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16 February 2025
 
  » arxiv » 1508.0079

 Article overview



Multi-Switch: a Tool for Finding Potential Edge-Disjoint $1$-factors
Tyler Seacrest ;
Date 1 Aug 2015
AbstractLet $n$ 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 ($lfloor k/2 floor + 2$) edge-disjoint $1$-factors, but in this case cannot prove the existence of a large regular graph.
Source arXiv, 1508.0079
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