Restricted cycle factors and arc-decompositions of digraphs

We study the complexity of finding 2-factors with various restrictions as well as edge-decompositions in (the underlying graphs of) digraphs. In particular we show that it is NP-complete to decide whether the underlying undirected graph of a digraph D has a 2-factor with cycles ^C1, ^C2,..., ^Ck such that at least one of the cycles ^Ci is a directed cycle in D (while the others may violate the orientation back in D). This solves an open problem from J. Bang-Jensen et al., Vertex-disjoint directed and undirected cycles in general digraphs, JCT B 106 (2014), 1-14. Our other main result is that it is also NP-complete to decide whether a 2-edge-colored bipartite graph has two edge-disjoint perfect matchings such that one of these is monochromatic (while the other does not have to be). We also study the complexity of a number of related problems. In particular we prove that for every even k≥2, the problem of deciding whether a bipartite digraph of girth k has a k-cycle-free cycle factor is NP-complete. Some of our reductions are based on connections to Latin squares and so-called avoidable arrays.