在图序列的包装之间导航
Navigating Between Packings of Graphic Sequences
摘要：
设$π_1=(d_1^{(1)}, \ldots,d_n^{(1)})$和$π_2=(d_1^{(2)},\ldots,d_n^{(2)})$是图形序列。我们称它们为\emph{pack}，如果在顶点集$\{v_1,\dots,v_n\}$上分别存在$π_1$和$π_2$的边不相交实现$G_1$和$G_2$，使得对于$j\in\{1,2\}$，对于所有$i\in\{1,\ldots,n\}$都有$d_{G_j}(v_i)=d_i^{(j)}$。在这种情况下，我们说$(G_1,G_2)$是$(π_1,π_2)$ - \textit{packing}。图形序列$π_1$和$π_2$打包的一个明确的必要条件是$π_1+π_2$，它们的分量和，也是图形的。然而，已知的是，该条件是不充分的，并且此外，确定两个序列包是否是$NP$ -完全的一般问题。S.~ Kundu在1973年证明了如果$π_2$几乎正则，即每个元素都来自$\{k-1, k\}$，那么$π_1$和$π_2$包当且仅当$π_1+π_2$是图形的。本文考虑图序列$π$，它具有$π+\mathbf{1}$是图的性质.根据Kundu定理，序列$π$和$\mathbf{1}$是压缩的，并且存在边不相交的实现$G$和$\mathcal{I}$，其中$\mathcal{I}$是1-因子.我们称这样的一个$(π,\mathbf{1})$包装为{ \em昆都实现}。假设$π$是一个图形序列，其中每个项最多为$n/24$，它与$\mathbf{1}$打包。本文包含两个结果。一方面，度序列$π+\mathbf{1}$的任意两个Kundu实现可以通过交换操作通过其他Kundu实现的序列相互转换。另一方面，相同的条件确保任何特定的1-因子都可以成为$π+\mathbf{1}$的Kundu实现的一部分。
英文摘要：
Let $π_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $π_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $π_1$ and $π_2$, 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 $(π_1,π_2)$-\textit{packing}. A clear necessary condition for graphic sequences $π_1$ and $π_2$ to pack is that $π_1+π_2$, 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 $π_2$ is almost regular, that is each element is from $\{k-1, k\}$, then $π_1$ and $π_2$ pack if and only if $π_1+π_2$ is graphic. In this paper we will consider graphic sequences $π$ with the property that $π+\mathbf{1}$ is graphic. By Kundu's theorem, the sequences $π$ 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 $(π,\mathbf{1})$ packing a {\em Kundu realization}. Assume that $π$ 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 $π+\mathbf{1}$ 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 $π+\mathbf{1}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
