Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is infinite?
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asked Sep 22, 2018 at 17:34
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If all degrees are infinite, it contains a perfect matching: just add edges one by one covering all vertices.
answered Sep 22, 2018 at 17:51