Opuscula Math. 37, no. 4 (2017), 501-508

Opuscula Mathematica

Spanning trees with a bounded number of leaves

Junqing Cai
Evelyne Flandrin
Hao Li
Qiang Sun

Abstract. In 1998, H. Broersma and H. Tuinstra proved that: Given a connected graph \(G\) with \(n\geq 3\) vertices, if \(d(u)+d(v)\geq n-k+1\) for all non-adjacent vertices \(u\) and \(v\) of \(G\) (\(k\geq 1\)), then \(G\) has a spanning tree with at most \(k\) leaves. In this paper, we generalize this result by using implicit degree sum condition of \(t\) (\(2\leq t\leq k\)) independent vertices and we prove what follows: Let \(G\) be a connected graph on \(n\geq 3\) vertices and \(k\geq 2\) be an integer. If the implicit degree sum of any \(t\) independent vertices is at least \(\frac{t(n-k)}{2}+1\) for (\(k\geq t\geq 2\)), then \(G\) has a spanning tree with at most \(k\) leaves.

Keywords: spanning tree, implicit degree, leaves.

Mathematics Subject Classification: 05C07.

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Cite this article as:
Junqing Cai, Evelyne Flandrin, Hao Li, Qiang Sun, Spanning trees with a bounded number of leaves, Opuscula Math. 37, no. 4 (2017), 501-508, http://dx.doi.org/10.7494/OpMath.2017.37.4.501

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