A Tutte-type characterization for graph factors

Let G be a connected general graph. For any vertex v ∈ V (G) and any function f : V(G) → Z⁺, we introduce a set J∗_f (v) consisting of the integer f(v) and all odd integers less than f(v), including all negative odd integers. In this paper, we shows that the graph G satisfies the general Tutte-type condition o(G - S) ≤ ∑_v∈S f(v) for any nonempty set S ⊃ V (G) if and only if either G has a colored J∗_f-factor for any 2-end-coloring, or G is of odd order and is J∗_f-critical for any 2-end-coloring. This characterization solves a problem posed by Akiyama and Kano, as well as a problem of Cui and Kano's.