发表机构
Charles University in Prague; Faculty of Mathematics and Physics, Charles University in Prague（布拉格查理大学; 布拉格查理大学数学物理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究每个度序列的实现图\(\mathcal{G_F}(d)\)是否有哈密顿路径，通过证明得出对于任意度序列\(d\)，\(\mathcal{G_F}(d)\)都有哈密顿路径，还得出具有规定行和列和的\((0,1)\)矩阵的交换图有哈密顿路径，回答了Brualdi的问题。
AI 中文摘要
给定一个度序列\(d\)，实现图\(\mathcal{G_F}(d)\)的顶点是\(d\)的所有带标签的实现，若两个实现相差一个\(2 -\)切换则它们相邻。我们证明对于每个度序列\(d\)，\(\mathcal{G_F}(d)\)都有一条哈密顿路径。该问题由Arikati和Peled于1999年提出，他们表明当\(d\)的优化差距为\(1\)时，\(\mathcal{G_F}(d)\)包含一个哈密顿圈。后来Barrus（2016年）和Mütze（2023年）独立提出对于每个度序列\(d\)，\(\mathcal{G_F}(d)\)中是否存在哈密顿路径或圈。结果，我们得出具有规定行和列和的\((0,1)\)矩阵的交换图有一条哈密顿路径，从而回答了Brualdi（1980年）的一个问题。
英文摘要
For a degree sequence $d$, the realization graph $\mathcal{G_F}(d)$ is the graph whose vertices are the labeled realizations of $d$, two of which are adjacent if they differ by a single $2$-switch. We prove that for every degree sequence $d$ and every realization $S$ of $d$, the graph $\mathcal{G_F}(d)$ contains a Hamilton path starting at $S$. This answers a question of Barrus (2016), which was also raised independently by Mütze (2023) in his survey of combinatorial Gray codes. As a consequence, an embedding observation of Arikati and Peled (1999) implies that for any vectors $R$ and $C$ of non-negative integers, the interchange graph $\mathcal{A}_{\mathcal F}(R,C)$ of $(0,1)$-matrices with row sums $R$ and column sums $C$ contains a Hamilton path starting at any prescribed matrix, thereby resolving a question of Brualdi (1980).
Comments17 pages, 6 figures
