The connectivity of friends-and-strangers graphs on complete multipartite graphs

TL;DR

This work provides a complete connectivity characterization for the friends-and-strangers graphs $\,\mathsf{FS}(X,Y)$ when $Y$ is a complete multipartite graph $K_{k_1,\ldots,k_t}$. It introduces a stopwatch subgraph and snake-tongue techniques to control local exchangeability and propagate it to global connectivity, and proves a group-theoretic stopwatch result that underpins the non-tree cases. The paper resolves Wang–Lu–Chen’s conjecture for the bipartite target case ($K_{k,n-k}$) and extends the analysis to multipartite targets, also classifying when exactly two components occur for bipartite $X$. It identifies exceptional configurations (e.g., $\theta_0$, $T_6,T_7,T_8$) and clarifies the role of $(n-k_t)$-bridges, offering a foundation for further explorations of component counts in FS graphs.

Abstract

For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $σ: V(X) \to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $σ(a), σ(b) \in V(Y)$ are adjacent in $Y$. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. We further extend this result to when $Y$ is a complete multipartite graph. We also determine when $\mathsf{FS}(X, Y)$ has exactly two connected components where $X$ is bipartite and $Y$ is a complete bipartite graph.

Figures (17)

• Figure 1: The $\theta_0$ graph.
• Figure 2: The three exceptions $T_6$, $T_7$, and $T_8$
• Figure 3: People are nodes filled in red and empty spots white. A dotted line is a fire exit. Blue arrows indicate movement of people. Numbers on blue arrows indicate the order of the movement. In this diagram, $p_1$ can evacuate through the fire exit. (Of course, $p_1$ may not reach the empty spot. Rather, $p_1$ will push everyone down the fire exit by one step.)
• Figure 4: The fireman $f_1$ is currently at distance $\ell - 1$ from $C$. The goal is to bring $f_1$ closer.
• Figure 8: An example where $k_1 = k_2 = 2$ and $k_3 = 3$. The swap across $\{w_1, w_{5}\}$ induces the permutations $\alpha = ((2 \; 3), (1 \; 2 \; 3 \; 4)^{-1})$ and $\beta = ((1 \; 2 \; 3)^{-1}, (2 \; 3 \; 4))$ on $S_{k_3} \times S_{k_1 + k_2}$.

Theorems & Definitions (50)

• Definition 1.1: defant2021
• Definition 1.2: wang2023
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: brunck2023
• Lemma 2.1: defant2021
• Lemma 2.2: godsil2001