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

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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 languageEnglish
Pages (from-to)273-281
Number of pages9
JournalAustralasian Journal of Combinatorics
Publication statusPublished - 2002
Externally publishedYes


ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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