A characterization on $(g,f)$-parity orientations

Let G be a graph and g,f:V(G)→2N be two set functions such that g(v)≤ f(v) and g(v)≡ f(v) 2 for every v∈ V(G). An orientation O of G is called a $(g,f)$-parity orientation if g(v)≤ d⁺O(v)≤ f(v) and g(v)≡ d⁺O(v) 2 for every v∈ V(G). In this paper, we give a Tutte-type characterization for a graph to have a $(g,f)$-parity orientation.