A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Abstract

Let G be a connected simple graph, let XV (G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d T (x) for all xX, where d T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d T (x) ≥ f(x) for all xX if and only if for any nonempty subset SX, |N G (S) − S| − f(S) + 2|S| − ω G (S) ≥, where ω G (S) is the number of components of the subgraph induced by S.

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References

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    A. Frank and A. Gyárfás: How to orient the edges of a graph? Colloq. Math. Soc. János Bolyai 18 (1976), 353–364.

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Correspondence to Kenta Ozeki.

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Egawa, Y., Ozeki, K. A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees. Combinatorica 34, 47–60 (2014). https://doi.org/10.1007/s00493-014-2576-7

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Mathematics Subject Classification (2000)

  • 05C05
  • 05C70