A New Sufficient Condition for a Graph To Be (g, f, n)-Critical

Abstract

AbstractLet G be a graph of order p, let a, b, and n be nonnegative integers with 1 ≤ a < b, and let g and f be two integer-valued functions defined on V(G) such that a ≤ g(x) < f (x) ≤ b for all x ∈ V(G). A (g, f )-factor of graph G is a spanning subgraph F of G such that g(x) ≤ dF(x) ≤ f (x) for each x ∈ V(F). Then a graph G is called (g, f, n)-critical if after deleting any n vertices of G the remaining graph of G has a (g, f )-factor. The binding number bind(G) of G is the minimum value of |NG(X)|/|X| taken over all non-empty subsets X of V(G) such that NG(X) ≠ V(G). In this paper, it is proved that G is a (g, f, n)-critical graph ifFurthermore, it is shown that this result is best possible in some sense.