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) ≤ degF (x) ≤ f(x) for all x ∈ V (G) and degF (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. 1 Partial parity (g, f)-factors We consider a finite graph G which may have multiple edges and loops, and so a graph means such a graph throughout this paper. Let V (G) and E(G) denote the set of vertices and that of edges of G, respectively. For two disjoint subsets S and T of V (G), we write eG(S, T ) for the number of edges of G joining S to T . For a vertex v of G, we denote by degG(v) the degree of v in G, and by NG(v) the neighborhood of v. Let Z and Z + denote the set of integers and that of non-negative integers, respectively. For a function f : V (G) → Z, a spanning subgraph F of G is called an f -factor if degF (x) = f(x) for all x ∈ V (G). For two functions g, f : V (G) → Z such that g(x) ≤ f(x) for all x ∈ V (G), a spanning subgraph H of G is called a (g, f)-factor if g(x) ≤ degH(x) ≤ f(x) for all x ∈ V (G). Note that when we consider (g, f)-factors, we allow that g(x) < 0 and/or degG(y) < f(y) for some vertices x and y of G, and this relaxation will play an important technical role. ∗Present address: Department of Mathematics, Kyusyu Tokai University, Choyo, Aso, Kumamoto, 865-1404 Japan