Spanning f-trees in n-connected graphs

For a graph [Formula: see text] and a positive-integer-valued function [Formula: see text] on [Formula: see text], we define a tree [Formula: see text] of [Formula: see text] as an [Formula: see text]-tree if [Formula: see text] for each vertex [Formula: see text], by which we prove the following theorem. Let [Formula: see text] be an [Formula: see text]-connected graph and [Formula: see text] a function. Suppose [Formula: see text] and [Formula: see text] are two nonadjacent vertices in [Formula: see text] whose degrees satisfy [Formula: see text] then [Formula: see text] contains a spanning [Formula: see text]-tree if and only if [Formula: see text] also contains a spanning [Formula: see text]-tree. This generalizes the theorem in [M. Kano and H. Kishimoto, Spanning [Formula: see text]-trees of [Formula: see text]-connected graphs, Graphs Combin. 27 (2011) 413–418].