For suitable positive integers *n* and *k* let *m*(*n, k*) denote the maximum number of edges in a graph of order *n* which has a unique *k*-factor. In 1964, Hetyei and in 1984, Hendry proved \( m{\left( {n,1} \right)} = \frac{{n^{2} }} {4} \) for even *n* and \( m{\left( {n,2} \right)} = {\left\lfloor {\frac{{n{\left( {n + 1} \right)}}} {4}} \right\rfloor } \), respectively. Recently, Johann confirmed the following conjectures of Hendry: \( m{\left( {n,k} \right)} = \frac{{nk}} {2} + {\left( {{}^{{n - k}}_{2} } \right)} \) for\( k > \frac{n} {2} \) and *kn* even and \( m{\left( {n,k} \right)} = \frac{{n^{2} }} {4} + {\left( {k - 1} \right)}\frac{n} {4} \) for *n* = 2*kq*, where *q* is a positive integer. In this paper we prove \( m{\left( {n,k} \right)} = k^{2} + {\left( {{}^{{n - k}}_{2} } \right)} \) for \( \frac{n} {3} \leqslant k < \frac{n} {2} \) and *kn* even, and we determine *m*(*n*, 3).

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Volkmann, L. The Maximum Size Of Graphs With A Unique*k*- Factor.
*Combinatorica* **24, **531–540 (2004). https://doi.org/10.1007/s00493-004-0032-9

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*Mathematics Subject Classification (2000):*

- 05C70