloured spherical triangulation G of a simple graph G. In Section 3 considering all such G with a fixed number of faces in each colour class yields the improved upper bound. In Section 4 we provide a construction for face 2-coloured spherical triangulations for which the associated graphs D I , where I ∈ {R, C, S}, have many spanning arborescences, obtaining a lower bound. Before doing so we will discuss the construction of face 2-coloured spherical triangulations that yield specific canonical groups. By Lemma 1, there exists a face 2-coloured spherical triangulation, with a vertex 3-colouring given by the sets R, C and S where D = D I for some I ∈ {R, C, S}. It is easy to see that in this case the triangulation is of a simple graph.

Constructing abelian groups
We will use recursive applications of the following elementary lemma to prove Proposition 2. Proof. Let v 1 ∈ V (D) and v 2 ∈ V (D 2 ) be the vertices identified to form D and denote the identified vertex as v. As D 1 and D 2 are strongly connected and Eulerian, D is also strongly connected and Eulerian. Let L ′ (D 1 ) (respectively L ′ (D 2 ), L ′ (D)) be the reduced asymmetric Laplacians with sink v 1 in D 1 (respectively v 2 in D 2 and v in D). Then applying, possibly trivial, row and column permutations to L ′ (D) yields

Proposition 2. Consider an arbitrary finite abelian group
Then there exists a face 2-coloured spherical triangulation with canonical group isomorphic to Z m 1 ⊕ · · · ⊕ Z m k .
Proof. Using Proposition 1, construct graphs D i for 1 ≤ i ≤ k where S(D i ) = Z m i . Take any spherical embedding of a tree with k edges, labelled e 1 , . . . , e k , and replace each edge with D i . It is easy to see that this can be done so that the resulting embedded digraph, D, is a strongly connected directed Eulerian spherical digraph. The resulting graph can also be obtained by recursive applications of Lemma 3 and hence has an abelian sand-pile group isomorphic to Z m 1 ⊕ · · · ⊕ Z m k . Therefore, by Lemma 1, there exists a face 2-coloured triangulation that has Z m 1 ⊕ · · · ⊕ Z m k as its canonical group. Figure 4 illustrates the construction used in the proof of Proposition 2 in the case where the canonical group of the face 2-coloured triangulation is isomorphic to Z 2 ⊕ Z 2 . r 0 r 1 r 2 Figure 4: A face 2-coloured spherical triangulation whose canonical group is isomorphic to Z 2 ⊕ Z 2 . A vertex has been placed at infinity and the digraph D R , where R = {r 0 , r 1 , r 2 }, is shown with dashed arcs.
Note that the construction used in the proof of Proposition 2 yields triangulations of graphs that are not simple, i.e. they do not correspond to latin bitrades.
Let A = Z m 1 ⊕ · · · ⊕ Z m k−1 where, without loss of generality, m i > 1 for 1 ≤ i ≤ k − 1. The construction in the proof of Proposition 2 yields a triangulation G with a proper vertex three colouring given by R, C and S such that D ∼ = D R (G) and neither D C (G) nor D S (G) contain any cut vertices. Hence a set T of nonisomorphic trees on k vertices yields |T | nonisomorphic face 2-coloured triangulations all of which have canonical groups isomorphic to A. (Otter [18] showed that the number of nonisomorphic trees on k vertices is asymptotically 0.4399237(2.95576) k k 3/2 .)

Improving the upper bound
In this section all the face 2-coloured spherical triangulations will be of simple graphs. Moreover, as we are concerned with the behaviour of m t as t → ∞, in the following discussion, we take t ≥ 4. Hence every vertex in any triangulation considered is contained in at least four faces.
