semiregular subgroups of transitive permutation groups › content › villanova › ... · sci...
TRANSCRIPT
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular Subgroups of Transitive PermutationGroups
Dragan Marusic
University of Primorska, Slovenia
Villanova, June 2014
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
University of Primorska, Koper, Slovenia
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
University of Primorska, Koper, Slovenia
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
University of Primorska, Koper, Slovenia
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
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Two important open problems in vertex-transitive graphs:
Existence of Hamiltonian paths/cycles.
Existence of semiregular automorphisms.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
1 Definitions
2 Hamiltonicity of vertex-transitive graphs
3 Semiregular automorphisms in vertex-transitive graphs
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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 .)
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Vertex-transitive graphs
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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.)
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Hamiltonicity of vertex-transitive graphs
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Vertex-transitive graphs without HC
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
The truncation of the Petersen graph
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Hamiltonicity of vertex-transitive graphs
Essential ingredients in proof methods
(Im)primitivity of transitive permutation groups.
Existence of semiregular automorphisms in vertex-transitivegraphs.
Graph covering techniques.
Graph embeddings.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Hamiltonicity of vertex-transitive graphs
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Hamiltonicity of vertex-transitive graphs
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).
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Hamiltonicity of vertex-transitive graphs
Most recent results for cubic Cayley graphs
A Cayley graph of A5.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular automorphisms in vertex-transitive graphs
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Examples
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Connection to hamiltonicity of VTG
The Pappus configuration & the Pappus graph
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Connection to hamiltonicity of VTG
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Connection to hamiltonicity of VTG
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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).
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular elements
(A) Automorphism groups of vertex-transitive (di)graphs;
(B) 2-closed transitive permutation groups;
(C) Transitive permutation groups.
A A B C
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular elements
(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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular elements
(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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Semiregular elements
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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).
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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).
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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).
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
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.
Dragan Marusic University of Primorska, Slovenia
OutlineDefinitions
Hamiltonicity of vertex-transitive graphsSemiregular automorphisms in vertex-transitive graphs
Thank you!
Dragan Marusic University of Primorska, Slovenia