Vertex-Transitive Digraphs of Order p 5 are Hamiltonian

We prove that connected vertex-transitive digraphs of order p5 (where p is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup G of prime power order such that G′ is generated by two elements or elementary abelian is Hamiltonian.


Introduction
One of the most famous problems in vertex-transitive graphs theory is the problem of existence of Hamilton paths/cycles (that is, simple paths/cycles going through all vertices) in finite connected vertex-transitive graphs (or digraphs).Graphs (or digraphs) which have Hamilton cycles are called Hamiltonian.The interest in this problem grew out of a question posed byLovász [13], who asked whether every finite connected vertex-transitive graph has a Hamilton path.In fact, there are only four known nontrivial connected vertex-transitive graph that do not possess Hamilton cycles.These four graphs are the Petersen graph, the Coxeter graph and the two graphs obtained from them by replacing each vertex by a triangle.The fact that none of these four graphs is a Cayley graph has led to a folklore conjecture that every connected Cayley graph with order greater than 2 has a Hamilton cycle.A large number of articles directly or indirectly related to this problem (for the list of relevant references and a detailed description of the status of this problem see [8]), have appeared in the literature, affirming the existence of such paths in some special vertex-transitive graphs and, in some cases, also the existence of Hamilton cycles.Since the publication of the survey paper [8], some recent improvements on this subject appeared, see [23,4,9,6,5,22] and so forth.
Let p be a prime number.It is known that connected vertex-transitive graphs of order kp, where k 4, and p i , where i 4, and 2p 2 (except for the Petersen graph and the Coxeter graph) contain a Hamilton cycle; see [1,17,7,14,3,15].A Hamilton path is known to exist in connected vertex-transitive graphs of order 5p, 6p and order 10p removing some special cases; see [16,11,10].As for Cayley graphs, perhaps the biggest achievement on this subject is due to Witte (now Morris) who proved that a connected Cayley digraph of any p-group has a Hamilton cycle [21].
It seems to be quite a challenge to generalize Witte's theorem on Hamilton cycles in Cayley digraphs of p-groups to arbitrary vertex-transitive digraphs of prime power order.The first successful attempt using this approach is due to Chen [3], who proved that vertextransitive digraphs of order p 4 are Hamiltonian.In this paper, we give some conditions under which one can obtain Hamilton cycles of vertex-transitive digraphs of prime power order by lifting Hamilton cycles from their quotient graphs.Using our methods, one can affirm the existence of Hamilton cycles of many connected vertex-transitive digraphs of prime power order.In particular, we obtain the following two theorems: Theorem 1.1.Connected vertex-transitive digraphs of order p 5 are Hamiltonian.
Theorem 1.2.Let Γ be a connected digraph of which the automorphism group contains a finite vertex-transitive subgroup G of prime power order.Let G be the derived subgroup of G. Then Γ is Hamiltonian if one of the following two conditions hold: (i) G is generated by two elements; (ii) G is elementary abelian.
The paper is organized as follows.In Section 2, notations and lemmas in group theory for later use are introduced.In section 3, we review the concept of coset digraphs and the representations of paths and cycles in coset digraphs.In section 4, the main theorems of this paper are proved.

Notations and preliminary lemmas in group theory
In this section, we fix some notations and introduce some lemmas for later use.The following standard group-theoretic notations will be used throughout the rest of the paper.
Remark.The Frattini subgroup Φ(G) of G is defined to be the intersection of all maximum subgroups of G.An element g of G is said to be a non-generator if G = X whenever G = g, X , where X is a subset of G.It is well known that Φ(G) is the set of non-generators of G.
Below we introduce four lemmas that will be needed in the proof of our main results.Proof.For any g ∈ G and any i > 1, since we have [g, Then, by the Basis theorem of Burnside (see [19,Theorem 1.16 of Chapter 2] for example), there exists a minimal generating set {[x, w], y 1 , . . ., y d−1 } of K. Furthermore, by (1), we can choose The following lemma is a direct corollary of [2, Theorem 1].
Lemma 2.4.Let G be a finite p-group for which G is generated by two elements.Then any subgroup of G can be generated by at most two elements.

