Vertex-Transitive and Cayley Graphs

Mingyao Xu Peking University

January 18, 2011

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 1 / 16 DEFINITIONS

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 2 / 16 G = Aut(Γ). Transitivity of graphs: vertex-, edge-, arc-transitive graphs. Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)).

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 Automorphism group G = Aut(Γ). Transitivity of graphs: vertex-, edge-, arc-transitive graphs. Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)). Automorphisms

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 Transitivity of graphs: vertex-, edge-, arc-transitive graphs. Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)). Automorphisms Automorphism group G = Aut(Γ).

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)). Automorphisms Automorphism group G = Aut(Γ). Transitivity of graphs: vertex-, edge-, arc-transitive graphs.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)). Automorphisms Automorphism group G = Aut(Γ). Transitivity of graphs: vertex-, edge-, arc-transitive graphs. Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 DEFINITIONS

Graphs Γ = (V (Γ),E(Γ)). Automorphisms Automorphism group G = Aut(Γ). Transitivity of graphs: vertex-, edge-, arc-transitive graphs. Cayley graphs: Let G be a finite group and S a subset of G not containing the identity element 1. Assume S−1 = S. We define the Cayley graph Γ = Cay(G, S) on G with respect to S by

V (Γ) = G,

E(Γ) = {(g, sg) g ∈ G, s ∈ S}.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 3 / 16 Cayley graphs

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 4 / 16 Let Γ = Cay(G, S) be a Cayley graph on G with respect to S. (1) Aut(Γ) contains the right regular representation R(G) of G, so Γ is vertex-transitive. (2) Γ is connected if and only if G = hSi. A graph Γ = (V,E) is a Cayley graph of a group G if and only if Aut(Γ) contains a regular subgroup isomorphic to G. Above Proposition =⇒ Cayley graphs are just those vertex-transitive graphs whose full automorphism groups have a regular subgroup.

Properties of Cayley graphs

Cayley graphs

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 5 / 16 A graph Γ = (V,E) is a Cayley graph of a group G if and only if Aut(Γ) contains a regular subgroup isomorphic to G. Above Proposition =⇒ Cayley graphs are just those vertex-transitive graphs whose full automorphism groups have a regular subgroup.

Cayley graphs

Properties of Cayley graphs Let Γ = Cay(G, S) be a Cayley graph on G with respect to S. (1) Aut(Γ) contains the right regular representation R(G) of G, so Γ is vertex-transitive. (2) Γ is connected if and only if G = hSi.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 5 / 16 Cayley graphs

Properties of Cayley graphs Let Γ = Cay(G, S) be a Cayley graph on G with respect to S. (1) Aut(Γ) contains the right regular representation R(G) of G, so Γ is vertex-transitive. (2) Γ is connected if and only if G = hSi. A graph Γ = (V,E) is a Cayley graph of a group G if and only if Aut(Γ) contains a regular subgroup isomorphic to G. Above Proposition =⇒ Cayley graphs are just those vertex-transitive graphs whose full automorphism groups have a regular subgroup.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 5 / 16 Cayley graphs

Properties of Cayley graphs Let Γ = Cay(G, S) be a Cayley graph on G with respect to S. (1) Aut(Γ) contains the right regular representation R(G) of G, so Γ is vertex-transitive. (2) Γ is connected if and only if G = hSi. A graph Γ = (V,E) is a Cayley graph of a group G if and only if Aut(Γ) contains a regular subgroup isomorphic to G. Above Proposition =⇒ Cayley graphs are just those vertex-transitive graphs whose full automorphism groups have a regular subgroup.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 5 / 16 Outline of a proof: If it is Cayley, then the group has order 10. There are two non-isomorphic groups: cyclic and dihedral. The of Cayley graphs on abelian groups are 3 or 4. So G = D . 10 5 2 −1 G = ha, b a = b = 1, bab = a i. S = {a, a−1, b} or three involutions. For the former, since abab is a cyclic of size 4, this is not the case. For the latter, the product of any two involutions is 1 or of order 5, a contradiction. Note: For a positive integer n, if every transitive group has a regular subgroup then every vertex-transitive graph is Cayley. (Example: n = p, a prime.)

The is not Cayley

Cayley graphs

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 6 / 16 Note: For a positive integer n, if every transitive group has a regular subgroup then every vertex-transitive graph is Cayley. (Example: n = p, a prime.)

