A TIGHT NEIGHBORHOOD UNION CONDITION ON FRACTIONAL (g, f,n′,m)-CRITICAL DELETED GRAPHS

A graph G is called a fractional (g, f, n', m)-critical deleted graph if it remains a fractional (g, f, m)-deleted graph after deleting any n' vertices. We prove that if G is a graph of order n, 1 <= a <= g(x) <= f(x) <= b for any x is an element of V (G), delta(G) >= b(2)/a+n'+2m, n > ((a vertical bar b)(2(a vertical bar b) 2m - 1) vertical bar bn')/a, and | N-G(x(1)) U N-G (X-2)| >= b(n vertical bar n')/(a vertical bar b) for any nonadjacent vertices x(1) an x(2), then G is a fractional (g, f, n', m)-critical deleted graph. The result is tight on the neighborhood union condition in some sense.