Partial parity (g, f)-factors and subgraphs covering given vertex subsets

Let G be a graph and W a subset of V(G). Let g, f : V(G) → Z be two integer-valued functions such that g(x) ≤ f(x) for all x ∈ V(G) and g(y) = f(y) (mod 2) for all y ∈ W. Then a spanning subgraph F of G is called a partial parity (g, f)-factor with respect to W if g(x) ≤ deg_F(x) ≤ f(x) for all x ∈ V(G) and deg_F(y) ≡ f(y) (mod 2) for all y ∈ W. We obtain a criterion for a graph G to have a partial parity (g, f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.