Abstract
In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set \({S \subseteq V(G)}\) of cardinality n(k−1) + c + 2, there exists a vertex set \({X \subseteq S}\) of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c −1. Then G has a spanning tree T with maximum degree at most \({k+\lceil c/n\rceil}\) and \({\sum_{v\in V(T)}\max\{d_T(v)-k,0\}\leq c}\) .
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This paper is dedicated to Prof. H. Enomoto on the occasion of his 60th birthday.
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Fujisawa, J., Matsumura, H. & Yamashita, T. Degree Bounded Spanning Trees. Graphs and Combinatorics 26, 695–720 (2010). https://doi.org/10.1007/s00373-010-0941-x
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DOI: https://doi.org/10.1007/s00373-010-0941-x