I am trying to show something about groups acting transitively on infinite graphs and my proof only works for graphs that that do not have an upper bound on the size of simple closed loops, so I would like to get a characterisation of infinite vertex graphs that have a bound on the size of simple loops.
More formally: suppose that $\Gamma = (V\Gamma, E\Gamma)$ is a locally finite simplicial graph, i.e. $V\Gamma$ is a set and the set of edges $E\Gamma \subseteq \binom{V\Gamma}{2}$ consists of unordered tuples of vertices and that for every vertex $v \in V\Gamma$ the neighbourhood $N(v) = \{u \in V\Gamma\ \mid \{u,v\} \in E\Gamma \}$ is finite. Also, assume that $\Gamma$ is vertex transitive, meaning that for every pair of vertices $u,v \in V\Gamma$ there is a graph automorphism $\alpha \in \mathop{Aut}(\Gamma)$ such that $\alpha(v) = u$. This implies that all vertices are of the same degree $d \in \mathbb{N}$.
I am inclined to believe that the following are equivalent:
(i) there is a uniform bound on the length of simple closed loops in $\Gamma$, i.e. there is $C \in \mathbb{N}$ such that if $L \subset E\Gamma$ is a simple loop (each vertex is contained in exactly two edges) then $|L| \leq C$;
(ii) every vertex $v \in V\Gamma$ is a bridge, i.e. the graph $\Gamma \setminus \{v\}$ is not connected for every $v \in V\Gamma$.
I can prove (i) -> (ii) but I am not sure how to show the implication in the opposite direction, my knowledge of graph theory is rather superficial.