# Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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# Towards the Weiss conjecture and beyond

Presented by Dr. Primož POTOčNIK
Type: Oral presentation
Track: Group Actions

## Content

Throughout this abstract, let $\Gamma$ denote a finite connected $G$-arc-transitive graph, let $G_v$ be the stabiliser of a vertex $v$, and let $G_v^{\Gamma(v)}$ be the permutation group induced by the action of $G_v$ on the neighbourhood $\Gamma(v)$. If the valence of $\Gamma$ is $3$, then a celebrated theorem of Tutte states that the order of $G_v$ is bounded above by $48$. Motivated by this surprising result, let us call a transitive permutation group $L$ {\em graph-restrictive} if there exists a constant $c(L)$ such that whenever $G_v^{\Gamma(v)}$ is permutation isomorphic to $L$, it follows that the order of $G_v$ is at most $c(L)$. In this terminology, Tutte's result states that both transitive permutation groups of degree $3$ are graph-restrictive. Following Tutte, several classes of permutation groups have been proved and some other classes have been conjectured to be graph-restrictive. However, no attempts have been made to characterise the class of graph-restricitive groups. Such a characterisation will be addressed in a series of three talks by Pablo Spiga, Gabriel Verret and myself. Several new results will be presented, including the proof of left to right'' direction of the following conjecture: \medskip {\bf Conjecture} A transitive permutation group $L$ is graphs restrictive if and only if it is semiprimitive (that is, if and only if every normal subgroup of $L$ is either transitive or semiregular). \medskip This conjecture generalises several well-known conjectures, for example, a conjecture of Richard Weiss which claims that every primitive permutation group is graph-restrictive, as well as a conjecture by Cheryl Praeger which claims the same for the class of quasiprimitive groups. This is joint work with Pablo Spiga and Gabriel Verret.

## Place

Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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