# The Maximum Size Of Graphs With A Uniquek- Factor

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 = 2kq, 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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Correspondence to Lutz Volkmann.

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

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