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\ge ۲‎, ‎n',m\ge۰‎, ‎n\ge۸k+n'+۴m-۷‎, ‎and \delta(G)\ge k+n'+m‎, ‎if‎ ‎|N_{G}(x)\cup N_{G}(y)|\ge\frac{n+n'}{۲}‎ ‎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 ‎\delta(G) of G are all sharp‎.

Keywords:

‎Graph‎ , ‎fractional‎ ‎factor‎ , ‎fractional (k , n&#۰۳۹ , , m)-critical deleted graph‎ , ‎neighborhood‎ ‎union condition