Coset digraphs and cycles
The digraphs considered in this paper are finite, connected, with no loops or multiple edges.For a vertex-transitive digraphs Γ, the following proposition gives a nice way to represent it by using subgroups of its automorphism group.For proof and comments of this proposition, see [12] and [3] respectively.As for the vertex-transitive digraph of prime power order, we have the following proposition.

Proposition 3.2 ([3]
).If Γ is a vertex-transitive digraph of order p n where p is a prime and n is a positive integer, then Γ admits a representation Cos(G, H, Ω) such that G is a p-group, H Z(G) = {1} and H < Φ(G).
Remark.By Lemma 2.1, the subgroup H of G in Proposition 3.2 is core-free, that is, H G = {1}.Therefore, G acts vertex-transitively on Γ by left multiplication.In fact, G can be chosen as a minimum vertex-transitive p-subgroup of Aut(Γ).
The digraph Γ = Cos(G, H, Ω) defined in Proposition 3.1 is usually called a coset digraph on G/H, which is actually a generalized orbital graph of G acting on G/H (the definition of generalized orbital graph can be found in many publications, see [18] for example).Particularly, if H = {1}, then Γ = Cos(G, H, Ω) is a Cayley digraph and denoted by Cay(G, Ω).Consider the action of G/H G on G/H by left multiplication.If K the electronic journal of combinatorics 22(1) (2015), #P1.76 is a subgroup of G which contains H, then K/H is coincident with a block for G/H G .It follows that K induces a quotient digraph Γ K of Γ: the vertices set of Γ K is the system of blocks containing K/H, and for any two such blocks ∆ 1 and ∆ 2 , (∆ 1 , ∆ 2 ) is an arc of Γ K if and only if there exist g 1 H ∈ ∆ 1 and g 2 H ∈ ∆ 2 such that (g 1 H, g 2 H) is an arc of Γ.The following proposition gives a representation of Γ K .Proposition 3.3.Let G be a finite group, H a subgroup of G, and Γ = Cos(G, H, Ω) a coset digraph on G/H.Let K be a subgroup of G which contains H, and Γ K the quotient digraph of Γ induced by K.
for all g 1 , g 2 ∈ G, we obtain a one to one mapping ) is an arc of Γ, and then (g By the above discussions, we get Γ K ∼ = Cos(G, K, Λ).
For a finite group G and a subgroup H of G, we use (x In particular, if Γ = Cos(G, H, Ω) is a coset digraph on G/H and Hx i H ∈ Ω for all i = 1, 2, . . ., n, then (x 1 , x 2 , . . ., x n ) • H is a walk visiting the vertices of Γ in the order The following proposition is apparent and we omit the proof.

Main results
Throughout this section, we assume p is a prime.Let Γ be a connected vertex-transitive digraph of order a power of p.Then, by Proposition 3. (ii) [G, w] is an elementary abelian group; the electronic journal of combinatorics 22(1) (2015), #P1.76 Proof.Let (x 1 , x 2 , . . ., x u ) • K be a Hamilton cycle of Σ.Note that [G, w] < K G .By Proposition 3.4, we can assume that Hx i H ∈ Ω for all i = 1, . . ., u, and furthermore . By the hamiltonicity of (x 1 , x 2 , . . ., x u ) • K, for any g ∈ G − K, there exists a positive integer i < u such that Proposition 3.4 (ii).Then, by Proposition 3.4 (i) and (iii), we get a Hamilton cycle (x 1 , . . ., .
, we can eliminate all the factors of x 1 • • • x u by finite steps of replacements.
It follows that we can assume In the following discussions, we always assume that (

