The maximum size of graphs with a unique k-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(n,1) = n(2)/4 for even n and m(n,2) = [n(n+1)/4], respectively. Recently, Johann confirmed the following conjectures of Hendry: m(n,k) = nk/2 + ((n-k)(2)) for k > n/2 and kn even and m(n,k) = n2/4 + (k-1)n/4 for n = 2kq, where q is a positive integer. In this paper we prove m(n,k) = k(2) + ((n-k)(2)) for n/3 less than or equal to k < n/2 and kn even, and we determine m(n, 3).