Multi-switch: A tool for finding potential edge-disjoint 1-factors

Abstract

Let n be even,  let π = (d₁, ... , d_n) be a graphic degree sequence, and let π - k = (d₁-k, ... , d_n-k) also be graphic.  Kundu proved that π has a realization G containing a k-factor, or k-regular graph.  Another way to state the conclusion of Kundu's theorem is that π potentially contains a k-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: π potentially contains k edge-disjoint 1-factors.  Along these lines, they proved π 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 π potentially has edge-disjoint copies of a (k-4)-factor and four 1-factors. We also prove that π potentially has (⌊k/2⌋+2) edge-disjoint 1-factors, but in this case cannot prove the existence of a large regular graph.