Yun Gao - Department of Editorial, Yunnan Normal University
Mohammad Reza Farahani - Department of Applied Mathematics, Iran University of Science and Technology
Wei Gao - School of Information and Technology, Yunnan Normal University

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.

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