Tutte 类型条件刻画与图因子

Let G be a graph. For any vertex v ∈ V(G) and any function [Formula presented], denote by J_f(v) the set consisting of the integer f(v) and all positive odd integers less than f(v), and by J_f^o(v) the set of positive odd integers no greater than f(v) + 1. In this paper, we show that a graph G satisfies the Tutte-type condition [Formula presented] for any nonempty set S ⊂ V(G), v∈S if and only if G contains an H-factor for any H ∈ H, where [Formula presented] for each v ∈ V(G)}. This is a new characterization on the open problem proposed by Akiyama and Kano (2011). Moreover, we also characterize toughness conditions in terms of graph factors.