Let Formula: see text be a graph, let Formula: see text be an integer and let Formula: see text = Formula: see text. A Formula: see text-factor of Formula: see text is a spanning subgraph of Formula: see text, in which each component is isomorphic to a member in Formula: see text. Let Formula: see text, Formula: see text and Formula: see text be three nonnegative integers. For any Formula: see text with Formula: see text, assume Formula: see text has a Formula: see text-factor, then Formula: see text is a Formula: see text-factor critical graph. For any Formula: see text with Formula: see text, assume Formula: see text has a Formula: see text-factor, then Formula: see text is a Formula: see text-factor deleted graph. For any Formula: see text with Formula: see text and any Formula: see text, assume Formula: see text contains a Formula: see text-factor, then Formula: see text is Formula: see text-factor critical avoidable. In this paper, we obtain existence theorems on a Formula: see text-factor deleted graph(Formula: see text-factor critical graph) is regarding the binding number and a Formula: see text-factor critical avoidable graph is regarding the tight toughness and the isolated toughness for a graph, respectively. Furthermore, by constructing extremal graphs, we show that the bounds are best possible.
Ren et al. (Fri,) studied this question.