A sufficient condition for graphs to be fractional (k,m)(k,m)-deleted graphs

Let GG be a graph, and kk a positive integer. 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 kk-factor of GG with indicator function hh where Fh={e∈E(G):h(e)>0}Fh={e∈E(G):h(e)>0}. A graph GG is called a fractional (k,m)(k,m)-deleted graph if there exists a fractional kk-factor G[Fh]G[Fh] of GG with indicator function hh such that h(e)=0h(e)=0 for any e∈E(H)e∈E(H), where HH is any subgraph of GG with mm edges. In this paper, we use a binding number to obtain a sufficient 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 Zhou’s previous results.