12.87 (1992)

Let $\Gamma$ be a connected undirected graph without loops or multiple edges and suppose that the automorphism group $\operatorname{Aut}(\Gamma)$ acts transitively on the vertex set of $\Gamma$. Is it true that at least one of the following assertions holds?
$\qquad$ 1. The stabilizer of a vertex of $\Gamma$ in $\operatorname{Aut}(\Gamma)$ is finite.
$\qquad$ 2. The group $\operatorname{Aut}(\Gamma)$ as a permutation group on the vertex set of $\Gamma$ admits an imprimitivity system $\sigma$ with finite blocks for which the stabilizer of a vertex of the factor-graph $\Gamma/\sigma$ in $\operatorname{Aut}(\Gamma/\sigma)$ is finite.
$\qquad$ 3. There exists a natural number $n$ such that the graph obtained from $\Gamma$ by adding edges joining distinct vertices the distance between which in $\Gamma$ is at most $n$ contains a regular tree of valency 3.