The edge-regular complete maps

Theorem 1.1

A complete graph K n has an orientable edge-regular embedding if and only if n = p d , where p is an odd prime and d is a positive integer such that p d ≡ 3 ( mod 4 ) . Furthermore, K p d has p d − 3 4 d ϕ ( p d − 1 2 ) non-isomorphic edge-regular embeddings, where ϕ is Euler’ s function, and each of the maps has full automorphism group isomorphic to Z p d : Z p d − 1 2 < AGL ( 1 , p d ) .

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.

2. A map ℳ is called a Cayley map if Aut ℳ contains a subgroup G which is regular on the vertices. Furthermore, if G ⊲ Aut ℳ , then ℳ is called a normal map. Thus, all the maps in Theorem 1.1 are normal maps.

3. A Cayley map ℳ is called balanced if each generated element s and its inverse element s − 1 of s are placed on the antipodal points, see [10]. It is known that an arc-transitive Cayley map ℳ is balanced if and only if ℳ is a normal map. Thus, the maps in Theorem 1.1 are not necessary balanced.

Corollary 1.2

A normal edge-regular Cayley map is not necessarily a balanced map.

Construction 2.1

Use the aforementioned notations.

1. Label the end points of the arcs emitting from the vertex 0 ∈ F + as

   β 0 , β 1 , β 2 , β 3 , … , β k − 1 .

2. For each vertex x ∈ F + , label the end points of the arcs emitting from x as

   β 0 x , β 1 x , β 2 x , β 3 x , … , β k − 1 x .

3. Define a rotation of the end points of arcs emitting from x by

   R x = ( β 0 x , β 1 x , β 2 x , β 3 x , … , β k − 1 x ) ,

   where x ∈ V = F + , and let R ( Γ ) = ∏ x ∈ F + R x .

4. Observing that the rotation system R ( Γ ) is uniquely determined by β 0 , β 1 , and ρ , we denote the orientable map defined by the rotation system as follows:

Lemma 2.2

There exist positive integers ℓ and m such that β 1 = ( β 0 − 1 ) ρ ℓ = β 2 ℓ − 1 and β 0 − 1 = β 2 m + 1 , where m = p d − 1 2 − ℓ .

Proof

Assume that β i and β i − 1 lie in the same orbit of 〈 ρ 〉 , where 0 ≤ i < p d − 1 . Then β i − 1 = β I ρ t with 1 ≤ t < p d − 1 2 , and ( β i ρ t ) ρ t = ( β i − 1 ) ρ t = ( β i ρ t ) − 1 = ( β i − 1 ) − 1 = β i . Thus, ( ρ t ) 2 = 1 . A contradiction occurred since ρ = p d − 1 2 is odd. So β i and β i − 1 lie in different orbits of 〈 ρ 〉 , where 0 ≤ i < p d − 1 . Hence, β 0 − 1 lies in the orbit of β 1 under 〈 ρ 〉 . It follows that there exists a positive integer ℓ such that β 1 = ( β 0 − 1 ) ρ ℓ = ( β 0 ρ ℓ ) − 1 = β 2 ℓ − 1 .

Furthermore, β 0 − 1 = β 1 ρ − ℓ = β 1 ρ p d − 1 2 − ℓ = β 2 ( p d − 1 2 − ℓ ) + 1 . Thus, letting m = p d − 1 2 − ℓ , we have β 0 − 1 = β 2 m + 1 .□

Lemma 2.3

Let ℳ = ℳ ( β 0 , β 1 , ρ ) and G = F + : 〈 ρ 〉 , as defined above. Then the following statements hold.

1. G ≤ Aut ℳ , and G is transitive on the edges of ℳ ;

2. ℳ is a Cayley map of F + .

Proof

By the definition of the rotation system ∏ x ∈ F + R x , each element of F + is an automorphism of ℳ . Since F + acts regularly on the vertices of ℳ , ℳ is a Cayley map of F + , and the underlying graph of ℳ is a complete graph of order p d . Furthermore, for each element x ∈ F + , the conjugation ρ x is such that, for 0 ≤ i ≤ k − 1 ,

reading the subscripts modulo k. Thus, ρ is an automorphism of the map ℳ , refer to [1, pp. 109–110]. Since G is transitive on the edges of the underlying graph Γ , we have that the map ℳ is G-edge-transitive.□

Lemma 2.4

A map ℳ ( β 0 , β 1 , ρ ) is balanced if and only if β 0 − 1 = β p d − 1 2 and β 1 − 1 = β p d + 1 2 .

Proof

Assume that ℳ ( β 0 , β 1 , ρ ) is balanced. Then for every i with 0 ≤ i < p d − 1 2 , the vertex β i − 1 is placed at the antipodal position of the vertex β i . Thus, β 0 − 1 = β p d − 1 2 and β 1 − 1 = β p d + 1 2 .