Similarly to the approach taken by Ribó Mor in [17], the improved upper bound for lim sup t→∞ (m t ) 1/t is obtained using a probabilistic argument based on Suen's Inequality. However, the results in [17] are concerned with the growth of the number of spanning trees in terms of the number of vertices in the graph, rather than the number of arcs. A vital component of Ribó Mor's argument is the addition of edges to a planar graph to obtain a triangulation. Thus, although the beginning of our argument follows that of [17] (in setting up the use of a refinement of Suen's Inequality), as we are interested in the growth rate as the number of arcs increases, the remainder necessarily follows a different approach.
Let {I i } i∈I be a finite family of Bernoulli random variables each with success probability p i ; i.e. P(I i ) = p i . A simple graph Γ where V (Γ) = I is called a dependency graph for {I i } i∈I if when two disjoint subsets of I, A and B say, are mutually independent, there is no edge between any vertex in A and any vertex in B. In particular two distinct variables I i and I j are independent unless there is an edge between i and j. For ease of notation, when discussing a dependency graph, if there exists an edge between vertices i and j, we write i ∼ j.
We will make use of the following refinement to Suen's Inequality (note that in our case both Suen's Inequality and that presented in Theorem 1 yield the same bound).
Theorem 1 (Janson [16]). Let I i , where i ∈ I be a finite family of Bernoulli random variables with success probability p i , having a dependency graph Γ. Let S = i∈I I i ; µ = E(S) = i∈I p i ; ∆ = 1 2 i∈I j∈I,i∼j E(I i I j ); and δ = max i∈I k∼i p k . Then Let G be a properly face 2-coloured triangulation of a simple graph G with t faces of each colour such that the order of its canonical is maximum over all such triangulations. Fix a vertex i 0 of D I (G) (for the remainder of this section we will write D I for D I (G)). Let R be a random selection of incoming arcs, one for each vertex of V (D I ) − i 0 . Then, denoting the subgraph of D I induced by the arcs of R as D I [R], we have is a spanning arborescence rooted at i 0 Equivalently We can now use Theorem 1 to provide an upper bound for the probability that D I [R] does not contain a directed cycle (as R contains exactly one incoming arc for each vertex not equal to i 0 if the underlying graph contains a cycle, it must be directed).
Let D I−i 0 denote the set of all directed cycles in D I that do not contain the vertex i 0 . For each γ ∈ D I−i 0 , define: Applying Theorem 1, with µ = γ∈D I−i 0 p γ , we have: Let D I denote the set of all directed cycles in D I and D i 0 denote the set of all directed cycles in D I that contain i 0 . Then, by Inequality 1 we have: Proof. We show that − ln(deg D I (i 0 )) + Now i 0 is necessarily a vertex in each cycle passing through it and the minimum (out-)degree of any vertex in D I is 2, so By Lemma 2, T (D R ) = T (D C ) = T (D S ), so, by Lemma 4, we have: Denote the set of faces in D I , for I ∈ {R, C, S}, by F I ; and, in a slight abuse of notation, the set of vertices on a face f by V (f ). As the triangulation is of a simple graph it is piecewise linear. Hence, the facial walk of any face in F I is a cycle and so F I ⊆ D I . As deg D I (i) = 1 2 deg G (i), we have .
Rewriting the upper bound (2) in terms of the n k 's, and bounding the second summation in terms of α and β we have As the average degree of a vertex in G is 6 − 12/n, for each vertex of degree 2i > 6 we can associate (2i − 6)/2 degree four vertices. Hence we have that i.e., when n 6 /n = 3/5. Hence, (2)) 5 A family of face 2-coloured spherical triangulations that has attracted recent interest, see [5] and [6], are triangulations that contain precisely six degree 4 vertices and all the other vertices have degree 6, i.e. near-homogeneous face 2-coloured spherical triangulations. Part of the motivation for their study comes from their connection to a solved case of Barnette's Conjecture [12]. When restricting ourselves to the near-homogeneous case we can significantly improve the upper bound.

