Some Sufficient Conditions for Component Factors Deleted or Critical Graphs

Let [Formula: see text] be a family of connected graphs. A [Formula: see text]-factor is a spanning subgraph [Formula: see text] of a graph [Formula: see text], where every connected component of [Formula: see text] is isomorphic to some graph in [Formula: see text]. Let [Formula: see text] be four integers with [Formula: see text] and [Formula: see text]. A graph [Formula: see text] is called a [Formula: see text]-factor deleted graph if [Formula: see text] admits a [Formula: see text]-factor for any [Formula: see text] with [Formula: see text]. A graph [Formula: see text] is called a [Formula: see text]-factor critical graph if [Formula: see text] admits a [Formula: see text]-factor for any [Formula: see text] with [Formula: see text]. Write [Formula: see text] and [Formula: see text], where [Formula: see text] is a special family of trees. Let [Formula: see text] be a [Formula: see text]-connected graph with [Formula: see text]. In this paper, we prove that if (i) [Formula: see text]; or (ii) [Formula: see text]; or (iii) [Formula: see text], then [Formula: see text] is a [Formula: see text]-factor deleted graph except for [Formula: see text]. Moreover, we obtain that if (i) [Formula: see text]; or (ii) [Formula: see text]; or (iii) [Formula: see text], then [Formula: see text] is a [Formula: see text]-factor critical graph. Finally, we also show that those bounds are the best possible.