Conversely, assume that β 0 − 1 = β p d − 1 2 and β 1 − 1 = β p d + 1 2 . Then we have

So β j − 1 = β p d − 1 2 + j is at the antipodal position of β j , for all j with 0 ≤ j < p d − 1 2 , and therefore, ℳ ( β 0 , β 1 , ρ ) is balanced.□

Lemma 2.5

The orientable map ℳ ( β 0 , β 1 , ρ ) is not reflexible, and the automorphism group Aut ℳ is a subgroup of AGL ( 1 , p d ) .

Proof

Let ℳ = ℳ ( β 0 , β 1 , ρ ) , and let A = Aut ℳ . Let A + be the group of elements of A which preserves the orientation of the supporting surface of ℳ . 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, C A ( G ) ≤ G , and A ≤ Aut ( G ) = AΓL ( 1 , p d ) = ( F + : F × ) : Z d . By Frattini’s argument, A α ≤ F × : Z d for any α ∈ V = F + . Since A α ≤ D 2 ( p d − 1 ) and d is odd, it follows that A α ≤ F × ≅ Z p d − 1 and A ≤ AGL ( 1 , p d ) .

If ℳ is reflexible, then there exists an involution τ ∈ A \ A + which reverses ρ . However, note that d is odd, then this is not possible. So ℳ is not reflexible.□

Lemma 2.6

Let ℳ = ℳ ( β 0 , β 1 , ρ ) , a map as constructed in Construction 2.1. Then either

1. ℳ is chiral (arc-regular), β 1 = β 0 ξ , and Aut ℳ = AGL ( 1 , p d ) ; or

2. ℳ is edge-regular, and Aut ℳ = G .

Proof

Assume that ℳ is arc-transitive. Then Aut ℳ = AGL ( 1 , p d ) = F + : F × = F + : 〈 ξ 〉 by Lemma 2.5. Since β 1 and β 0 lie in different orbits of 〈 ρ 〉 = 〈 ξ 2 〉 , there exists an odd integer ℓ with 1 ≤ ℓ < p d − 1 such that β 1 = β 0 ξ ℓ . Thus, β 0 ξ 2 = β 0 ρ = β 2 = β 1 ξ ℓ = β 0 ξ 2 ℓ . Since 〈 ξ 〉 is regular on F + \ { 0 } , it follows that ℓ = 1 , and so β 1 = β 0 ξ .

Assume that ℳ is not arc-transitive. Then by Lemma 2.5, Aut ℳ = G , which is regular on the edge set of ℳ .□

Lemma 2.7

A complete graph K p d with p d ≡ 3 ( mod 4 ) has precisely p d − 3 4 d ϕ ( p d − 1 2 ) non-isomorphic orientable edge-regular embeddings, and 1 d ϕ ( p d − 1 2 ) non-isomorphic orientable chiral embeddings.

Proof

As argued above, we may fix β 0 . Then there are precisely p d − 1 2 choices for β 1 , and there are precisely ϕ ( p d − 1 2 ) choices for ρ , where ϕ is Euler’s function. Thus, there are exactly p d − 1 2 ϕ ( p d − 1 2 ) different triples for ( β 0 , β 1 , ρ ) , so the number of different maps of the form ℳ ( β 0 , β 1 , ρ ) is equal to p d − 1 2 ϕ ( p d − 1 2 ) .

By Lemma 2.6, a map ℳ ( β 0 , β 1 , ρ ) is arc-transitive if and only if β 1 = β 0 ξ , and if and only if it is arc-regular. Thus, the number of different arc-transitive embeddings of the form ℳ ( β 0 , β 0 ξ , ρ ) of K p d equals the number of choices of ξ with | ξ | = p d − 1 , which is equal to ϕ ( p d − 1 2 ) since p d − 1 2 is odd and ϕ ( p d − 1 ) = ϕ ( p d − 1 2 ) × ϕ ( 2 ) = ϕ ( p d − 1 2 ) . The others are all edge-regular, and the number of them equals p d − 1 2 ϕ ( p d − 1 2 ) − ϕ ( p d − 1 2 ) = p d − 3 2 ϕ ( p d − 1 2 ) .

Now, we determine isomorphism classes of the p d − 1 2 ϕ ( p d − 1 2 ) different maps described above. Let

Let ℳ 1 , ℳ 2 ∈ S which are isomorphic, and let φ be an isomorphism from ℳ 1 to ℳ 2 . Then φ induces an isomorphism from Aut ℳ 1 to Aut ℳ 2 .

Case 1. First, assume that ℳ 1 is edge-regular. Then so is ℳ 2 . By Lemma 2.6, we have

Hence, G φ = (Aut ℳ 1 ) φ = Aut ℳ 2 = G , namely, φ is an automorphism of G. Since F + is a normal Sylow p-subgroup of G, it follows that F + is a characteristic subgroup of G, and φ normalizes F + ≅ Z p d . Thus, φ ∈ Aut ( Z p d ) = GL ( d , p ) , and φ ∈ N GL ( d , p ) ( 〈 ρ 〉 ) . We may choose φ to normalize 〈 ρ 〉 . By [21, Theorem 7.3 (a) of Chapter 2],

where τ : ξ ↦ ξ p is the Frobenius automorphism of GF ( p d ) .

