Spanning trees of bounded degree

Andrzej Czygrinow, Genghua Fan, Glenn Hurlbert, Henry Kierstead, William T. Trotter

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

Dirac's classic theorem asserts that if G is a graph on n vertices, and δ(G) ≥ n/2, then G has a hamilton cycle. As is well known, the proof also shows that if deg(x) + deg(y) ≥ (n - 1), for every pair x, y of independent vertices in G, then G has a hamilton path. More generally, S. Win has shown that if k ≥ 2, G is connected and ∑x∈I deg(x) ≥ n - 1 whenever I is a k-element independent set, then G has a spanning tree T with Δ(T) ≤ k. Here we are interested in the structure of spanning trees under the additional assumption that G does not have a spanning tree with maximum degree less than k. We show that apart from a single exceptional class of graphs, if ∑x∈I deg(x) ≥ n - 1 for every k-element independent set, then G has a spanning caterpillar T with maximum degree k. Furthermore, given a maximum path P in G, we may require that P is the spine of T and that the set of all vertices whose degree in T is 3 or larger is independent in T.

Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalElectronic Journal of Combinatorics
Volume8
Issue number1 R
StatePublished - Dec 1 2001

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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  • Cite this

    Czygrinow, A., Fan, G., Hurlbert, G., Kierstead, H., & Trotter, W. T. (2001). Spanning trees of bounded degree. Electronic Journal of Combinatorics, 8(1 R), 1-12.