A generalization of Tutte's 1-factor theorem to countable graphs (Q800941)

\textit{W. T. Tutte's} well-known theorem on 1-factors [Can. J. Math. 6, 347-352 (1954; Zbl 0055.171)] states that a nontrivial finite graph G has a 1-factor (perfect matching) if and only if for every proper subset S of V(G), the number of odd components of \(G-S\) does not exceed \(| S|\). The author presents a characterization of countable graphs possessing perfect matchings that reduces to Tutte's theorem in the finite case. He conjectures that the result is valid for general graphs.