Since ρ = ξ 2 is an automorphism of ℳ ∈ S , it fixes all maps in S . Let S 1 be the subset of S which consists of edge-regular maps. So ρ = ξ 2 fixes each element of S 1 . It follows that 〈 ξ 〉 : 〈 τ 〉 / 〈 ξ 2 〉 ≅ 〈 σ 〉 × 〈 τ 〉 , where σ = ξ p d − 1 2 . Thus, N GL ( d , p ) ( 〈 ρ 〉 ) = 〈 ξ 〉 : 〈 τ 〉 acting on S 1 is isomorphic to 〈 σ 〉 × 〈 τ 〉 . As Aut ℳ = G , the group 〈 σ 〉 × 〈 τ 〉 acts semiregularly on S 1 . Obviously, two maps ℳ 1 and ℳ 2 are isomorphic if and only if they lie in the same orbit of 〈 σ 〉 × 〈 τ 〉 acting on S 1 . Since | S 1 | = p d − 3 2 ϕ ( p d − 1 2 ) , it follows that there are exactly 1 2 d ⋅ p d − 3 2 ϕ ( p d − 1 2 ) non-isomorphic edge-regular maps in S .

Case 2. Assume now that ℳ 1 and ℳ 2 are arc-regular. By Lemma 2.6, we have

where σ = ξ p d − 1 2 . So ( G . 〈 σ 〉 ) φ = ( Aut ℳ 1 ) φ = Aut ℳ 2 = G . 〈 σ 〉 , namely, φ is an automorphism of G. Arguing as in the previous case shows that φ ∈ Γ L ( 1 , p d ) = Z p d − 1 : Z d = 〈 ξ 〉 : 〈 τ 〉 , where τ : ξ ↦ ξ p . Since for any ℳ ( β 0 , β 1 , ρ i ) ∈ S ,

and ℳ ( β 1 , β 2 , ρ i ) ∈ S and ℳ ( β 1 p , β 2 p , ρ i p ) ∈ S , it follows that each element of 〈 ξ 〉 : 〈 τ 〉 is an isomorphism among maps in S . Let S 2 be the subset of S consisting of arc-regular maps. As ξ is an automorphism of each map in S 2 , the induced action of 〈 ξ 〉 : 〈 τ 〉 is isomorphic to 〈 τ 〉 ≅ Z d and 〈 τ 〉 is semiregular on S 2 . It follows that there are precisely 1 d ϕ ( p d − 1 2 ) non-isomorphic arc-regular maps in S .□

Lemma 3.1

Let Γ = ( V , E ) = K n and G ≤ AutΓ . Assume that G is edge-transitive but not arc-transitive on Γ , then the following statements hold:

1. n = p d ≡ 3 (mod 4 ) , where p is an odd prime and d is a positive integer;

2. G ≅ Z p d : Z p d − 1 2 < AGL ( 1 , p d ) .

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 [22, Theorem 9.4B]), G has a unique minimal normal subgroup N ≅ Z p d for some odd prime p and positive integer d such that p d ≡ 3 (mod  4) , and G α = Z p d − 1 2 , where α is a vertex. Thus, G = N : G α ≅ Z p d : Z p d − 1 2 < AGL ( 1 , p d ) .□

Proof of Theorem 1.1

Let ℳ = ( V , E , F ) be a 2-cell embedding of Γ = K n , and let G = Aut ℳ . Then, for a vertex α ∈ V , the stabilizer G α is a cyclic or dihedral group. Assume that G is edge-transitive but not arc-transitive on ℳ . We need to prove that ℳ is a map as given in Construction 2.1.

By Lemma 3.1, there exist an odd prime p and a positive integer d such that G = N : G α = Z p d : Z p d − 1 2 , where n = p d ≡ 3 ( mod 4 ) , and N is regular on the vertex set V. Thus, we may identify the vertex set V with N. Furthermore, we identify N with the additive group F + of the field F = GF( p d ) . Let the multiplicative group F × of F be generated by ξ , namely, F × = 〈 ξ 〉 ≅ Z p d − 1 2 . Let ρ = ξ 2 , and let α be the vertex of ℳ corresponding to 0 ∈ F + . Then G α = 〈 ρ 〉 ≅ Z p d − 1 2 .

The action of G α = 〈 ρ 〉 on V \ { α } is semiregular and divides V \ { α } into two orbits. Since ρ fixes the vertex 0 and preserves the supporting surface, the rotation of the end points of the arcs emanating from 0 has the form

where k = p d − 1 such that

reading the subscripts i + 2 modulo k. Thus, the two orbits of 〈 ρ 〉 acting on Γ ( α ) are { β 0 , β 2 , … , β k − 2 } and { β 1 , β 3 , … , β k − 1 } . Therefore, the rotation system for ℳ is the same as the one constructed in Construction 2.1, and so the map ℳ is as constructed in Construction 2.1.□