On Isomorphisms of Vertex-transitive Graphs

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.


Introduction
Throughout this paper, by (di)graph we mean finite digraph without loops or multiedges, and all groups are assumed to be finite. Deciding whether two graphs are isomorphic is fundamental for the study of graphs, especially for determining isomorphism classes of graphs. A graph is said to be G-vertex-transitive if the subgroup G of its full automorphism group acts transitively on the vertex set. One would expect to determine the isomorphisms between two G-vertex-transitive graphs by the information of the group G. For Cayley graphs, such an approach was initiated by a conjecture ofÁdám in 1967 [1], and has been extensively studied over the past decades, see for example [2,4,8,11,13,21,22,23,24] and more references listed in the survey [15]. Since a large number of vertex-transitive graphs are not Cayley graphs, it is natural to extend the study from Cayley graphs to vertextransitive graphs. The isomorphism problem for metacirculants (not necessarily Cayley graphs) has been considered by Dobson [9].
To be precise, we need the concept of coset graphs. Let Γ = (V, E) be a G-vertextransitive graph, α be a vertex of Γ and S be the set of elements of G which maps α to its (out) neighbors. Then Γ is uniquely determined by the triple (G, G α , S) in the following sense: writing H = G α and identifying the vertex set V with the set [G:H] of right cosets of H in G, the action of G on V is equivalent to the action of G on [G:H] by right multiplication. In particular, if α is identified with H ∈ [G:H] then the neighborhood Γ(α) consists of Hg with g ∈ S, and moreover, Hx ∼ Hy if and only if yx −1 ∈ HSH. This defines a coset graph representation of Γ, denoted by Cos(G, H, HSH). Note that H is core-free in G (that is, H does not contain any nontrivial normal subgroup of G) since G is a transitive permutation group on V , and S ⊆ G \ H since Γ has no loops. Clearly, for any automorphism τ ∈ Aut(G) we have Cos(G, H, HSH) ∼ = Cos(G, H τ , H τ S τ H τ ). Definition 1.1. The G-vertex-transitive graph Γ = Cos(G, H, HSH) is called a GIgraph ('GI' stands for 'Group automorphism inducing Isomorphism') of G if for any graph Σ = Cos(G, H, HT H) with T ⊆ G \ H and Γ ∼ = Σ, there exists τ ∈ Aut(G) such that H τ = H and HS τ H = HT H. A group G is called a DGI-group ('D' emphasizes that our graph may be Directed) if each G-vertex-transitive graph is a GI-graph of G. A group G is called a GI-group if each undirected G-vertex-transitive graph is a GI-graph of G.
Note that Cos(G, 1, S) is a Cayley graph of G, and the GI-graphs of G with H = 1 are exactly the so called CI-graphs of G. If each Cayley graph of G is a CI-graph of G, then G is called a DCI-group. If each undirected Cayley graph of G is a CI-graph of G, then G is called a CI-group. Clearly, a DGI-group is necessarily a GI-group, and a DGI-group (GI-group) is necessarily a DCI-group (CI-group). A small list of candidates for DCI and CI-groups has been obtained, through the effort of many mathematicians, see [15,Theorem 8.7] and [17,Corollary 1.5]. However, determining which groups in the list are indeed DCI or CI-groups is not easy and largely open. As being DGI-groups (GI-groups) is more restrictive than being DCIgroups (CI-groups), the explicit list of DGI-groups (GI-groups) would be smaller than that of DCI-groups (CI-groups). Thus we propose the problem: Problem 1.2. Classify the finite DGI-groups (GI-groups).
In the literature, a crucial step to solve a conjecture of Babai and Frankel [5] stating that CI-groups are solvable was to determine whether there exists a non-CI-Cayley graph of A 5 . After 20 years since Babai-Frankel conjecture was posed, a non-CI-Cayley graph of A 5 of valency 29 was constructed by Li [14], thus completing the proof of the conjecture. Although some other non-CI-Cayley graphs of A 5 was later constructed in [6,26], Li's graph is the only known connected symmetric non-CI-graph of A 5 yet. Here a graph Γ is called G-symmetric for some G Aut(Γ) if G acts transitively on the arc set of Γ, and Γ is simply called symmetric if Γ is Aut(Γ)-symmetric. In general, constructing connected symmetric non-GI-graphs is not easy. Due to the significance of non-CI-Cayley graphs of A 5 , one would ask: Problem 1.3. Does there exist a connected symmetric non-GI-graph of A 5 other than Li's?
