Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ of pairwise disjoint edges such that $\bigcup M = V$)?



Infinite graphs with large degree but no perfect matching
Consider the ordering on $V$ such that for any vertex $v$ there exist less than $|V|$ vertices $y<x$. Construct the matching inductively. 


