On a Spanning k-Tree in which Specified Vertices Have Degree Less Than k

摘要

A k-tree is a tree with maximum degree at most k. In this paper, we give a degree sum condition for a graph to have a spanning k-tree in which specified vertices have degree less than k. We denote by σ_(k)(G) the minimum value of the degree sum of k independent vertices in a graph G. Let k ≥ 3 and s ≥ 0 be integers, and suppose G is a connected graph and σ_(k)(G) ≥ |V (G)|+s?1. Then for any s specified vertices, G contains a spanning k-tree in which every specified vertex has degree less than k. The degree condition is sharp.