Multi-Switch: a Tool for Finding Potential Edge-Disjoint $1$-factors

Abstract: Let $n$ be even, let $\pi = (d_1, \ldots, d_n)$ be a graphic degree sequence, and let $\pi - k = (d_1 - k, \ldots, d_n - k)$ also be graphic. Kundu proved that $\pi$ has a realization $G$ containing a $k$-factor, or $k$-regular graph. Another way to state the conclusion of Kundu's theorem is that $\pi$ \emph{potentially} contains a $k$-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: $\pi$ potentially contains $k$ edge-disjoint $1$-factors. Along these lines, they proved $\pi$ would potentially contain edge-disjoint copies of a $(k-2)$-factor and two $1$-factors. We follow the methods of Busch et al.\ but introduce a new tool which we call a multi-switch. Using this new idea, we prove that $\pi$ potentially has edge-disjoint copies of a $(k-4)$-factor and four $1$-factors. We also prove that $\pi$ potentially has ($\lfloor k/2 \rfloor + 2$) edge-disjoint $1$-factors, but in this case cannot prove the existence of a large regular graph.