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

Let GG be a graph of order nn, and let k≥2k≥2 and m≥0m≥0 be two integers. 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, it is proved that GG is a fractional (k,m)(k,m)-deleted graph if δ(G)≥k+2mδ(G)≥k+2m, n≥8k2+4k−8+8m(k+1)+4m−2k+m−1 and ∣NG(x)∪NG(y)∣≥n2 for any two nonadjacent vertices xx and yy of GG such that NG(x)∩NG(y)≠0̸NG(x)∩NG(y)≠0̸. Furthermore, it is shown that the result in this paper is best possible in some sense.