# Spanning Trees with few Branch and end Vertices

## Keywords:

Spanning tree, End vertex, k-ended tree, Branch vertex, Degree sum of the branch vertices, Ore-type condition## Abstract

For a graph G, let σ2 be the minimum degree sum of two nonadjacent vertices in G. A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. We consider: (*) connected graphs on n vertices such that σ2 ≥ n - k + 1 for some positive integer k. In 1976, it was proved (by the author) that every graph satisfying (*), has a spanning tree with at most k end vertices. In this paper we first show that every graph satisfying (*), has a spanning tree with at most k +1 branch and end vertices altogether. The next result states that every graph satisfying (*), has a spanning tree with at most 1/2 (k - 1) branch vertices. The third result states that every graph satisfying (*), has a spanning tree with at most 3/2 (k -1) degree sum of branch vertices. All results are sharp.

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*Mathematical Problems of Computer Science*,

*46*, 18–25. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/143

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