The layout of this paper is as follows. After this introduction, we give the criterion for GI-graph in Section 2, which enables us to construct GI and non-GI-graphs, respectively, in Section 3. In particular, we prove the theorem below by Example 3.3. Then in Section 4 we establish some results on Problem 1.2. The final section is devoted to Problem 1.3, where it is shown that a connected A 5 -symmetric graph is necessarily GI if its full automorphism group is almost simple or vertex-primitive.

Criterion for GI-graph
As mentioned in the introduction, G-vertex-transitive graphs can be represented as coset graphs of G: for a core-free subgroup H of G and a subset S ⊆ G \ H, define Γ = Cos(G, H, HSH) to be the graph with vertex set V := [G:H] such that Hx ∼ Hy if and only if yx −1 ∈ HSH. For any g ∈ G, the right multiplication of g on the cosets in [G:H] gives an element of Sym(V ), denoted byĝ. Moreover, denoteĜ = {ĝ | g ∈ G}. (The reader should be aware that this also depends on the subgroup H although theˆsymbol does not indicate.) We list here some basic facts concerning coset graphs. In this case, the valency of Γ is equal to |H|/|H g ∩ H|.
Let X and Y be permutation groups on Ω and ∆, respectively. We say that X is permutation isomorphic to Y if there exist a bijection σ : Ω → ∆ and a group isomorphism ϕ : X → Y such that (α x ) σ = (α σ ) ϕ(x) for any α ∈ Ω and x ∈ X. The following folklore theorem is an extension of the criterion for a Cayley graph to be a CI-graph [3,4] to those vertex-transitive graphs. The proof goes along the same lines as that of the CI-graph criterion, so we omit it. Based on Theorem 2.2, we establish a sufficient condition on GI-graphs as follows.
Theorem 2.3. Suppose that G is a finite group of odd order, p is the smallest prime divisor of |G|, Γ is a G-vertex-transitive graph and A is the full automorphism group of Γ. For any vertex α of Γ, if gcd(|G|, |A α |) = 1, then Γ is a GI-graph of G. In particular, if Γ is connected of valency less than p, then Γ is a GI-graph of G.
Proof. Since G is transitive on the vertices of Γ we have A = GA α . Assume that gcd(|G|, |A α |) = 1. Then G is a Hall π-subgroup of A, where π is the set of the prime divisors of |G|. Note that π is a set of odd primes as |G| is odd. Then for any σ ∈ Sym(V ) with G σ A, one deduces from [12, Theorem A] that G and G σ are conjugate in A as they are Hall π-subgroups of A. Hence according to Theorem 2.2, Γ is a GI-graph of G. Now assume that Γ is connected of valency less than p. It suffices to prove that gcd(|G|, |A α |) = 1. Suppose for a contradiction that there exists a prime number r dividing gcd(|G|, |A α |) and that R is a Sylow r-subgroup of A α . Since Γ is connected, there exist a neighbor β of α and an element x ∈ R such that β x = β. It follows that the orbit of β under x has length at least r, contrary to our assumption that the valency of Γ is less than p r.
Below is a necessary condition for GI-graphs.

Examples
First of all, the complete graphs and their complements are GI-graphs. We regard them as trivial GI-graphs. An observation of [16] says that every finite group of order greater than two has non-trivial CI-graphs. Thus we know that every finite group of order greater than two has non-trivial GI-graphs. Given a finite group G of odd order, recall that as Theorem 2.3 asserts, every G-vertex-transitive graph Γ of valency less than the smallest prime divisor of |G| is a GI-graph of G. This provides us with more examples of GI-graphs.
A 2-arc of a graph Γ is a triple (α, β, γ) of pairwise distinct vertices of Γ such that α ∼ β and β ∼ γ. A graph is said to be (G, 2)-arc-transitive for some G Aut(Γ) if G acts transitively on the set of 2-arcs. Recall that the socle of a group G is the product of all its minimal normal subgroups, denoted by Soc(G). We call a group almost simple if its socle is nonabelian simple. It is readily seen that the almost simple groups with a given socle T are precisely those groups G satisfying T G Aut(T ), whence G/T is a solvable group by the well-known Schreier conjecture. The next example follows from [10, Theorem 1.3] and the criteria in Theorem 2.2.
