Almost resolvable P k ‐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 λ K n is a partition of the edges of λ K n into almost P k ‐factors. We prove that necessary and sufficient conditions for the existence of an almost resolvable P k ‐decomposition of λ K n are n ≡ 1 (mod k) and λnk /2 ≡ 0 (mod k −1).