Factor domination and minimum degree

For a graph G=(V,E), a factor F of G is a subgraph of G on the same vertex set, V. A subset S⊆V is a dominating set of G if every vertex v∈V-S is adjacent, in G, to a vertex of S. Let F ₁,F ₂,..,F _k be factors of G. A set S⊆V that is, simultaneously, a dominating set of F _i for each i with 1ik is called a factor dominating set of F ₁,F ₂,..,F _k. The cardinality of a smallest such set is called the factor domination number of F ₁,F ₂,..,F _k and denoted by γ(F ₁,F ₂,..,F _k). In this paper, we give bounds on γ(F ₁,F ₂,..,F _k) in terms of the minimum degrees of the F _i.