For a graph (undirected or directed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and algorithmic complexity. In this work, we study four variants of the problem of finding a cycle-factor subject to the following parity constraints: (1) all cycles are odd, (2) all cycles are even, (3) at least one cycle is odd, and (4) at least one cycle is even. We show that all the variants of the problem are NP-complete both in undirected and directed graphs, even if each vertex is incident to at most three edges. We also prove that the first two variants are NP-complete even for planar directed graphs.

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