The Connectedness of the Friends-and-Strangers Graph of a Lollipop and Others

Abstract

Let X and Y be any two graphs of order n. The friends-and-strangers graph $\textsf{FS}(X,Y)$ of X and Y is a graph with vertex set consisting of all bijections $\sigma :V(X) \rightarrow V(Y)$, in which two bijections $\sigma$, $\sigma '$ are adjacent if and only if they differ precisely on two adjacent vertices of X, and the corresponding mappings are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let $\textsf{Lollipop}_{n-k,k}$ be a lollipop graph of order n obtained by identifying one end of a path of order $n-k+1$ with a vertex of a complete graph of order k. Defant and Kravitz started to study the connectedness of $\textsf{FS}(\textsf{Lollipop}_{n-k,k},Y)$. In this paper, we give a sufficient and necessary condition for $\textsf{FS}(\textsf{Lollipop}_{n-k,k},Y)$ to be connected for all $2\le k\le n$, which interpolates between two previous results on paths and complete graphs.