Degree conditions for path-factor critical deleted or covered graphs

A path-factor of a graph ?? is a spanning subgraph of ?? whose components are paths. A ??=??-factor of a graph ?? is a path-factor of ?? whose components are paths with at least ?? vertices, where ?? is an integer with ?? = 2. A graph ?? is ??=??-factor covered if for any ?? ? ??(??), ?? admits a ??=??-factor including ??. A graph ?? is (??=??, ??)-factor critical deleted if for any ?? ? ?? (??) with |??| = ?? and any ?? ? ??(?? - ??), ?? - ?? - ?? has a ??=??-factor. A graph ?? is (??=??, ??)-factor critical covered if for any ?? ? ?? (??) with |??| = ??, ?? - ?? is a ??=??-factor covered graph. In this paper, we verify that (i) an (?? + ?? + 2)-connected graph ?? of order ?? with ?? = 4?? + ?? + 7 is (??=3, ??)-factor critical deleted if max{????(??1), ??(??)(??(2)), . . . , ????(??(2??+1))} =( ??+2??) /(3) for any independent set {??1, ??2, . . . , ??2??+1} of ??, where ?? and ?? are two nonnegative integers with ?? = 1; (ii) an (?? + ?? + 1)-connected graph ?? of order ?? with ?? = 4?? + ?? + 5 is (??=3, ??)-factor critical covered if max{????(??1), ????(??2), . . . , ????(??2??+1)} =( ??+2??+2)/(3) for any independent set {??(1), ??(2), . . . , ??(2??+1)} of ??, where ?? and ?? are two nonnegative integers with ?? = 1.