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

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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 | a 2 = b 2 = c 2 = 1 } , or S = { a , x , x − 1 | a 2 = x s = 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 | a 2 = 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 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 A 5 . 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

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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 S V 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