Utilizing Theorem 2.4, we are able to construct some disconnected non-GI-graphs.  g ∈ G such that G α , g = G and |G α |/|G g α ∩ G α | = 12. Let Γ = Cos(G, G α , G α gG α ). Then Γ is a connected G-symmetric and G-vertex-primitive graph of order |Ω| = 40 and valency 12. Moreover, G has two conjugacy classes of subgroups isomorphic to S 6 , fused in X, and the groups in both conjugacy classes are transitive on Ω. Take P to be a group in one of these two conjugacy classes, and Q be a group in the other. Since P and Q are conjugate in X, they are permutation isomorphic. We claim that Γ is a non-Cayley non-GI-graph of P .
In fact, the conclusion that Γ is not a Cayley graph is obvious as |P | = 40. Denote Y = Aut(Γ). In light of Theorem 2.2, it suffices to show that P and Q are not conjugate in Y . If Soc(Y ) = Soc(G) = PSU 4 (2), then Y = G since G Y and G = Aut(PSU 4 (2)), which indicates that P and Q are not conjugate in Y , as desired. Assume next that Soc(G) = Soc(Y ). Then there exists a subgroup H of Y such that Soc(G) = Soc(H) and G is maximal in H. By [18], either G is maximal in A 40 G, or H is almost simple with socle PSL 4 (3). For the latter, H α = C 3 3 :PSL 3 (3) or C 3 3 :(PSL 3 (3) × C 2 ) since H is primitive on 40 points, but then H α does not have a subgroup of index 12, violating the requirement that Γ is H-symmetric as Γ is G-symmetric. Therefore, G is maximal in A 40 G, and hence Y ∩ A 40 G = G or A 40 G. Because Γ is not a complete graph, we have Y A 40 . It follows that Y ∩ A 40 G = G. If G A 40 , then A 40 G = S 40 and thus Y = Y ∩ S 40 = G, contrary to our assumption that Soc(G) = Soc(Y ). Consequently, G A 40 , and so G has index two in Y . Since Soc(G) is a minimal normal subgroup of Y and Soc(G) = Soc(Y ), we conclude that Y has a minimal normal subgroup other than Soc(G), say N. Viewing that N G, we have N = C 2 and Y = G × N. Hence P and Q are not conjugate in Y , proving our claim.

GI-groups
A group G is said to be Hamiltonian if every subgroup of G is normal. It is obvious that abelian groups are all Hamiltonian, but the converse is not true (for instance, the quaternion group Q 8 is Hamiltonian but not abelian). Proof. For any coset graph Cos (G, H, HSH) of G, the condition that H is core-free in G forces H = 1 since G is Hamiltonian. This means that each coset graph of G is a Cayley graph of G. Hence the concepts of DGI (GI) and DCI (CI) coincide.
Lemma 4.1 immediately shows up some DGI-groups (GI-groups) from the list of DCI-groups (CI-groups). For example, since the groups C k , C 2k and C 4k , where k is odd square-free, are Hamiltonian and DCI [19,20] simultaneously, we know that they are DGI-groups.  [4]. Hence we assume that H = 1. As H is core-free in G, we conclude that H = C 2 and |V | = |G|/|H| = p. Let X be a subgroup of Aut(Γ) such that X = ϕ −1Ĝ ϕ for some ϕ ∈ Sym(V ), and Y be a Sylow p-subgroup of X. Then Y = C p , and by the Sylow theorem, there exists τ ∈ Aut(Γ) such that Y = τ −1N τ . It derives from X = ϕ −1Ĝ ϕ that Y = ϕ −1N ϕ. Thereby we obtain ϕ −1N ϕ = τ −1N τ , or equivalently, ϕτ −1 ∈ N Sym(V ) (N ). Note that N Sym(V ) (N ) N Sym(V ) (Ĝ). This leads to ϕτ −1 ∈ N Sym(V ) (Ĝ) and thus Now appealing Theorem 2.2 we know that Γ is a DGI-graph of G, which proves the lemma.