Cayley graphs

The Petersen graph is not Cayley Outline of a proof: If it is Cayley, then the group has order 10. There are two non-isomorphic groups: cyclic and dihedral. The girth of Cayley graphs on abelian groups are 3 or 4. So G = D . 10 5 2 −1 G = ha, b a = b = 1, bab = a i. S = {a, a−1, b} or three involutions. For the former, since abab is a cyclic of size 4, this is not the case. For the latter, the product of any two involutions is 1 or of order 5, a contradiction.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 6 / 16 Cayley graphs

The Petersen graph is not Cayley Outline of a proof: If it is Cayley, then the group has order 10. There are two non-isomorphic groups: cyclic and dihedral. The girth of Cayley graphs on abelian groups are 3 or 4. So G = D . 10 5 2 −1 G = ha, b a = b = 1, bab = a i. S = {a, a−1, b} or three involutions. For the former, since abab is a cyclic of size 4, this is not the case. For the latter, the product of any two involutions is 1 or of order 5, a contradiction. Note: For a positive integer n, if every transitive group has a regular subgroup then every vertex-transitive graph is Cayley. (Example: n = p, a prime.)

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 6 / 16 Cayley graphs

The Petersen graph is not Cayley Outline of a proof: If it is Cayley, then the group has order 10. There are two non-isomorphic groups: cyclic and dihedral. The girth of Cayley graphs on abelian groups are 3 or 4. So G = D . 10 5 2 −1 G = ha, b a = b = 1, bab = a i. S = {a, a−1, b} or three involutions. For the former, since abab is a cyclic of size 4, this is not the case. For the latter, the product of any two involutions is 1 or of order 5, a contradiction. Note: For a positive integer n, if every transitive group has a regular subgroup then every vertex-transitive graph is Cayley. (Example: n = p, a prime.)

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 6 / 16 A Question

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 7 / 16 Question: NR = NC?. Answer: NR % NC. For example, 12 ∈/ N C, but 12 ∈ N R since M11, acting on 12 points, has no regular subgroup.

Exercise: 6 is the smallest number in NR\NC since A6 has no regular subgroups.

Let

NR = {n ∈ N there is a transitive group of n without a regular subgroup}

NC = {n ∈ N there is a vertex-transitive graph of order n which is non-Cayley}

Then NR k NC.

A Question

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 8 / 16 Answer: NR % NC. For example, 12 ∈/ N C, but 12 ∈ N R since M11, acting on 12 points, has no regular subgroup.

Exercise: 6 is the smallest number in NR\NC since A6 has no regular subgroups.

A Question

Let

NR = {n ∈ N there is a transitive group of degree n without a regular subgroup}

NC = {n ∈ N there is a vertex-transitive graph of order n which is non-Cayley}

Then NR k NC. Question: NR = NC?.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 8 / 16 Exercise: 6 is the smallest number in NR\NC since A6 has no regular subgroups.

A Question

Let

NR = {n ∈ N there is a transitive group of degree n without a regular subgroup}

NC = {n ∈ N there is a vertex-transitive graph of order n which is non-Cayley}

Then NR k NC. Question: NR = NC?. Answer: NR % NC. For example, 12 ∈/ N C, but 12 ∈ N R since M11, acting on 12 points, has no regular subgroup.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 8 / 16 A Question

Let

NR = {n ∈ N there is a transitive group of degree n without a regular subgroup}

NC = {n ∈ N there is a vertex-transitive graph of order n which is non-Cayley}

Then NR k NC. Question: NR = NC?. Answer: NR % NC. For example, 12 ∈/ N C, but 12 ∈ N R since M11, acting on 12 points, has no regular subgroup.

Exercise: 6 is the smallest number in NR\NC since A6 has no regular subgroups.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 8 / 16 A Question

Let

NR = {n ∈ N there is a transitive group of degree n without a regular subgroup}

NC = {n ∈ N there is a vertex-transitive graph of order n which is non-Cayley}

Then NR k NC. Question: NR = NC?. Answer: NR % NC. For example, 12 ∈/ N C, but 12 ∈ N R since M11, acting on 12 points, has no regular subgroup.

