# Trees through specified vertices

Research output: Contribution to journalArticle

1 Citation (Scopus)

### Abstract

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac's theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n - 2 if G is a connected graph with A ⊂ V (G) such that dG (v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T) ⊃ A and dT (v) ≤ ⌈ frac(n - 1, k) ⌉ for all v ∈ A.

Original language English 2749-2754 6 Discrete Mathematics 309 9 https://doi.org/10.1016/j.disc.2008.06.032 Published - 6 May 2009

### Fingerprint

Hamiltonians
Hamiltonian circuit
Connected graph
Theorem

### Keywords

• Subtrees through specified vertices

### Cite this

Cutler, Jonathan. / Trees through specified vertices. In: Discrete Mathematics. 2009 ; Vol. 309, No. 9. pp. 2749-2754.
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In: Discrete Mathematics, Vol. 309, No. 9, 06.05.2009, p. 2749-2754.

Research output: Contribution to journalArticle

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AU - Cutler, Jonathan

PY - 2009/5/6

Y1 - 2009/5/6

N2 - We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac's theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n - 2 if G is a connected graph with A ⊂ V (G) such that dG (v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T) ⊃ A and dT (v) ≤ ⌈ frac(n - 1, k) ⌉ for all v ∈ A.

AB - We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac's theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n - 2 if G is a connected graph with A ⊂ V (G) such that dG (v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T) ⊃ A and dT (v) ≤ ⌈ frac(n - 1, k) ⌉ for all v ∈ A.

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