Proof of (i).
For the case that [G, w] is generated by only one element.Set [G, w] = y where y = [g, w] for some g ∈ G−K.As in the above paragraph, set g −1 = x 1 , . . ., x i for some positive integer i < u.Then we get a Hamilton cycle (x 1 , . . ., Now we deal with the case when [G, w] is generated by two elements.By Lemma 2.2, let [G, w] = y, z where y = [g, w] and z = [x j , w] for some g ∈ G − K and 1 j u.Since x 1 • • • x u = 1, by Proposition 3.4 (iv), we have that (x j+1 , . . ., x u , x 1 , . . ., x j ) • K is also a Hamilton cycle of Σ. Therefore we can assume j = u without loss of generality.Since g ∈ G \ K, again we set g −1 = x 1 • • • x i for some 1 i < u.Now we have [G, w] = y, z = y, yz −1 and get two Hamilton cycles of Σ: By the main result of Witte [21], the Cayley digraph Cay([G, w], {y, yz −1 }) is Hamiltonian.Then, by Lemma 3.5, Γ is Hamiltonian.

Proof of (ii).
Assume [G, w] is an elementary abelian group of order p d for some integer d.Recall that (x 1 , x 2 , . . ., x u )•K is a Hamilton cycle of Σ such that x 1 x 2 • • • x u = 1 and Hx i H ∈ Ω for all i = 1, . . ., u. Set g −1 i = x 1 • • • x i for all i = 1, . . ., u.Let u 1 be the smallest integer such that [g u 1 , w] = 1, and let u i be the smallest integer such that [g u i , w] / ∈ [g u 1 , w], . . ., [g u i−1 , w] for any i 2. Then 1 (f) for any 0 To complete the proof, we need to prove that (b i,1 , . . ., b i,p i u ) p • H are cycles of Γ for all i = 0, 1, . . ., d − 1.
Suppose to the contrary that there is an integer j d − 1 such that (b i,1 , . . ., b i,p i u ) p • H are cycles of Γ for all i ∈ {0, 1, . . ., j −1}, but (b j,1 , . . ., b j,p j u ) p •H is not a cycle of Γ.Then there exist distinct integer pairs (r, s) and (e, f ) with 0 r, e p − 1 and 1 s, f p j u such that (y Proof of Theorem 1.2.Let Γ be a connected digraph of which the automorphism group contains a finite vertex-transitive subgroup G of prime power order. (i) Suppose that G is generated by two elements.By Lemma 2.4, we can assume that G is a minimum vertex-transitive p-subgroup of Aut(Γ) without loss of any generality.Let H be a vertex stabilizer in G. Assume that the order of Γ is p n .Then |G : H| = p n , H ∩ Z(G) = {1}, H < Φ(G) and Γ admits a representation Γ = Cos(G, H, Ω) for some Ω ⊆ H\G/H − {H}.
We proceed the remainder proof by induction on the order of Γ. Assume that the assertion holds for any such vertex-transitive digraph of order a power of p smaller than

Lemma 2 . 1 (Lemma 2 . 2 .
[19]).Let G be a finite p-group and N be a normal subgroup of G. Then N ∩ Z(G) = {1} if and only if N = {1}.Let G be a finite p-group and w be an element of G with centralizer C G (w) < G. Let X be a generating set of G. Then [G, w ] = [G, w].Furthermore, there exists a minimal generating set {[x, w], [g 1 , w], . . ., [g d−1 , w]} of [G, w] where x ∈ X and g i ∈ G − [G, w]C G (w) for all i = 1, . . ., d − 1.
p n .Let w be an element of order p in the center of H such that [G, w] ∩ H = {1}.Set K = [G, w]H and Λ = {KxK | HxH ∈ Ω}.By Proposition 3.1, the automorphism group of Σ = Cos(G, K, Λ) contains a vertex-transitive subgroup isomorphic to G/K G .Since G is generated by two elements, we have (G/K G ) can be generated by two elements.By induction hypothesis, Σ is Hamiltonian.By Lemma 2.4, [G, w] can be generated by at most two elements.It follows from Lemma 4.1 (i) that Γ is Hamiltonian.the electronic journal of combinatorics 22(1) (2015), #P1.76