a neighborhood union condition for fractional (k,n′,m)-critical deleted graphs

A graph g is called a fractional‎ ‎(k,n′,m)-critical deleted graph if any n′ vertices are removed‎ ‎from g the resulting graph is a fractional (k,m)-deleted‎ ‎graph‎. ‎in this paper‎, ‎we prove that for integers k≥2‎, ‎n′,m≥0‎, ‎n≥8k+n′+4m−7‎, ‎and δ(g)≥k+n′+m‎, ‎if‎ ‎|ng(x)∪ng(y)|≥n+n′2‎ ‎for each pair of non-adjacent vertices x‎, ‎y of g‎, ‎then g‎ ‎is a fractional (k,n′,m)-critical deleted graph‎. ‎the bounds for‎ ‎neighborhood union condition‎, ‎the order n and the minimum degree‎ ‎δ(g) of g are all sharp‎.