We close this section with a theorem stating that being DGI-groups (GI-groups) is inherited by subgroups. Proof. Suppose that G is a DGI-group (GI-group). Let Γ = Cos(H, K, KSK) and Σ = Cos(H, K, KT K) be two isomorphic (undirected) coset graphs of H, where K is a core-free subgroup of H and S, T are subsets of H \ K. Clearly, K is also core-free in G. Without loss of generality we assume that S and T are both unions of double cosets of K.
First assume that K, S = H. Then Γ is connected, and so is Σ since Γ ∼ = Σ. Noticing that Cos(G, K, KSK) and Cos(G, K, KT K) are |G|/|H| copies of Γ and Σ, respectively, we have Cos(G, K, KSK) ∼ = Cos(G, K, KT K). Then as G is a DGIgroup (GI-group), there exists τ ∈ Aut(G) such that K τ = K and KS τ K = KT K. It follows that Hence Cos(G, K, KSK) ∼ = Cos(G, K, KT K), and there exists τ ∈ Aut(G) such that K τ = K and KS τ K = KT K since G is a DGI-group (GI-group). As a consequence, showing that τ induces an automorphism of H. Moreover, Thereby we conclude that there always exists τ ∈ Aut(G) such that K τ = K and KS τ K = KT K. This implies that H is a DGI-group (GI-group).

GI-properties of connected A 5 -symmetric graphs
For a group G, the expression G = HK with proper subgroups H and K of G is called a factorization of G. The lemma below can be read off from [25].
Theorem 5.2. Let G = A 5 and Γ be a connected symmetric coset graph of G. If Aut(Γ) is almost simple, then Γ is a GI-graph of G.
Proof. Suppose on the contrary that Γ is not a GI-graph of G. By Theorem 2.2, Aut(Γ) = A 5 or S 5 . Let α be a vertex of Γ, X = Aut(Γ) and T be the socle of X. Then T = A 5 , X =ĜX α andĜ ∩ T is a normal subgroup ofĜ. It follows that G ∩ T = 1 orĜ sinceĜ ∼ = G is simple. IfĜ ∩ T = 1, then A 5 =Ĝ ∼ =ĜT /T X/T , contrary to Schreier conjecture. HenceĜ ∩ T =Ĝ, or equivalently,Ĝ T . Thereby we have the factorization T =ĜT α , which is classified in Lemma 5.1. If T acts 2-transitively on [T :T α ], then Γ is the complete graph on n vertices and X = S n , which implies that Γ is a GI-graph of G by Theorem 2.2, contrary to our assumption. Consequently, T does not act 2-transitively on [T :T α ], and so we deduce from Lemma 5.1 that one of the following three cases appears: (i) T = A 6 and T α = A 4 or S 4 ; (ii) T = PSL 2 (11) and T α = C 11 ; (iii) T = PSL 2 (29) and T α = C 29 :C 7 . First suppose that case (i) appears. As X should have at least two conjugacy classes of subgroups isomorphic to A 5 by Theorem 2.2, the only possibilities for X are A 6 and S 6 . If X = A 6 , then A 4 X α S 4 and hence X has only one conjugacy class of vertex-transitive subgroups isomorphic to A 5 , which leads to a contradiction that Γ is a GI-graph of G by Theorem 2.2. If X = S 6 , then A 4 X α S 4 × S 2 and hence X has at most one conjugacy class of vertex-transitive subgroups isomorphic to A 5 , again a contradiction.
