arXiv 2203.103372022-03-19math.CO

关于图的结构方面

On Structural Aspects of Friends-And-Strangers Graphs

Ryan Jeong

备注：: 32 pages, 18 figures

摘要：

给定两个顶点数相同的图$X$和$Y$，朋友-陌生人图$\mathsf{FS}(X, Y)$的顶点是从$V(X)$到$V(Y)$的所有$n!$双射，双射$σ, τ$相邻当且仅当它们在$V(X)$的两个元素上不同，其映射在$Y$中相邻。本文研究了某个集合上的所有图$X$的$\mathsf{FS}(X, Y)$连通的充要条件。在设置中，我们采取$X$从所有的双连通图的集合，我们证明了，$\mathsf{FS}(X, Y)$是连接的所有双连通$X$当且仅当$\overline{Y}$是一个森林与树的共同互质大小，这解决了一个猜想的Defant和Kravitz。我们还发起，并取得重大进展，确定周长的$\mathsf{FS}(X, \text{Star}_n)$连接图$X$，特别是专注于必要的轨迹，中心顶点的$\text{Star}_n$采取周围所有这样的图$X$，以实现周长。

英文摘要：

Given two graphs $X$ and $Y$ with the same number of vertices, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all $n!$ bijections from $V(X)$ to $V(Y)$, with bijections $σ, τ$ adjacent if and only if they differ on two elements of $V(X)$, whose mappings are adjacent in $Y$. In this article, we study necessary and sufficient conditions for $\mathsf{FS}(X, Y)$ to be connected for all graphs $X$ from some set. In the setting that we take $X$ to be drawn from the set of all biconnected graphs, we prove that $\mathsf{FS}(X, Y)$ is connected for all biconnected $X$ if and only if $\overline{Y}$ is a forest with trees of jointly coprime size; this resolves a conjecture of Defant and Kravitz. We also initiate and make significant progress toward determining the girth of $\mathsf{FS}(X, \text{Star}_n)$ for connected graphs $X$, and in particular focus on the necessary trajectories that the central vertex of $\text{Star}_n$ takes around all such graphs $X$ to achieve the girth.

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。

摘要：

英文摘要：

相关论文