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

Let G   be a graph. Let h:E(G)→[0,1]h:E(G)→[0,1] be a function. If ∑e∋xh(e)=k∑e∋xh(e)=k holds for each x∈V(G)x∈V(G), then we call G[Fh]G[Fh] a fractional k-factor of G with indicator function h   where Fh={e∈E(G):h(e)>0}Fh={e∈E(G):h(e)>0}. A graph G   is called a fractional (k,m)(k,m)-deleted graph if for every e∈E(H)e∈E(H), there exists a fractional k  -factor G[Fh]G[Fh] of G with indicator function h   such that h(e)=0h(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)(k,m)-deleted graph. This result is best possible in some sense, and it is an extension of Liuʼs previous result.

► We study the fractional factor problem in graphs, which has wide-range applications in many areas.
► We obtain a toughness condition for a graph to be a fractional (k,m)(k,m)-deleted graph.
► Finally, we show that the toughness condition in the main theorem is sharp.