An existence theorem on fractional deleted graphs

Let r≥ 2 , β≥ 0 and 2≤ a≤ b-β be integers, and let G be a graph of order n with n≥(a+b)(a+b+1+(r-2)(b-β ))/a+β and δ (G)≥(r-1)(b-β )(b+1)/a+β , and let g and f be two integer-valued functions defined on V(G) satisfying a≤ g(x)≤ f(x)-β≤ b-β for each x∈ V(G) . In this paper, it is proved that G is a fractional (g,f) -deleted graph if G satisfies max{d_G(x_1),d_G(x_2),… ,d_G(x_r)}≥(b-β )n/a+b for any independent subset {x_1,x_2,… ,x_r} in G . Furthermore, it is shown that the result in this paper is best possible in some sense.