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 (G) k + n ′ + m, if jNG(x) [ NG(y)j 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 inimum degree (G) of G are all sharp.