Semiregular Subgroups of Transitive Permutation Groups › content › villanova › ... · SCI journal Ars Mathematica Contemporanea Dragan Maru si c University of Primorska, Slovenia

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Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs

Semiregular Subgroups of Transitive PermutationGroups

Dragan Marusic

University of Primorska, Slovenia

Villanova, June 2014

[email protected]

Dragan Marusic University of Primorska, Slovenia

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University of Primorska, Koper, Slovenia


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UP FAMNIT – Faculty of Mathematics, Natural Sciences andInformation Technologies

International faculty / students

Conferences, workshops, annual PhD summer schools (RoglaMountain)

EU projects, bilateral project, international cooperation, ...

3 Young researchers positions starting in October 2014

SCI journal Ars Mathematica Contemporanea


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Two important open problems in vertex-transitive graphs:

Existence of Hamiltonian paths/cycles.

Existence of semiregular automorphisms.


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1 Definitions

2 Hamiltonicity of vertex-transitive graphs

3 Semiregular automorphisms in vertex-transitive graphs


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Vertex-transitive graphs

A graph X = (V ,E ) is vertex-transitive if for any pair of verticesu, v there exists an automorphism α such that α(u) = v . (Aut(X )is transitive on V .)


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Vertex-transitive graphs


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Cayley graphs

Cayley graph

Given a group G and a subset S of G \ {1}, S = S−1, the Cayleygraph Cay(G ,S)

has vertex set G and

edges of the form {g , gs} for all g ∈ G and s ∈ S .

A vertex-transitive graph is a Cayley graph provided there exists asubgroup G of Aut(X ) such that for any pair of vertices u, v thereexists a unique automorphism α ∈ G such that α(u) = v . (Thetransitivity is achieved with a minimal number of automorphisms.)


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Hamiltonicity of vertex-transitive graphs


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Tying together two seemingly unrelated concepts:traversability and symmetry

Lovasz question, ’69

Does every connected vertex-transitive graph have a Hamiltonpath?

Lovasz problem is usually referred to as the Lovasz conjecture,presumably in view of the fact that, after all these years, aconnected vertex-transitive graph without a Hamilton path is yetto be produced.


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VT graphs without Hamilton cycle

Only four connected VTG (having at least three vertices) nothaving a Hamilton cycle are known to exist:

the Petersen graph,

the Coxeter graph,

and the two graphs obtained from them by truncation.

All of these are cubic graphs, suggesting that no attempt to resolvethe problem can bypass a thorough analysis of cubic VTG.

None of these four graphs is a Cayley graph, leading to theconjecture that every connected Cayley graph has a Hamilton cycle.


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Vertex-transitive graphs without HC


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The truncation of the Petersen graph


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Essential ingredients in proof methods

(Im)primitivity of transitive permutation groups.

Existence of semiregular automorphisms in vertex-transitivegraphs.

Graph covering techniques.

Graph embeddings.


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If Cay(G , S) is a cubic Cayley graph then |S | = 3, and either

S = {a, b, c | a2 = b2 = c2 = 1}, or

S = {a, x , x−1 | a2 = x s = 1} where s ≥ 3.


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Most recent results for cubic Cayley graphs

Glover, Kutnar, Malnic, DM, 2007-11

A Cayley graph Cay(G , S) on a groupG = 〈a, x | a2 = x s = (ax)3 = 1, . . .〉, where S = {a, x , x−1}, has

a Hamilton cycle when |G | is congruent to 2 modulo 4,

a Hamilton cycle when |G | ≡ 0 (mod 4) and either s is odd ors ≡ 0 (mod 4), and

a cycle of length |G | − 2, and also a Hamilton path, when|G | ≡ 0 (mod 4) and s ≡ 2 (mod 4).


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Most recent results for cubic Cayley graphs

A Cayley graph of A5.


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Semiregular automorphisms in vertex-transitive graphs


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Semiregular automorphisms in VTG

DM, 1981; for transitive 2-closed groups, Klin, 1996

Does every vertex-transitive graph have a semiregularautomorphism?

An element of a permutation group is semiregular, more precisely(m, n)-semiregular, if it has m orbits of size n and no other orbit.

