Page 300 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 300
GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
Some small progress on the PSV Conjecture
Luke Morgan, Luke.morgan@famnit.upr.si
University of Primorska, Slovenia
The subject of the talk is the problem of bounding the number of automorphisms of an arc-
transitive graph in terms of the valency of the graph. More specifically, we consider the prob-
lem for groups acting arc-transitively on graphs such that the local action (that induced on the
neighbours of a vertex by the stabiliser of that vertex) is semiprimitive. This problem was first
considered by Weiss for the case of primitive local action and then generalised by Praeger to
the case of quasiprimitive local action.
I will report on some recent small progress on the first type - that of semiprimitive local
action, where the action is that of the symmetric group Sn on the set of n(n − 1) ordered pairs.
The result is akin to Tutte’s famous result on cubic s-arc transitive graphs where the number of
automorphisms fixing a vertex is bounded by 3∗2s−1. Tutte’s proof was elegant, elementary and
self-contained. Our recent progress relies on some group theoretical tools that were developed
for use in the Classification of the Finite Simple Groups - and some tricks to allow us to patch
things together. I’ll try to present these results in a friendly fashion, as well as keeping in mind
the “big picture” concerning where progress now stands on these conjectures.
On Cayley isomorphism property for abelian groups
Grigory Ryabov, gric2ryabov@gmail.com
Sobolev Institute of Mathematics and Novosibirsk State University, Russian Federation
Coauthor: István Kovács
A finite group G is called a DCI-group if every two isomorphic Cayley digraphs over G are
Cayley isomorphic, i.e. there exists an isomorphism between these digraphs that is also an
automorphism of G. One of the motivations to study DCI-groups comes from the Cayley graph
isomorphism problem. Suppose that G is a DCI-group. Then to determine whether two Cayley
digraphs Cay(G, S) and Cay(G, T ) are isomorphic, we only need to check the existence of
ϕ ∈ Aut(G) with Sϕ = T . The latter, usually, is much easier.
The definition of a DCI-group goes back to Ádám who conjectured [1], in our terms, that
every cyclic group is DCI. This conjecture was proved to be false. The problem of determining
all finite DCI-groups was raised by Babai and Frankl [2]. One of the crucial steps towards
the classification of all DCI-groups is to determine abelian DCI-groups. It was proved that
every abelian DCI-group is the direct product of groups of coprime orders each of which is
elementary abelian or isomorphic to Z4 (see [7, Theorem 8.8]). However, the classification of
abelian DCI-groups is far from complete. In the talk we discuss on new infinite families of
abelian DCI-groups and approaches to determining whether a given group is DCI.
References
[1] A. Ádám, Research Problem 2-10, J. Combin. Theory, 2 (1967), 393.
[2] L. Babai, P. Frankl, Isomorphisms of Cayley graphs I, Colloq. Math. Soc. János Bolyai,
18, North-Holland, Amsterdam (1978), 35–52.
[3] C. H. Li, On isomorphisms of finite Cayley graphs – a survey, Discrete Math., 256, Nos. 1-
2 (2002), 301–334.
298
Some small progress on the PSV Conjecture
Luke Morgan, Luke.morgan@famnit.upr.si
University of Primorska, Slovenia
The subject of the talk is the problem of bounding the number of automorphisms of an arc-
transitive graph in terms of the valency of the graph. More specifically, we consider the prob-
lem for groups acting arc-transitively on graphs such that the local action (that induced on the
neighbours of a vertex by the stabiliser of that vertex) is semiprimitive. This problem was first
considered by Weiss for the case of primitive local action and then generalised by Praeger to
the case of quasiprimitive local action.
I will report on some recent small progress on the first type - that of semiprimitive local
action, where the action is that of the symmetric group Sn on the set of n(n − 1) ordered pairs.
The result is akin to Tutte’s famous result on cubic s-arc transitive graphs where the number of
automorphisms fixing a vertex is bounded by 3∗2s−1. Tutte’s proof was elegant, elementary and
self-contained. Our recent progress relies on some group theoretical tools that were developed
for use in the Classification of the Finite Simple Groups - and some tricks to allow us to patch
things together. I’ll try to present these results in a friendly fashion, as well as keeping in mind
the “big picture” concerning where progress now stands on these conjectures.
On Cayley isomorphism property for abelian groups
Grigory Ryabov, gric2ryabov@gmail.com
Sobolev Institute of Mathematics and Novosibirsk State University, Russian Federation
Coauthor: István Kovács
A finite group G is called a DCI-group if every two isomorphic Cayley digraphs over G are
Cayley isomorphic, i.e. there exists an isomorphism between these digraphs that is also an
automorphism of G. One of the motivations to study DCI-groups comes from the Cayley graph
isomorphism problem. Suppose that G is a DCI-group. Then to determine whether two Cayley
digraphs Cay(G, S) and Cay(G, T ) are isomorphic, we only need to check the existence of
ϕ ∈ Aut(G) with Sϕ = T . The latter, usually, is much easier.
The definition of a DCI-group goes back to Ádám who conjectured [1], in our terms, that
every cyclic group is DCI. This conjecture was proved to be false. The problem of determining
all finite DCI-groups was raised by Babai and Frankl [2]. One of the crucial steps towards
the classification of all DCI-groups is to determine abelian DCI-groups. It was proved that
every abelian DCI-group is the direct product of groups of coprime orders each of which is
elementary abelian or isomorphic to Z4 (see [7, Theorem 8.8]). However, the classification of
abelian DCI-groups is far from complete. In the talk we discuss on new infinite families of
abelian DCI-groups and approaches to determining whether a given group is DCI.
References
[1] A. Ádám, Research Problem 2-10, J. Combin. Theory, 2 (1967), 393.
[2] L. Babai, P. Frankl, Isomorphisms of Cayley graphs I, Colloq. Math. Soc. János Bolyai,
18, North-Holland, Amsterdam (1978), 35–52.
[3] C. H. Li, On isomorphisms of finite Cayley graphs – a survey, Discrete Math., 256, Nos. 1-
2 (2002), 301–334.
298
