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

Abstract

Let n be even, let π = ( d 1 , ... , d n ) be a graphic degree sequence, and let π - k = ( d 1 - 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.