Independence number, connectivity and -critical graphs

Let G be a graph, and let a , b and k be nonnegative integers with 1 ≤ a ≤ b . An [ a , b ] -factor of graph G is defined as a spanning subgraph F of G such that a ≤ d F ( x ) ≤ b for each x ∈ V ( G ) . 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 if κ ( G ) ≥ max { ( a + 1 ) b + 2 k 2 , ( a + 1 ) 2 α ( G ) + 4 b k 4 b } , then G is an ( a , b , k ) -critical graph. Furthermore, it is showed that the result in this paper is best possible in some sense.