# Degree conditions for the existence of [k, k + 1]-factors containing a given Hamiltonian cycle

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9 Citations (Scopus)

### Abstract

Let k ≥ 2 be an integer and G a 2-connected graph of order {pipe}G{pipe} ≥ 3 with minimum degree at least k. Suppose that {pipe}G{pipe} ≥ 8k - 16 for even {pipe}G{pipe} and {pipe}G{pipe} ≥ 6k - 13 for odd {pipe}G{pipe}. We prove that G has a [k, k + 1]-factor containing a given Hamiltonian cycle if max{degG(x), degG(y)} ≥ {pipe}G{pipe}/2 for each pair of nonadjacent vertices x and y in G. This is best possible in the sense that there exists a graph having no k-factor containing a given Hamiltonian cycle under the same conditions. The lower bound of {pipe}G{pipe} is also sharp.

Original language English 273-281 9 Australasian Journal of Combinatorics 26 Published - 2002 Yes

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics

### Cite this

In: Australasian Journal of Combinatorics, Vol. 26, 2002, p. 273-281.

Research output: Contribution to journalArticle

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title = "Degree conditions for the existence of [k, k + 1]-factors containing a given Hamiltonian cycle",
abstract = "Let k ≥ 2 be an integer and G a 2-connected graph of order {pipe}G{pipe} ≥ 3 with minimum degree at least k. Suppose that {pipe}G{pipe} ≥ 8k - 16 for even {pipe}G{pipe} and {pipe}G{pipe} ≥ 6k - 13 for odd {pipe}G{pipe}. We prove that G has a [k, k + 1]-factor containing a given Hamiltonian cycle if max{degG(x), degG(y)} ≥ {pipe}G{pipe}/2 for each pair of nonadjacent vertices x and y in G. This is best possible in the sense that there exists a graph having no k-factor containing a given Hamiltonian cycle under the same conditions. The lower bound of {pipe}G{pipe} is also sharp.",
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AB - Let k ≥ 2 be an integer and G a 2-connected graph of order {pipe}G{pipe} ≥ 3 with minimum degree at least k. Suppose that {pipe}G{pipe} ≥ 8k - 16 for even {pipe}G{pipe} and {pipe}G{pipe} ≥ 6k - 13 for odd {pipe}G{pipe}. We prove that G has a [k, k + 1]-factor containing a given Hamiltonian cycle if max{degG(x), degG(y)} ≥ {pipe}G{pipe}/2 for each pair of nonadjacent vertices x and y in G. This is best possible in the sense that there exists a graph having no k-factor containing a given Hamiltonian cycle under the same conditions. The lower bound of {pipe}G{pipe} is also sharp.

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