# Degree sequences and the existence of k-factors

Bauer, D., Broersma, H. J., van den Heuvel, J., Kahl, N. and Schmeichel, E. (2009) Degree sequences and the existence of k-factors. . arXiv.

Full text not available from this repository.

## Abstract

We consider sufficient conditions for a degree sequence $\pi$ 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 $\pi$ to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most $\beta\ge0$. These theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for $\pi$ 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 $\pi$ to be forcibly $k$-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any $k\ge2$, 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.

Item Type: Monograph (Report) http://arxiv.org/ © 2009 The authors Mathematics Q Science > QA Mathematics Departments > Mathematics 09 Apr 2010 13:18 17 Jul 2020 23:22 http://eprints.lse.ac.uk/id/eprint/27679