It is known that each finite transitive permutation group contains afixed-point-free element of prime power order, but not necessarily afixed-point-free element of prime order and, hence, no semiregularelement.


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Examples


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Connection to hamiltonicity of VTG

The Pappus configuration & the Pappus graph


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Semiregular elements

(A) Automorphism groups of vertex-transitive (di)graphs;

(B) 2-closed transitive permutation groups;

(C) Transitive permutation groups.

The 2-closure G (2) of a permutation group G is the largest subgroup of

the symmetric group SV having the same orbits on V × V as G . The

group G is said to be 2-closed if it coincides with G (2).


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(A) Automorphism groups of vertex-transitive (di)graphs;

(B) 2-closed transitive permutation groups;

(C) Transitive permutation groups.

A A B C


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(B) but not (A):

Regular action of H = (Z2)2 = {id , (12)(34), (13)(24), (14)(23)}on V = {1, 2, 3, 4}. Each of the orbital graphs has a dihedralautomorphism group intersecting in H; so H is 2-closed but notthe automorphism group of a (di)graph.


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(C) but not (B):

AGL(1, p2), for p = 2k − 1 a Mersenne prime, acting on the set ofp(p + 1) lines of the affine plane AG (2, p).

Let p = 2k − 1 be a Mersenne prime. Affine groupG = {g | g(x) = ax + b, a ∈ GF ∗(p2), b ∈ GF (p2)}acting on cosets of the subgroupH = {g | g(x) = ax + b, a ∈ GF ∗(p), b ∈ GF (p)}.Every prime order element of G fixes a point.


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The now commonly accepted, and slightly more general, version ofthe semiregularity problem involves the whole class of 2-closedtransitive groups (the polycirculant conjecture).

A permutation group with no semiregular elements is called elusive.

The 2-closures of all known elusive groups are non-elusive, thussupporting the polycirculant conjecture.


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Known results - graphs of a particular valency

All cubic VTG have SA (DM, Scapellato, ’93).

Every arc-transitive graph (AGT) of prime valency has SA (Xu, ’07).

All quartic VTG have SA (Dobson, Malnic, DM, Nowitz, ’07).

All VTG of valency p + 1 admitting a transitive {2, p}-group for podd have SA (Dobson, Malnic, DM, Nowitz, ’07).

All ATG with valency pq, p, q primes, such that Aut(X) has anonabelian minimal normal subgroup N with at least 3 vertex orbits,have SA (Xu, ’08).

All ATG with valency 2p (Verret, Giudici, ’13).

All ATG with valency 8 (Verret, ’13).


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Known results - graphs of a particular order

All transitive permutation groups of degree pk or mp, for someprime p and m < p, have SE of order p (DM, ’81).

All VTD of order 2p2 have SA of order p (DM, Scapellato, ’93).

There are no elusive 2-closed groups of square-free degree (Dobson,Malnic, DM, Nowitz, ’07).


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Known results - other

All vertex-primitive graphs have SA (Giudici, ’03).

All vertex-quasiprimitive graphs have SA (Giudici, ’03).

All vertex-transitive bipartite graphs where only system ofimprimitivity is the bipartition, have SA (Giudici, Xu, ’07).

Every 2-arc-transitive graph has SA (Xu, ’07).

Every VT, edge-primitive graph has SA (Giudici, Li, ’09).

All distance-transitive graphs have SA (Kutnar, Sparl, ’09).

All generalized Cayley graphs (Hujdurovic, Kutnar, DM, ’13).


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Semiregular automorphisms

Typical situa,on: G solvable.

Summary -‐ Current situa/on X = VT graph

G transi,ve subgroup of Aut(X)

Giudici Every normal subgroup

of G transi,ve

? G has an intransi,ve normal subgroup


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Semiregular automorphisms

The main steps towards a possible complete solution of theproblem would have to consist of a proof of the existence ofsemiregular automorphisms in vertex-transitive graphs admitting atransitive solvable group.

Even for small valency graphs this is not easy. For example,valency 5 is still open.


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Thank you!


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Semiregular Subgroups of Transitive Permutation Groups › content › villanova › ... · SCI journal Ars Mathematica Contemporanea Dragan Maru si c University of Primorska, Slovenia