Independence number, connectivity and (a,b,k)(a,b,k)-critical graphs

Let GG be a graph, and let a,ba,b and kk be nonnegative integers with 1≤a≤b1≤a≤b. An [a,b][a,b]-factor of graph GG is defined as a spanning subgraph FF of GG such that a≤dF(x)≤ba≤dF(x)≤b for each x∈V(G)x∈V(G). Then a graph GG is called an (a,b,k)(a,b,k)-critical graph if after deleting any kk vertices of GG the remaining graph of GG has an [a,b][a,b]-factor. In this paper, it is proved that if κ(G)≥max{(a+1)b+2k2,(a+1)2α(G)+4bk4b}, then GG is an (a,b,k)(a,b,k)-critical graph. Furthermore, it is showed that the result in this paper is best possible in some sense.