Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $G$ be a connected general graph. For any vertex $v\in V(G)$ and any function $f: V(G)\to\mathbb{Z}^+$,  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. ,  shows that the graph $G$ satisfies the general Tutte-type condition $o(G-S)\le \sum_{v\in S}f(v)$ for any nonempty set $S\subset V(G)$ 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 ll as a problem of Cui and Kano's.
['Hongliang Lu', 'David G. L. Wang']
I am writing a research paper on a idea A Tutte-type characterization for graph factors
[{"from": "human", "value": "I am writing a research paper on a idea A Tutte-type characterization for graph factors"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $G$ be a connected general graph. For any vertex $v\\in V(G)$ and any function $f: V(G)\\to\\mathbb{Z}^+$,  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. ,  shows that the graph $G$ satisfies the general Tutte-type condition $o(G-S)\\le \\sum_{v\\in S}f(v)$ for any nonempty set $S\\subset V(G)$ 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 ll as a problem of Cui and Kano's."}]