The connectedness of friends-and-strangers graphs about graph parameters and others

TL;DR

This work studies when FS$(X,Y)$ is connected or highly connected by establishing a main sufficient condition: if $Δ(X)+κ(Y)\ge n+s-1$ (with $Y$ not equal to $C_n$), FS$(X,Y)$ is $s$-connected or has a controlled two-component structure. The authors prove the result by reducing to the tree case and using induction, then apply it to obtain several applications, including sharper connectivity results when $Y$ is complete and concrete outcomes for DL-type graphs. They also completely resolve Problem p1 for large $n$ and provide a broad κ-sum criterion that guarantees 2-connectivity or connectivity, depending on parity, culminating in a near-complete classification for $X\in DL_{n-k,k}$. These findings refine Bangachev’s conjectures, extend prior work on FS graphs, and give a comprehensive framework for predicting FS$(X,Y)$ connectivity across broad graph families.

Abstract

Let $X$ and $Y$ be two graphs of order $n$. The friends-and-strangers graph $\textup{FS}(X,Y)$ of $X$ and $Y$ is a graph whose vertex set consists of all bijections $σ: V(X)\rightarrow V(Y)$, in which two bijections $σ$ and $ σ'$ are adjacent if and only if they agree on all but two adjacent vertices of $X$ such that the corresponding images are adjacent in $Y$. The most fundamental question about these friends-and-strangers graphs is whether they are connected. In this paper, we provide a sufficient condition regarding the maximum degree $Δ(X)$ and vertex connectivity $κ(Y)$ that ensures the graph $\textup{FS}(X,Y)$ is $s$-connected. As a corollary, we improve upon a result by Bangachev and partially confirm a conjecture he proposed. Furthermore, we completely characterize the connectedness of $\textup{FS}(X,Y)$, where $X\in\textup{DL}_{n-k,k}$.

Figures (1)

• Figure 1: Graphs $\theta_0$ and $\theta_1$.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1