r -regular families of graph automorphisms Robert Jajcay Comenius - - PowerPoint PPT Presentation

r-regular families

• f

graph automorphisms

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work with Gareth Jones October 11, 2016

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Vertex-Transitive Graphs

Definition

A graph Γ = (V , E) is said to be vertex-transitive if the group Aut(Γ) of automorphisms of Γ acts transitively on V , i.e., u, v ∈ V ∃ϕ ∈ Aut(Γ) : ϕ(u) = v

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Cayley Graphs

Definition

A graph Γ = (V , E) is said to be Cayley if the group Aut(Γ) of automorphisms of Γ contains a subgroup that acts regularly on V , i.e., u, v ∈ V ∃ unique ϕ ∈ Aut(Γ) : ϕ(u) = v Conjecture (?): lim

n→∞

# of Cayley graphs of order ≤ n # of v-t graphs of order ≤ n > 0

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Definition (Sabidussi, Godsil)

The Cayley deficiency of a vertex-transitive graph Γ, d(Γ), is the

• rder of the vertex-stabilizer of a smallest vertex-transitive

automorphism group of Γ.

◮ Cayley graphs are exactly the vertex-transitive graphs with

d(Γ) = 1;

◮ the Petersen graph P of order 10 admits a vertex-transitive

group of automorphism of order 20 and none smaller, hence d(P) = 2;

◮ there exist vertex-transitive graphs that are arbitrarily far from

being Cayley: Kneser and Johnson graphs [Jon&RJ]

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Quasi-Cayley Graphs

Definition (Gauyacq)

A vertex-transitive graph Γ = (V , E) is quasi-Cayley iff there exists a regular family of graph automorphisms F satisfying the property that for each pair of vertices u, v ∈ V there exists a unique automorphism ϕ ∈ F mapping u to v, ϕ(u) = v.

◮ every quasi-Cayley graph is vertex-transitive ◮ every Cayley graph is quasi-Cayley ◮ Petersen graph is not Cayley but it is quasi-Cayley

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Construction of quasi-Cayley graphs

◮ take a Latin square L; ◮ let the rows of L define permutations ϕi, 1 ≤ i ≤ n, of

{1, 2, . . . , n}: ϕi(j) = Li,j

◮ take G = ϕ1, ϕ2, . . . , ϕn ◮ take E to be a union of the orbits of G in the action on

n

2

• ◮ the graph ({1, 2, . . . , n}, E) is quasi-Cayley but it can also be Cayley

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

r-regular families

Definition (RJ,Jones)

A subset F ⊆ SX of permutations of a set of elements X is said to form an r-regular family of permutations on X if for every pair

• f elements x, y ∈ X there exist exactly r permutations ϕ ∈ F

mapping x to y, ϕ(x) = y.

Definition (RJ, Jones)

The quasi-Cayley deficiency of a vertex-transitive graph Γ, r(Γ), is the smallest r for which there exists an r-regular family of automorphisms of Γ.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Quasi-Cayley Deficiency

Observation: If Γ = (V , E) is a vertex-transitive graph, then Γ admits an r-regular family of automorphisms where r is smaller or equal to the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of Γ. r(Γ) ≤ d(Γ) r(P) = 1, d(P) = 2

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Vertex-transitive graphs which are not quasi-Cayley graphs

• 1. the Johnson graph J = J(n, k) has as its vertex set the set

V = N

k

• f k-element subsets of an n-element set N, with

vertices v and w adjacent if and only if |v ∩ w| = k − 1

• 2. the Kneser graph K = K(n, k) has the same vertex-set as

J(n, k) with the adjacency defined between any two k-element subsets that are disjoint

• 3. the distance i graph J(n, k)i has the same vertex set and with

adjacency defined by |v ∩ w| = k − i

• 4. the merged Johnson graph J(n, k)I = ∪i∈IJ(n, k)i for some

non-empty proper subset I of {2, . . . , k}

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Vertex-transitive graphs not admitting r-regular families for small r’s

Lemma (RJ,Jones)

Let M be an r · C(n, k) × n matrix with entries from {1, 2, . . . , n} satisfying properties:

• 1. each row contains all of {1, 2, . . . , n}, or, in other words, each

i ∈ {1, 2, . . . , n} appears exactly once in each row;

• 2. the matrix formed by any subset of k columns of M contains

each k-subset of {1, 2, . . . , n} (as a row) exactly r times. Then

• 1. k | r

n−1

k−1

• 2. k | 2r

n−2

k−2

• Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk ..............................

