On Cui-Kano's Characterization Problem on Graph Factors

An $H_n$-factor of a graph $G$ is defined to be a spanning subgraph $F$ of $G$ such that each vertex has degree belonging to the set $\{1,3,5,...,2n-1,2n\}$ in $F$. In this paper, we investigate $H_n$-factors of graphs by using Lov\'asz's structural descriptions to the degree prescribed subgraph problem. We find some sufficient conditions for the existence of an $H_n$-factor of a graph. In particular, we make progress on the characterization problem for a special family of graphs proposed by Cui and Kano in 1988.