On the structure of graphs with a unique k-factor

We study the structure of graphs with a unique k-factor. Our results imply a conjecture of Hendry on the maximal number m(n, k) of edges in a graph G of order n with a unique k-factor: For k > n/2,kn even, we prove m(n,k) = nk2 + n−k2 and construct all corresponding extremal graphs. For k ≤ n/2,kn even, we prove m(n,k) ≤ + (k − 1)n/4. For n = 2kl, l ∈ N, this bound is sharp, and we prove that the corresponding extremal graph is unique up to isomorphism.