Joint work r-regular families of graph automorphisms

Vertex-transitive graphs not admitting r-regular families for small r’s

Corollary (RJ,Jones)

Let p be a prime, k = p and n = ℓp for a positive integer ℓ > 2. Then J(n, k) does not admit any r-regular family for 1 ≤ r < k.

Corollary (RJ,Jones)

For every k ≥ 1, there exist infinitely many vertex-transitive graphs Γ for which the smallest r(Γ) > k.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Classification of merged Johnson graphs that are Cayley

Theorem (Jones, RJ)

Let 2 ≤ k ≤ n/2 and let I be a non-empty subset of {1, . . . , k}. Then the merged Johnson graph J = J(n, k)I is a Cayley graph of a group G if and only if one of the following holds:

• 1. n is a prime power, n ≡ 3 mod (4) and k = 2, with any I, and

G ∼ = AHL1(F) acting on some Dickson near-field N = F of

• rder n;
• 2. n = 8 and k = 3, with any I, and G ∼

= AGL1(8) acting on the finite field N = F8;

• 3. n = 32 and k = 3, with any I, and G ∼

= AΓL1(32) acting on the finite field N = F32;

• 4. I = {1, . . . , k}, with any n and k, and G any group of order

n

k

• , acting on itself by right multiplication;
• 5. k = n/2 and I = {k} or {1, . . . , k − 1}, with G any group of
• rder

n

k

• , acting on itself by right multiplication.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Theorem (Jones,RJ)

Let 2 ≤ k ≤ n/2 and let I be a non-empty subset of {1, . . . , k}. Then G is a 2-regular group of automorphisms of the graph J = J(n, k)I if and only if one of the following holds:

• 1. n is a prime power, k = 2, with any I, and G ∼

= AGL1(F) for some near-field N = F of order n;

• 2. n = 6 and k = 3, where G ∼

= AGL1(5) × S2, with any I, and AGL1(5) acting naturally on the projective line N = P1(F5), fixing ∞, and S2 generated by complementation in N;

• 3. n = 10, k = 5 and I = {1, 4}, {2, 3}, {1, 4, 5} or {2, 3, 5},

where G ∼ = PSL2(8)

• 4. n is even, k = n/2 and I = {k} or {1, . . . , k − 1}, where G is

any group of order 2 n

k

• with a non-normal subgroup H of
• rder 2, acting on the cosets of H;
• 5. I = {1, . . . , k}, where G is any group of order 2

n

k

• with a

non-normal subgroup H of order 2, acting on the cosets of H.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

If k ≥ 6, or if 2 ≤ k ≤ 5 and n avoids an easily described subset of

N of asymptotic density 0, then the only k-homogeneous groups of

degree n, and hence the only vertex-transitive groups of automorphisms of J(n, k), are the symmetric group Sn and the alternating group An. d(J(n, k)) ≥ n! 2 for the majority of Johnson graphs. However, we could not find an infinite family of graphs J(n, k) for which we could prove that r(J(n, k)) is much smaller than d(Γ).

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Praeger-Xu graphs

Theorem (RJ,Potoˇ cnik,Wilson)

• 1. Every Praeger-Xu graph is quasi-Cayley, r(PX(n, k)) = 1;
• 2. for every positive integer k, there exists a Praeger-Xu graph

Γ = PX(n, k) with the property d(PX(n, k)) > k.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Open Problems

• 1. What is the relation between the smallest r(Γ) and d(Γ) for

Kneser, Johnson and merged Johnson graphs?

• 2. Is there a result similar to the result for the generalized

Petersen graphs concerning the cubic bicirculants?

• 3. What is the largest r such that every vertex-transitive graph

with an r-regular family of automorphisms admits an automorphism whose orbits are all of the same size?

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms

Thank you.

Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work r-regular families of graph automorphisms