Navigating Between Packings of Graphic Sequences
Peter L. Erdos, Michael Ferrara, Stephen G. Hartke

TL;DR
This paper studies the structure and transformations of specific graph packings called Kundu realizations, where a graphic sequence and a 1-factor are packed without edge overlaps, providing methods to convert between realizations and include particular 1-factors.
Contribution
It proves that all Kundu realizations of a given degree sequence can be transformed into each other via swap operations and that any 1-factor can be incorporated into such a realization under certain conditions.
Findings
All Kundu realizations are connected through swap operations.
Any specific 1-factor can be included in a Kundu realization.
Results apply when the degree sequence terms are at most n/24.
Abstract
Let and be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations and of and , respectively, on vertex set such that for , for all . In this case, we say that is a -\textit{packing}. A clear necessary condition for graphic sequences and to pack is that , their componentwise sum, is also graphic. It is known, however, that this condition is not sufficient, and furthermore that the general problem of determining if two sequences pack is - complete. S.~Kundu proved in 1973 that if is almost regular, that is each element is from , then and pack if and only if is graphic. In…
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