a) How can we prove that a Cayley graph (or a vertex transitive graph) is connected and bridgeless?
b) It is known that a even ordered Cayley graph (or vertex transitive graph) has a perfect matching and a odd ordered Cayley graph G (or vertex transitive graph) has a perfect matching for G-v, where v is any vertex of graph G.
Is there a way to prove that the Cayley graph (or vertex transitive graph) remains connected and bridgeless even when we remove the edges corresponding to above mentioned perfect matching?
I observed that when we remove a perfect matching from a certain Cayley graph it remains connected and bridgeless, but I can't think of a way to prove it.
Actually the Cayley graph I'm considering is the Cayley graph of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ with respect to a generating set $S=\{u,t\}$, where $|u|=3, |t|=5$. When a certain perfect matching is removed the graph stays connected and bridgeless. But how can I prove it?
Please help me with these questions and guide me.
Thanks a lot in advance.