在图上行走的朋友和陌生人
Friends and Strangers Walking on Graphs
摘要：
给定图$X$和$Y$，其顶点集$V(X)$和$V(Y)$具有相同的基数，我们定义了一个图$\mathsf{FS}(X,Y)$，其顶点集由所有双射$σ:V(X)\to V(Y)$组成，其中两个双射$σ$和$σ'$是相邻的，如果它们在除了两个相邻顶点$a,b \in V(X)$之外的任何地方都一致，使得$σ(a)$和$σ(b)$在$Y$中是相邻的。这个设置，这有一个自然的解释方面的朋友和陌生人走在图，提供了一个共同的推广凯莱图的对称群所产生的换位，著名的$15$-难题，概括的$15$-难题的研究威尔逊，和工作的斯坦利有关的标志$h$-向量。我们得到了几个一般性的结果，图$\mathsf{FS}(X,Y)$之前，我们的注意力集中在一些特定的选择$X$。当$X$是一个路径图时，我们证明了$\mathsf{FS}(X,Y)$的连通分支对应于$Y$的补图的非循环方向。当$X$是一个循环，我们得到一个完整的描述的连通组件的$\mathsf{FS}(X,Y)$的复曲面非循环方向的补充$Y$。然后，我们得到各种必要和/或充分条件的图$X$和$Y$，保证连通性的$\mathsf{FS}(X,Y)$。最后，我们提出了几个有希望的进一步问题。
英文摘要：
Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, we define a graph $\mathsf{FS}(X,Y)$ whose vertex set consists of all bijections $σ:V(X)\to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $σ(a)$ and $σ(b)$ are adjacent in $Y$. This setup, which has a natural interpretation in terms of friends and strangers walking on graphs, provides a common generalization of Cayley graphs of symmetric groups generated by transpositions, the famous $15$-puzzle, generalizations of the $15$-puzzle as studied by Wilson, and work of Stanley related to flag $h$-vectors. We derive several general results about the graphs $\mathsf{FS}(X,Y)$ before focusing our attention on some specific choices of $X$. When $X$ is a path graph, we show that the connected components of $\mathsf{FS}(X,Y)$ correspond to the acyclic orientations of the complement of $Y$. When $X$ is a cycle, we obtain a full description of the connected components of $\mathsf{FS}(X,Y)$ in terms of toric acyclic orientations of the complement of $Y$. We then derive various necessary and/or sufficient conditions on the graphs $X$ and $Y$ that guarantee the connectedness of $\mathsf{FS}(X,Y)$. Finally, we raise several promising further questions.
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