Almost resolvable Pk-decompositions of complete graphs

An almost P_k-factor of G is a P_k-factor of G - { v } for some vertex v. An almost resolvable P_k-decomposition of λK_n is a partition of the edges of λK_n into almost P_k-factors. We prove that necessary and sufficient conditions for the existence of an almost resolvable P_k-decomposition of λK_n are n ≡ 1 (mod k) and λnk/2 ≡ 0 (mod k ?1).