A toughness condition for fractional (k,m) -deleted graphs

Let G be a graph. Let $h:E(G)\to [0,1]$ be a function. If ${\Sigma }_{e\ni x}h(e)=k$ holds for each $x\in V(G)$ , then we call $G[F_h]$ a fractional k‐factor of G with indicator function h where $F_h=\{e\in E(G):h(e)>0\}$ . A graph G is called a fractional $(k,m)$ ‐deleted graph if for every $e\in E(H)$ , there exists a fractional k‐factor $G[F_h]$ of G with indicator function h such that $h(e)=0$ , where H is any subgraph of G with m edges. In this paper, we obtain a toughness condition for a graph to be a fractional $(k,m)$ ‐deleted graph. This result is best possible in some sense, and it is an extension of Liu's previous result.