Degree Sequences and the Existence of $k$-Factors

TL;DR

This paper establishes new sufficient conditions for degree sequences to be forcibly $k$-factor graphical, extending classical results and proposing a more computationally feasible theorem for all $k \\ge 2$ based on Tutte's theorem.

Contribution

The paper introduces strong theorems for forcibly $k$-factor graphical degree sequences, including a general theorem for all $k \\ge 2$ with improved algorithmic complexity.

Findings

Theorems for forcibly 1-factor and 2-factor graphical degree sequences are as strong as Chvátal's Hamiltonian theorem.

Number of conditions for $k=2$ increases significantly, conjectured to grow superpolynomially with $k$.

A new theorem for any $k \\ge 2$ based on Tutte's factor theorem is presented, offering a more computationally efficient criterion.

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.