Is there a connected infinite graph $G=(V,E)$ such that $\text{deg}(v) \geq 2$ for all $v\in V$, and $G$ possesses no perfect matching?



Connected infinite graph $G$ with $\delta(G)\geq 2$ and no perfect matching
What did you try? $K\_{32}$ has no perfect matching. So start with any infinite connected graph with minimum degree $2$, pick two vertices $u,v$ and add three new vertices $A,B,C$ each of degree $2$ with edges going to $u$ and $v.$ No matching can have edges on all three of the new vertices.