Exercise: 6 is the smallest number in NR\NC since A6 has no regular subgroups.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 8 / 16 (Maruˇsiˇcıˇc)Any transitive group G of degree p2 on Ω has a regular subgroup, i.e., p2 ∈/ N R. Outline of a proof: Take a minimal transitive subgroup P of G. Then P is a p-group and every maximal subgroup M of P is intran- 2 2 sitive. For any α ∈ Ω, we have |Pα| = |P |/p and |Mα| > |M|/p , so Mα = Pα. It follows that Pα ≤ M and hence Pα ≤ Φ(P ). If |P : Φ(P )| = p, then P is cyclic and is regular. If |P : Φ(P )| = p2, then Pα = Φ(P ). Since Φ(P ) is normal in P and Pα is core-free, we ∼ 2 have Pα = 1 and hence P = Zp is regular.

p2 ∈/ N C

A Question

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 9 / 16 Outline of a proof: Take a minimal transitive subgroup P of G. Then P is a p-group and every maximal subgroup M of P is intran- 2 2 sitive. For any α ∈ Ω, we have |Pα| = |P |/p and |Mα| > |M|/p , so Mα = Pα. It follows that Pα ≤ M and hence Pα ≤ Φ(P ). If |P : Φ(P )| = p, then P is cyclic and is regular. If |P : Φ(P )| = p2, then Pα = Φ(P ). Since Φ(P ) is normal in P and Pα is core-free, we ∼ 2 have Pα = 1 and hence P = Zp is regular.

A Question

p2 ∈/ N C (Maruˇsiˇcıˇc)Any transitive group G of degree p2 on Ω has a regular subgroup, i.e., p2 ∈/ N R.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 9 / 16 A Question

p2 ∈/ N C (Maruˇsiˇcıˇc)Any transitive group G of degree p2 on Ω has a regular subgroup, i.e., p2 ∈/ N R. Outline of a proof: Take a minimal transitive subgroup P of G. Then P is a p-group and every maximal subgroup M of P is intran- 2 2 sitive. For any α ∈ Ω, we have |Pα| = |P |/p and |Mα| > |M|/p , so Mα = Pα. It follows that Pα ≤ M and hence Pα ≤ Φ(P ). If |P : Φ(P )| = p, then P is cyclic and is regular. If |P : Φ(P )| = p2, then Pα = Φ(P ). Since Φ(P ) is normal in P and Pα is core-free, we ∼ 2 have Pα = 1 and hence P = Zp is regular.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 9 / 16 A Question

p2 ∈/ N C (Maruˇsiˇcıˇc)Any transitive group G of degree p2 on Ω has a regular subgroup, i.e., p2 ∈/ N R. Outline of a proof: Take a minimal transitive subgroup P of G. Then P is a p-group and every maximal subgroup M of P is intran- 2 2 sitive. For any α ∈ Ω, we have |Pα| = |P |/p and |Mα| > |M|/p , so Mα = Pα. It follows that Pα ≤ M and hence Pα ≤ Φ(P ). If |P : Φ(P )| = p, then P is cyclic and is regular. If |P : Φ(P )| = p2, then Pα = Φ(P ). Since Φ(P ) is normal in P and Pα is core-free, we ∼ 2 have Pα = 1 and hence P = Zp is regular.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 9 / 16 (Maruˇsiˇcıˇc) p3 ∈/ N C. p3 ∈ N R (For p > 2). Let G be the following group of order p4

2 p p p p G = ha, b a = b = c = 1, [a, b] = c, [c, a] = a , [c, b] = 1i.

Let H = hci. Consider the transitive permutation representation ϕ of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree p3, and ϕ(G) has no regular subgroups.

p3 ∈ N R \ N C

A Question

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 10 / 16 p3 ∈ N R (For p > 2). Let G be the following group of order p4

2 p p p p G = ha, b a = b = c = 1, [a, b] = c, [c, a] = a , [c, b] = 1i.

Let H = hci. Consider the transitive permutation representation ϕ of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree p3, and ϕ(G) has no regular subgroups.

A Question

p3 ∈ N R \ N C (Maruˇsiˇcıˇc) p3 ∈/ N C.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 10 / 16 A Question

p3 ∈ N R \ N C (Maruˇsiˇcıˇc) p3 ∈/ N C. p3 ∈ N R (For p > 2). Let G be the following group of order p4

2 p p p p G = ha, b a = b = c = 1, [a, b] = c, [c, a] = a , [c, b] = 1i.

Let H = hci. Consider the transitive permutation representation ϕ of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree p3, and ϕ(G) has no regular subgroups.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 10 / 16 A Question

p3 ∈ N R \ N C (Maruˇsiˇcıˇc) p3 ∈/ N C. p3 ∈ N R (For p > 2). Let G be the following group of order p4

2 p p p p G = ha, b a = b = c = 1, [a, b] = c, [c, a] = a , [c, b] = 1i.