Next suppose that case (ii) appears. As X should have at least two conjugacy classes of subgroups isomorphic to A 5 by Theorem 2.2, it derives that X = PSL 2 (11) and so X α = C 11 . Since Γ is symmetric, there exists g ∈ X \ X α such that Γ ∼ = Cos(X, X α , X α gX α ). Let Y = PGL 2 (11) > X. One can take an involution t ∈ N Y (X α ) such that X α g t X α = X α gX α . Let H = X α , t = X α t , and note t ∈ X. Due to X α g t X α = X α gX α we have tgt ∈ X α gX α . For any h 1 , h 2 ∈ H, if h 1 gh 2 ∈ X, then either h 1 , h 2 ∈ X α or h 1 , h 2 / ∈ X α . Further, if h 1 , h 2 / ∈ X α , then h 1 t, th 2 ∈ X α and so h 1 gh 2 = (h 1 t)tgt(th 2 ) ∈ X α gX α . This shows that (HgH) ∩ X = X α gX α . Then the map is a graph isomorphism from Cos(X, X α , X α gX α ) to Cos (Y, H, HgH). However, this implies that Y = PGL 2 (11) is a group of automorphisms of Γ ∼ = Cos(Y, H, HgH), contrary to the condition that Aut(Γ) = X = PSL 2 (11). Finally suppose that case (iii) appears. As X should have at least two conjugacy classes of subgroups isomorphic to A 5 by Theorem 2.2, it derives that X = PSL 2 (29) and so X α = C 29 :C 7 . Thus Γ has order |X|/|X α | = 60. Take β to be a neighbor of α in Γ. Since Γ is X-symmetric, |X α |/|X αβ | equals the valency of Γ, which is less than 60. Hence X αβ = C 29 or C 7 . If X αβ = C 29 , then X αβ fixes each neighbor of α since X αβ ⊳ X α . This will cause a contradiction that X αβ = 1 due to the connectivity of Γ. Consequently, X αβ = C 7 and Γ is of valency |X α |/|X αβ | = 29. Note that X has a maximal subgroup K = C 29 :C 14 containing X α such that X acts 2-transitively on [X:K]. We deduce that X has an imprimitive block system B = {V 1 , V 2 , . . . , V 30 } on the vertex set of Γ, where |V 1 | = · · · = |V 30 | = 2, and the quotient graph of Γ with respect to the partition B is complete. Moreover, denoted by V k the block in B such that α ∈ V k , the action of X α on B \ {V k } is transitive. Therefore, distinct neighbors of α lie in distinct blocks in B, and so the induced graph Γ[V i ∪ V j ] is a perfect matching for any two blocks V i , V j in B. Now we see that interchanging the two vertices in each V i is an automorphism of Γ. Then the kernel of X acting on B is non-trivial, contrary to the fact that X = PSL 2 (29) is simple. Proof. Suppose on the contrary that Γ is not a GI-graph of G. A subgroup of G has order 1, 2, 3, 4, 5, 6, 10 or 12, whence the order of Γ is 60, 30, 20, 15, 12, 10, 6 or 5. In view of Theorem 5.2 we may assume that X := Aut(Γ) is not almost simple. Further, Theorem 2.2 requires X to have at least two conjugacy classes of transitive subgroups isomorphic to A 5 . Then by [7, Appendix B], X = Hol(G) or Soc(Hol(G)), where the symbol Hol denotes the holomorph of a group. Let N = Soc(Hol(G)) = G × G and D be the full diagonal subgroup of N. Then the vertex set of Γ can be viewed as [N:D], with the action of N by right multiplication. Moreover, let t be the permutation D(g 1 , g 2 ) → D(g 2 , g 1 ) for (g 1 , g 2 ) ∈ N on [N:D], α = D ∈ [N:D], H = X α , t and Y = X, t . Clearly, t is an involution in Y \ X. Since Γ is symmetric, there exists g ∈ X \ X α such that Γ ∼ = Cos(X, X α , X α gX α ).
First suppose that g ∈ Soc(Hol(G)). Then g = (g 1 , g 2 ) acts on [N:D] by right multiplication for some g 1 , g 2 ∈ G. Take h 1 ∈ G such that (g 1 g −1 2 ) h 1 = (g 1 g −1 2 ) −1 and write h 2 = g −1 1 h −1 1 g 2 . For i = 1, 2 set x i to be the right multiplication of (h i , h i ) on [N:D]. It is routine to verify that tgt = x 1 gx 2 ∈ X α gX α . Hence (HgH) ∩ X = X α gX α , and thus the map X α x → Hx for x ∈ X is a graph isomorphism from Cos(X, X α , X α gX α ) to Cos(Y, H, HgH). However, this implies that Y is a group of automorphisms of Γ ∼ = Cos(Y, H, HgH), contrary to the condition that Aut(Γ) = X < Y .
Consequently, (HgH) ∩ X = X α gX α , and so the map X α x → Hx for x ∈ X is a graph isomorphism from Cos(X, X α , X α gX α ) to Cos(Y, H, HgH). However, this implies that Y is a group of automorphisms of Γ ∼ = Cos(Y, H, HgH), contrary to the condition that Aut(Γ) = X < Y .