论文标题
蜘蛛和补充的朋友和纠缠中的连接性
Connectedness in Friends-and-Strangers Graphs of Spiders and Complements
论文作者
论文摘要
令$ x $和$ y $是两个带有顶点套装$ [n] $的图。他们的朋友和trangers Graph $ \ mathsf {fs}（x，y）$是一个图表，其顶点与$ s_n $的元素相对应，而两个排列$σ$和$σ'$如果由thresspositive $ \ \ \ y $ $ \ a，$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$σ'$相邻$σ（b）$以$ y $相邻。特定的朋友和纠纷图，例如$ \ MATHSF {fs}（\ Mathsf {path} _n，y）$和$ \ Mathsf {fs}（\ Mathsf {cyclesf {cycle} _n，y）$，并且已经研究了它们的连接组件，它们使用了各种等价关系等等。蜘蛛图是一组路径图的集合，它们都连接到单个中心点。在本文中，我们更深入地研究了$ \ mathsf {fs}（x，y）$何时在$ x $是蜘蛛时连接的，而$ y $是蜘蛛或tadpole的补充。
Let $X$ and $Y$ be two graphs with vertex set $[n]$. Their friends-and-strangers graph $\mathsf{FS}(X,Y)$ is a graph with vertices corresponding to elements of the group $S_n$, and two permutations $σ$ and $σ'$ are adjacent if they are separated by a transposition $\{a,b\}$ such that $a$ and $b$ are adjacent in $X$ and $σ(a)$ and $σ(b)$ are adjacent in $Y$. Specific friends-and-strangers graphs such as $\mathsf{FS}(\mathsf{Path}_n,Y)$ and $\mathsf{FS}(\mathsf{Cycle}_n,Y)$ have been researched, and their connected components have been enumerated using various equivalence relations such as double-flip equivalence. A spider graph is a collection of path graphs that are all connected to a single center point. In this paper, we delve deeper into the question of when $\mathsf{FS}(X,Y)$ is connected when $X$ is a spider and $Y$ is the complement of a spider or a tadpole.
