The connectedness of the friends-and-strangers graph of lollipop graphs and others

TL;DR

This paper investigates the conditions under which the friends-and-strangers graph of lollipop graphs and an arbitrary graph Y is connected, providing a complete characterization for all parameters.

Contribution

It establishes a necessary and sufficient condition for the connectedness of friends-and-strangers graphs involving lollipop graphs and any graph Y.

Findings

Characterization of connectedness for all k in 2 to n

Complete criteria for friends-and-strangers graphs of lollipop graphs

Extension of previous partial results on graph connectedness

Abstract

Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, 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 $\mathsf{Lollipop}_{n-k,k}$ 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 $\mathsf{FS}(\mathsf{Lollipop}_{n-k,k},Y)$. In this paper, we give a sufficient and necessary condition for $\mathsf{FS}(\mathsf{Lollipop}_{n-k,k},Y)$ to be connected for all $2\leq k\leq n$.