### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 149-166 |

Number of pages | 18 |

Journal | Graphs and combinatorics |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2012 |

### Keywords

- k-Factor
- EWI-22742
- MSC-05C
- Best monotone condition
- METIS-296437
- IR-83470
- Degree sequence

### Cite this

*Graphs and combinatorics*,

*28*(2), 149-166. https://doi.org/10.1007/s00373-011-1044-z

}

*Graphs and combinatorics*, vol. 28, no. 2, pp. 149-166. https://doi.org/10.1007/s00373-011-1044-z

**Degree sequences and the existence of k-factors.** / Bauer, D.; Broersma, Haitze J.; van den Heuvel, J.; Kahl, N.; Schmeichel, E.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Degree sequences and the existence of k-factors

AU - Bauer, D.

AU - Broersma, Haitze J.

AU - van den Heuvel, J.

AU - Kahl, N.

AU - Schmeichel, E.

N1 - eemcs-eprint-22742

PY - 2012/3

Y1 - 2012/3

N2 - We consider sufficient conditions for a degree sequence π to be forcibly k-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially k-factor graphical. We first give a theorem for π to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most β ≥ 0. These theorems are equal in strength to Chvátal’s well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for π to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from k = 1 to k = 2, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a k-factor will increase superpolynomially in k. This suggests the desirability of finding a theorem for π to be forcibly k-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any k ≥ 2, based on Tutte’s well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness.

AB - We consider sufficient conditions for a degree sequence π to be forcibly k-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially k-factor graphical. We first give a theorem for π to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most β ≥ 0. These theorems are equal in strength to Chvátal’s well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for π to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from k = 1 to k = 2, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a k-factor will increase superpolynomially in k. This suggests the desirability of finding a theorem for π to be forcibly k-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any k ≥ 2, based on Tutte’s well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness.

KW - k-Factor

KW - EWI-22742

KW - MSC-05C

KW - Best monotone condition

KW - METIS-296437

KW - IR-83470

KW - Degree sequence

U2 - 10.1007/s00373-011-1044-z

DO - 10.1007/s00373-011-1044-z

M3 - Article

VL - 28

SP - 149

EP - 166

JO - Graphs and combinatorics

JF - Graphs and combinatorics

SN - 0911-0119

IS - 2

ER -