Binding number and minimum degree for (a, b, k)-critical graphs

Abstract

Let a, b, k be nonnegative integers with 1 ≤a <b, and let G be a graph of order n. An [a, b]-factor of G is defined as a spanning subgraph F of G such that a < d _F(x) < b for each x ⋯ V(F). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if G satisfies n ≥(a+b-1) ²//b+bk/b-1, the binding number bind(G) ≥(a+b-1)(n-1)/bn-bk-1 and the minimum degree δ(G) ≥ 1 +(a-1)n+bk/a+b-1 Furthermore, it is showed that the result in this paper is best possible in some sense. graph,.