Existence theorems on star-factor deleted (critical) graph and critical avoidable graphs

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.