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

Tyler Seacrest


Let n be even,  let π = (d1, ... , dn) be a graphic degree sequence, and let π - k = (d1-k, ... , dn-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.


graph, degree sequence, matching, Kundu's theorem

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Arthur H. Busch, Michael J. Ferrara, Stephen G. Hartke, Michael S. Jacobson, Hemanshu Kaul, and Douglas B. West, Packing of graphic n-tuples, J. Graph Theory, 70(1):29--39, 2012. Preprint available at

Yong~Chuan Chen, A short proof of Kundu's k-factor theorem, Discrete Math., 71(2):177--179, 1988. Available at

Kundu, The k-factor conjecture is true, Discrete Math., 6:367--376, 1973. Available at

Daniel Leven and Zvi Galil, NP completeness of finding the chromatic index of regular graphs, J. Algorithms, 4(1):35--44, 1983. Available at

Julius Petersen, Die Theorie der regularen graphs, Acta Math., 15(1):193--220, 1891.

Douglas B. West, Introduction to graph theory, Prentice Hall Inc., Upper Saddle River, NJ, 1996.


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