Counting 1-factors in infinite graphs (Q1814587)

Let \(G\) be an \(n\)-connected graph having a 1-factor. Then for \(n\geq 3\), \(G\) has more than one 1-factor. For a finite such \(G\), the number of different 1-factors \(G\) must contain was determined for \(n\) odd by \textit{J. Zaks} [J. Comb. Theory, Ser. B 11, 169-180 (1971; Zbl 0219.05074)] and for \(n\) even by \textit{W. Mader} [Math. Ann. 201, 269-282 (1973; Zbl 0234.05115)]. By \textit{F. Bry} [J. Comb. Theory, Ser. B 34, 48-57 (1983; Zbl 0512.05049)] it was shown, that for infinite, locally finite, \(n\)- connected graphs \(G\) with a 1-factor, the minimum number \(f(n)\) of different 1-factors is finite, and lower and upper bounds for \(f(n)\) were given. In this paper, it is now proved that \(f(n)=n!\) holds for all integers \(n\geq 3\).