An extension of Tutte's 1-factor theorem

AbstractWe prove the following theorem: Let G be a graph with vertex-set V and ƒ, g be two integer-valued functions defined on V such that 0⩽g(x) ⩽1⩽ƒ(x) for all x ∈ V. Then G contains a factor F such that g(x)⩽dF(x)⩽ƒ(x) for all x ∈ V if and only if for every subset X of V, ƒ(X) is at least equal to the number of connected components C of G[V − X] such that either C = {x} and g(x) = 1, or |C| is odd ⩾3 and g(x) = ƒ(x) = 1 for all x ∈ C. Applications are given to certain combinatorial geometries associated with factors of graphs