Spanning k-tree with specified vertices

Abstract

A [Formula: see text]-tree is a tree with maximum degree at most [Formula: see text]. For a graph [Formula: see text] and [Formula: see text] with [Formula: see text], let [Formula: see text] be the cardinality of a maximum independent set containing [Formula: see text] and [Formula: see text]. For a graph [Formula: see text] and [Formula: see text], the local connectivity [Formula: see text] is defined to be the maximum number of internally disjoint paths connecting [Formula: see text] and [Formula: see text] in [Formula: see text]. In this paper, we prove the following theorem and show the condition is sharp. Let [Formula: see text], [Formula: see text] and [Formula: see text] be integers with [Formula: see text], [Formula: see text] and [Formula: see text]. For any two nonadjacent vertices [Formula: see text] and [Formula: see text] of [Formula: see text], we have [Formula: see text] and [Formula: see text]. Then for any [Formula: see text] distinct vertices of [Formula: see text], [Formula: see text] has a spanning [Formula: see text]-tree such that each of [Formula: see text] specified vertices has degree at most [Formula: see text]. This theorem implies H. Matsuda and H. Matsumura’s result in [on a [Formula: see text]-tree containing specified cleares in a graph, Graphs Combin. 22 (2006) 371–381] and V. Neumann-Lara and E. Rivera-Campo’s result in [Spanning trees with bounded degrees, Combinatorica 11 (1991) 55–61].