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A k-Tree Containing Specified Vertices

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Abstract

A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let \({S \subseteq V(G)}\) with κ(S) ≥ 1. Suppose that for every l > κ(S), there exists an integer t such that \({1 \le t \leq (k-1)l+2 - \lfloor \frac{l-1}{k} \rfloor}\) and the degree sum of any t independent vertices of S is at least n + tl − kl − 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.

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Authors and Affiliations

  1. Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Shuya Chiba

  2. Tokyo, Japan

    Ryota Matsubara & Masao Tsugaki

  3. Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-0061, Japan

    Kenta Ozeki

Authors
  1. Shuya Chiba
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  2. Ryota Matsubara
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  3. Kenta Ozeki
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  4. Masao Tsugaki
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Correspondence to Shuya Chiba.

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Chiba, S., Matsubara, R., Ozeki, K. et al. A k-Tree Containing Specified Vertices. Graphs and Combinatorics 26, 187–205 (2010). https://doi.org/10.1007/s00373-010-0903-3

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  • DOI: https://doi.org/10.1007/s00373-010-0903-3

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