Semiregular Subgroups of Transitive Permutation Groups Dragan Maru - - PowerPoint PPT Presentation

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Semiregular Subgroups of Transitive Permutation Groups

Dragan Maruˇ siˇ c

University of Primorska, Slovenia Villanova, June 2014 dragan.marusic@upr.si

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

University of Primorska, Koper, Slovenia

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

University of Primorska, Koper, Slovenia

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

University of Primorska, Koper, Slovenia

UP FAMNIT – Faculty of Mathematics, Natural Sciences and Information Technologies International faculty / students Conferences, workshops, annual PhD summer schools (Rogla Mountain) EU projects, bilateral project, international cooperation, ... 3 Young researchers positions starting in October 2014 SCI journal Ars Mathematica Contemporanea

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Two important open problems in vertex-transitive graphs: Existence of Hamiltonian paths/cycles. Existence of semiregular automorphisms.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

1 Definitions 2 Hamiltonicity of vertex-transitive graphs 3 Semiregular automorphisms in vertex-transitive graphs

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Vertex-transitive graphs

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

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Vertex-transitive graphs

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Cayley graphs

Cayley graph Given a group G and a subset S of G \ {1}, S = S−1, the Cayley graph 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 a subgroup G of Aut(X) such that for any pair of vertices u, v there exists a unique automorphism α ∈ G such that α(u) = v. (The transitivity is achieved with a minimal number of automorphisms.)

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Hamiltonicity of vertex-transitive graphs

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Tying together two seemingly unrelated concepts: traversability and symmetry

Lov´ asz question, ’69 Does every connected vertex-transitive graph have a Hamilton path? Lov´ asz problem is usually referred to as the Lov´ asz conjecture, presumably in view of the fact that, after all these years, a connected vertex-transitive graph without a Hamilton path is yet to be produced.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

VT graphs without Hamilton cycle

Only four connected VTG (having at least three vertices) not having 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 resolve the problem can bypass a thorough analysis of cubic VTG. None of these four graphs is a Cayley graph, leading to the conjecture that every connected Cayley graph has a Hamilton cycle.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Vertex-transitive graphs without HC

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

The truncation of the Petersen graph

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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-transitive graphs. Graph covering techniques. Graph embeddings.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 = xs = 1} where s ≥ 3.

• Dragan Maruˇ

siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Hamiltonicity of vertex-transitive graphs

Most recent results for cubic Cayley graphs Glover, Kutnar, Malniˇ c, DM, 2007-11 A Cayley graph Cay(G, S) on a group G = a, x | a2 = xs = (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 or s ≡ 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 Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Hamiltonicity of vertex-transitive graphs

Most recent results for cubic Cayley graphs A Cayley graph of A5.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Semiregular automorphisms in vertex-transitive graphs

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 semiregular automorphism? 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 a fixed-point-free element of prime power order, but not necessarily a fixed-point-free element of prime order and, hence, no semiregular element.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Examples

• Dragan Maruˇ

siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Connection to hamiltonicity of VTG

The Pappus configuration & the Pappus graph

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Connection to hamiltonicity of VTG

• Dragan Maruˇ

siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Connection to hamiltonicity of VTG

• Dragan Maruˇ

siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Semiregular elements

(B) but not (A): Regular action of H = (Z2)2 = {id, (12)(34), (13)(24), (14)(23)}

• n V = {1, 2, 3, 4}. Each of the orbital graphs has a dihedral

automorphism group intersecting in H; so H is 2-closed but not the automorphism group of a (di)graph.

• Dragan Maruˇ

siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 of p(p + 1) lines of the affine plane AG(2, p). Let p = 2k − 1 be a Mersenne prime. Affine group G = {g | g(x) = ax + b, a ∈ GF ∗(p2), b ∈ GF(p2)} acting on cosets of the subgroup H = {g | g(x) = ax + b, a ∈ GF ∗(p), b ∈ GF(p)}. Every prime order element of G fixes a point.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Semiregular elements

The now commonly accepted, and slightly more general, version of the semiregularity problem involves the whole class of 2-closed transitive 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, thus supporting the polycirculant conjecture.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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, Malniˇ c, DM, Nowitz, ’07). All VTG of valency p + 1 admitting a transitive {2, p}-group for p

• dd have SA (Dobson, Malniˇ

c, DM, Nowitz, ’07). All ATG with valency pq, p, q primes, such that Aut(X) has a nonabelian 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 Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Known results - graphs of a particular order

All transitive permutation groups of degree pk or mp, for some prime 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, Malniˇ c, DM, Nowitz, ’07).

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 of imprimitivity 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, ˇ Sparl, ’09). All generalized Cayley graphs (Hujdurovi´ c, Kutnar, DM, ’13).

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular 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 ¡

• f ¡G ¡transi,ve ¡

? ¡ G ¡has ¡an ¡intransi,ve ¡ normal ¡subgroup ¡

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Semiregular automorphisms

The main steps towards a possible complete solution of the problem would have to consist of a proof of the existence of semiregular automorphisms in vertex-transitive graphs admitting a transitive solvable group. Even for small valency graphs this is not easy. For example, valency 5 is still open.

Dragan Maruˇ siˇ c University of Primorska, Slovenia

Outline Definitions Hamiltonicity of vertex-transitive graphs Semiregular automorphisms in vertex-transitive graphs

Thank you!

Dragan Maruˇ siˇ c University of Primorska, Slovenia