On a conjecture that strengthens Kundu's $k$-factor Theorem

Abstract

Let $π=(d_{1},\ldots,d_{n})$ be a non-increasing degree sequence with even $n$. In 1974, Kundu showed that if $\mathcal{D}_{k}(π)=(d_{1}-k,\ldots,d_{n}-k)$ is graphic, then some realization of $π$ has a $k$-factor. For $r\leq 2$, Busch et al. and later Seacrest for $r\leq 4$ showed that if $r\leq k$ and $\mathcal{D}_{k}(π)$ is graphic, then there is a realization with a $k$-factor whose edges can be partitioned into a $(k-r)$-factor and $r$ edge-disjoint $1$-factors. We improve this to any $r\leq \min\{\lceil\frac{k+5}{3}\big\rceil,k\}$. In 1978, Brualdi and then Busch et al. in 2012, conjectured that $r=k$. The conjecture is still open for $k\geq6$. However, Busch et al. showed the conjecture is true when $d_{1}\leq \frac{n}{2}+1$ or $d_{n}\geq \frac{n}{2}+k-2$. We explore this conjecture by first developing new tools that generalize edge-exchanges. With these new tools, we can drop the assumption $\mathcal{D}_{k}(π)$ is graphic and show that if $d_{d_{1}-d_{n}+k}\geq d_{1}-d_{n}+k-1,$ then $π$ has a realization with $k$ edge-disjoint $1$-factors. From this we confirm the conjecture when $d_{n}\geq \frac{d_{1}+k-1}{2}$ or when $\mathcal{D}_{k}(π)$ is graphic and $d_{1}\leq \max \{n/2+d_{n}-k,(n+d_{n})/2\}$.