The edge-regular complete maps

Abstract A map is called edge-regular if it is edge-transitive but not arc-transitive. In this paper, we show that a complete graph K n {K}_{n} has an orientable edge-regular embedding if and only if n = p d > 3 n={p}^{d}\gt 3 with p an odd prime such that p d ≡ 3 {p}^{d}\equiv 3 ( mod 4 ) (\mathrm{mod}\hspace{.25em}4) . Furthermore, K p d {K}_{{p}^{d}} has p d − 3 4 d ϕ ( p d − 1 2 ) \tfrac{{p}^{d}-3}{4d}\hspace{0.25em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) non-isomorphic orientable edge-regular embeddings.


Introduction
An orientable map is a 2-cell embedding of a finite graph in an orientable surface, and thus a map is an incident triple of vertex set V, edge set E, and face set F, denoted by = ( ) V E F , , . The finite graph is called the underlying graph of the map, and the surface is called the supporting surface of the map. In this paper, we focus on the orientable supporting surface.
An automorphism of a map = ( ) V E F , , is a permutation of ∪ ∪ V E F that preserves the incidence relation between vertices, edges, and faces. A map is called edge-transitive (or an edge-transitive embedding of its underlying graph) or arc-transitive if the automorphism group Aut is transitive on the edge set or the arc set, respectively. Sometimes, the arc-transitive map is called a rotary map or a symmetrical map. For convenience, a map is called edge-regular if is edge-transitive but not arctransitive.
The purpose of this paper is to give a classification of orientable edge-regular maps with underlying graphs being complete graphs. For convenience, a map with underlying graph being a complete graph is called a complete map.
In general, a flag of a map is an incident triple of vertex, edge, and face. A map is called flagtransitive if Aut is transitive on the flag set. Since the action of Aut on the flag set is always semiregular, a flag-transitive map is flag-regular, and it is simply called regular. It is easy to see that if a subgroup ≤ G Aut is transitive on the edge set of , then the index of G in Aut is not more than four. Thus, an edge-transitive map is highly symmetrical.
The literature studying "symmetrical" maps is rich, refer to [1]. Recent investigation began with Biggs' study in [2,3] of arc-transitive complete maps. Among the arc-transitive embeddings of K n constructed by Biggs, the unique embeddings for = n 2, 3, and 4 are flag-regular, others with ≥ n 5 a prime power are all arc-regular. In the past four decades, plenty of results about symmetrical maps have been obtained, see [4][5][6][7][8][9][10] and references therein. In particular, arc-transitive complete maps are classified in [3,7,11]; vertextransitive complete maps are characterized in [12]. Very recently, some special families of edge-transitive maps with underlying complete bipartite graphs are classified in [13][14][15][16]. In this paper, we study the edgeregular complete maps. The main result of the paper is stated in the following theorem. .
Remarks on Theorem 1.1.
(1) The main results of Theorem 1.1 were obtained by James [17] and Jones [18], both of which used the methods of group theories. However, the research approach adopted in the present paper is more concise and easier.  We now construct edge-regular embeddings of the complete graph = K Γ p d . Let 0 be the identity of + F .
into two orbits, say Δ 0 and Δ 1 . We make labelings for the elements of Δ 0 and Δ 1 as follows.
Let ∈ v Δ 0 be a non-identity element of + F , and set Then the orbits . For a vertex α, a cyclic permutation of the neighbor set ( ) α Γ of α is called a rotation at α and denoted by R α . A rotation system ( ) R Γ of a graph Γ is the product of rotations at all vertices, that is, Then each rotation system ( ) R Γ defines an orientable embedding of Γ, refer to [1, pp. 104-108].

Construction 2.1
Use the aforementioned notations.
(1) Label the end points of the arcs emitting from the vertex ∈ + F 0 as (2) For each vertex ∈ + x F , label the end points of the arcs emitting from x as (3) Define a rotation of the end points of arcs emitting from x by Observing that the rotation system ( ) R Γ is uniquely determined by β β , 0 1 , and ρ, we denote the orientable map defined by the rotation system as follows: We next study the maps . The first lemma determines the relation between β i 's and their inverses.
It follows that there exists a positive integer ℓ such that ( ) For a vertex ∈ v V Γ, noting that the vertex rotations R v can be regarded as permutations not only of the set ( ) v Γ but also of the generating set S. So Cayley maps have another equivalent definitions, see [19]. Proof. By the definition of the rotation system ∏ ∈ + R x F x , each element of + F is an automorphism of . Since + F acts regularly on the vertices of , is a Cayley map of + F , and the underlying graph of is a complete graph of order p d . Furthermore, for each element ∈ + x F , the conjugation ρ x is such that, for ≤ ≤ − is not reflexible, and the automorphism group Aut is a subgroup of ( ) p AGL 1, d .
, and let = A Aut . Let + A be the group of elements of A which preserves the orientation of the supporting surface of . Then G is of index at most 2 in + A , and + A is of index at most 2 in A. Thus, ⊲ ⊲ + G A A, and the index | | = A G : 1, 2, or 4. Since G is of odd order, it follows that G is a characteristic subgroup of A. As G is primitive on the vertex set V, so is A. Hence, ( )  for all ∈ = + x V F is a circular permutation, namely, x β x  β  x  ,  ,  ,  , ,  ,  ,  ,  , ,  ,   k  x  i  i  i  i  i k  0  1  2  3  1  1  2  3  1 where ≤ ≤ − i k 0 1 and the subscripts are modulo k, we have , , .
Therefore, to enumerate different edge-transitive embeddings of K p d , we may fix the first element β 0 .
The following lemma gives a counting formula of the maps ( ) β β ρ , , . , , namely, φ is an automorphism of G. Arguing as in the previous case shows that ∈ ( ) = = 〈 〉 〈 〉 − φ p ξ τ ΓL 1, : : p . Since for any , it follows that each element of 〈 〉 〈 〉 ξ τ : is an isomorphism among maps in . Let 2 be the subset of consisting of arc-regular maps. As ξ is an automorphism of each map in 2 , the induced action of 〈 〉 〈 〉 ξ τ : is isomorphic to 〈 〉 ≅ τ d and 〈 〉 τ is semiregular on 2 . It follows that there are precisely  .
Proof. Since G is edge-transitive, but not arc-transitive, we have that ≤ G S n is a 2-homogeneous, but not 2-transitive permutation group on V. Noting that G α is a cyclic group or a dihedral group. By Kantor's classification (see [ . □ Now we are ready to prove our main theorem.
Proof of Theorem 1.
be a 2-cell embedding of = K Γ n , and let = G Aut . Then, for a vertex ∈ α V, the stabilizer G α is a cyclic or dihedral group. Assume that G is edge-transitive but not arctransitive on . We need to prove that is a map as given in Construction 2. 1 . Therefore, the rotation system for is the same as the one constructed in Construction 2.1, and so the map is as constructed in Construction 2.1. □