Improving the lower bound
In [13], Grubman and Wanless analyse the effect, to order of the canonical group of face 2-coloured spherical triangulations, of applying several recursive constructions. They obtain a lower bound on the growth rate of 5123 1/30 by using a construction that identifies a black triangle in one face 2-coloured spherical triangulation, G 1 say, with a white triangle in a second face 2-coloured spherical triangulation, G 2 say. When viewed as a recursive construction applied to edges of the related digraphs D I (G 1 ) and D I (G 2 ) this equates to removing an arc from a vertex u to a vertex u ′ in D I (G 1 ) and an arc from a vertex w to a vertex w ′ in D I (G 2 ), then identifying u and w ′ and adding an arc from w to u ′ . Denote the resulting strongly connected digraph as D, then, by considering the spanning arborescences rooted at u = w ′ , it follows that T (D) = T (D I (G 1 ))T (D I (G 2 )). Note that the construction can be applied so that D has a directed embedding in the sphere. By considering recursive constructions applied to faces, rather than the arcs, of D R , D C and D S , taking care to ensure the resulting related undirected triangulations are still simple, we will provide an improved lower bound. Then there exists a face 2-coloured spherical triangulation G ′ of a simple graph with t + 2k faces with a proper vertex 3-colouring given by the colour classes R ′ , C ′ and S ′ with canonical group C ′ such that: there exists a I ∈ {R ′ , C ′ , S ′ } where D I (G ′ ) contains a face of size k in which all the vertices have (out-)degree 2; and Proof. Denote the vertices of the face f by v 1 , v 2 , . . . , v k so that the arcs on the boundary of the face are from v i to v i+1 , where subscripts are taken modulo k. Insert a new vertex into the interior of f , call this vertex u, also add an arc from u to v j and an arc from v j to u, for all 1 ≤ j ≤ k, maintaining a directed Eulerian spherical embedding, D ′ say. (We have replaced a face of size k with k triangular faces and k digons.) We next calculate a lower bound for the number of spanning arborescences in D ′ . Let A be the set of all spanning arborescences in D rooted at For each arborescence in A, remove the ingoing arc with end vertex v ′ i , for all 1 ≤ i ≤ j. As deg D (v ′ i ) = 2 this yields at least 1 2 j |A| different subgraphs. Now, to each of these subgraphs, add the arc from v to u and the arcs from u to v ′ i for all 1 ≤ i ≤ j. This results in 1 2 j |A| spanning arborescences of D ′ rooted at x. There were k choices for v and k−1 j choices for the other j vertices. Hence we have at least To complete the proof we need to show that D ′ corresponds to a face 2coloured spherical embedding of a simple graph G ′ with a vertex 3-colouring with colour classes R ′ , C ′ and S ′ and that there exists a I ∈ {R ′ , C ′ , S ′ } such that D I (G ′ ) has a face of size k in which all the vertices have (out-)degree 2.
By Lemma 1, D ′ corresponds to a face 2-coloured spherical triangulation G ′ . To see that this new triangulation is also of a simple graph note the following. The triangulation G ′ can be obtained from G by first deleting the vertex of degree 2k that corresponds to f in D I and all the faces and edges incident to it and replacing them with a single face of size 2k. Denote the vertices of this new face by w 0 , w 1 , . . . , w 2k−1 so that the edges on the boundary of the face are from w i to w i+1 where subscripts are taken modulo 2k. Next insert 2k + 1 new vertices, z, z 0 . . . , z 2k−1 and edges into the new face so that the rotations at the new vertices are: ρ(z) = (z 0 , z 1 , . . . , z 2k−1 ) ρ(z i ) = (z, z i−1 , w i , z i+1 ), where 0 ≤ i ≤ 2k − 1 and subscripts are taken modulo 2k. Hence G ′ is also a triangulation of a simple graph.
By Observation 1, G ′ contains a vertex (z in the previous paragraph) with degree 2k whose neighbours are all contained in precisely four faces (two white and two black). Hence there exists a D I (G ′ ) with a face of size k in which all the vertices have (out-)degree 2. Similar base triangulations for Lemma 5 to be recursively applied to can easily be obtained for face sizes other than 4, but the resulting families have smaller growth rates.