Let H = hci. Consider the transitive permutation representation ϕ of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree p3, and ϕ(G) has no regular subgroups.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 10 / 16 A Question

p3 ∈ N R (For p = 2). Let

2 2 2 4 G = ha, b, c, d a = b = c = d = 1, [a, b] = [b, c] = [c, a] = 1, ad = ab, bd = bc, cd = ci.

∼ 3 5 2 Then G = Z2 o Z4 has order 2 . Let H = hb, d i and ϕ be the transitive permutation representation of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree 23 and has no regular subgroup.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 11 / 16 A Question

p3 ∈ N R (For p = 2). Let

2 2 2 4 G = ha, b, c, d a = b = c = d = 1, [a, b] = [b, c] = [c, a] = 1, ad = ab, bd = bc, cd = ci.

∼ 3 5 2 Then G = Z2 o Z4 has order 2 . Let H = hb, d i and ϕ be the transitive permutation representation of G acting on the coset space [G : H]. Then ϕ(G) is a transitive group of degree 23 and has no regular subgroup.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 11 / 16 Determine the set NR

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 12 / 16 Theorem Let n be a positive integer greater than 1. Then n ∈ N R unless n = p or p2 for a prime p.

Determine the set NR

Let p < q be two primes. Then pq ∈ N R.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 13 / 16 Theorem Let n be a positive integer greater than 1. Then n ∈ N R unless n = p or p2 for a prime p.

Determine the set NR

Let p < q be two primes. Then pq ∈ N R.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 13 / 16 Determine the set NR

Let p < q be two primes. Then pq ∈ N R.

Theorem Let n be a positive integer greater than 1. Then n ∈ N R unless n = p or p2 for a prime p.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 13 / 16 Determine the set NR

Let p < q be two primes. Then pq ∈ N R.

Theorem Let n be a positive integer greater than 1. Then n ∈ N R unless n = p or p2 for a prime p.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 13 / 16 Further Questions

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 14 / 16 Question 1: Is N2R = NC? Question 2: Is ND = NC?

Let

N2R = {n ∈ N there is a 2-closed transitive group of degree n without a regular subgroup}

ND = {n ∈ N there is a vertex-transitive digraph of order n which is non-Cayley}

Further Questions

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 15 / 16 Question 2: Is ND = NC?

Further Questions

Let

N2R = {n ∈ N there is a 2-closed transitive group of degree n without a regular subgroup}

ND = {n ∈ N there is a vertex-transitive digraph of order n which is non-Cayley}

Question 1: Is N2R = NC?

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 15 / 16 Further Questions

Let

N2R = {n ∈ N there is a 2-closed transitive group of degree n without a regular subgroup}

ND = {n ∈ N there is a vertex-transitive digraph of order n which is non-Cayley}

Question 1: Is N2R = NC? Question 2: Is ND = NC?

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 15 / 16 Further Questions

Let

N2R = {n ∈ N there is a 2-closed transitive group of degree n without a regular subgroup}

ND = {n ∈ N there is a vertex-transitive digraph of order n which is non-Cayley}

Question 1: Is N2R = NC? Question 2: Is ND = NC?

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 15 / 16 Determine the set PNR. Note: Different from the set NR, we know that pn ∈/ PN R for any prime p and any positive integer n. Thus, determining the set PNR should be much harder than NR.

Let

PNR = {n ∈ N there is a primitive group of degree n without a regular subgroup}

Further Questions

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 16 / 16 Note: Different from the set NR, we know that pn ∈/ PN R for any prime p and any positive integer n. Thus, determining the set PNR should be much harder than NR.

Further Questions

Let

PNR = {n ∈ N there is a primitive group of degree n without a regular subgroup}

Determine the set PNR.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 16 / 16 Further Questions

Let

PNR = {n ∈ N there is a primitive group of degree n without a regular subgroup}

Determine the set PNR. Note: Different from the set NR, we know that pn ∈/ PN R for any prime p and any positive integer n. Thus, determining the set PNR should be much harder than NR.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 16 / 16 Further Questions

Let

PNR = {n ∈ N there is a primitive group of degree n without a regular subgroup}

Determine the set PNR. Note: Different from the set NR, we know that pn ∈/ PN R for any prime p and any positive integer n. Thus, determining the set PNR should be much harder than NR.

Mingyao Xu Peking University () Vertex-Transitive and Cayley Graphs January 18, 2011 16 / 16