Navigating Between Packings of Graphic Sequences

Let $\pi_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $\pi_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $\pi_1$ and $\pi_2$, respectively, on vertex set $\{v_1,\dots,v_n\}$ such that for $j\in\{1,2\}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i\in\{1,\ldots,n\}$. In this case, we say that $(G_1,G_2)$ is a $(\pi_1,\pi_2)$-\textit{packing}. A clear necessary condition for graphic sequences $\pi_1$ and $\pi_2$ to pack is that $\pi_1+\pi_2$, 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 $NP$- complete. S.~Kundu proved in 1973 that if $\pi_2$ is almost regular, that is each element is from $\{k-1, k\}$, then $\pi_1$ and $\pi_2$ pack if and only if $\pi_1+\pi_2$ is graphic. In this paper we will consider graphic sequences $\pi$ with the property that $\pi+\mathbf{1}$ is graphic. By Kundu's theorem, the sequences $\pi$ and $\mathbf{1}$ pack, and there exist edge-disjoint realizations $G$ and $\mathcal{I}$, where $\mathcal{I}$ is a 1-factor. We call such a $(\pi,\mathbf{1})$ packing a {\em Kundu realization}. Assume that $\pi$ is a graphic sequence, in which each term is at most $n/24$, that packs with $\mathbf{1}$. This paper contains two results. On one hand, any two Kundu realizations of the degree sequence $\pi+\mathbf{1}$ can be transformed into each other through a sequence of other Kundu realizations by swap operations. On the other hand, the same conditions ensure that any particular 1-factor can be part of a Kundu realization of $\pi+\mathbf{1}$.