Diameter Bounds for Friends-and-Strangers Graphs

Consider two $n$-vertex graphs $X$ and $Y$, where we interpret $X$ as a social network with edges representing friendships and $Y$ as a movement graph with edges representing adjacent positions. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ is a graph on the $n!$ permutations $V(X)\to V(Y)$, where two configurations are adjacent if and only if one can be obtained from the other by swapping two friends located on adjacent positions. Friends-and-strangers graphs were first introduced by Defant and Kravitz, and generalize sliding puzzles as well as token swapping problems. Previous work has largely focused on their connectivity properties. In this paper, we study the diameter of the connected components of $\mathsf{FS}(X, Y)$. We extend the result of Kornhauser, Miller, and Spirakis on sliding puzzles to general graphs in two ways. First, we show that the diameter of $\mathsf{FS}(X, Y)$ is polynomially bounded when both the friendship and the movement graphs have large minimum degree. Second, when both the underlying graphs $X$ and $Y$ are Erdős-Rényi random graphs, we show that the distance between any pair of configurations is almost always polynomially bounded under certain conditions on the edge probabilities.

💡 Research Summary

The paper studies the diameter of friends‑and‑strangers graphs FS(X,Y), a combinatorial model that generalizes sliding puzzles and token‑swapping problems. A vertex of FS(X,Y) is a bijection σ:V(X)→V(Y) assigning each “person” (a vertex of X) to a distinct “location” (a vertex of Y). Two configurations σ and τ are adjacent if they differ by a single “friendly swap”: a pair of friends (an edge of X) occupying adjacent locations (an edge of Y) exchange places. The size of FS(X,Y) is n! and its connectivity has been extensively studied; however, the diameter of its connected components—i.e., the maximum number of swaps needed to transform one reachable configuration into another—has received far less attention.

The authors obtain three main families of polynomial‑time diameter bounds.

1. Minimum‑degree conditions.
   Building on Bangachev’s connectivity criteria, they prove two theorems.

   • Theorem 1.3. If min(δ(X),δ(Y)) + 2·max(δ(X),δ(Y)) ≥ 2n, then FS(X,Y) is connected and its diameter is O(n⁵).
   • Theorem 1.4. If δ(X)+δ(Y) ≥ 3n/2, then FS(X,Y) is connected and its diameter is at most 3n(n−1)/2 = O(n²).
     The proofs use a leaf‑removal lemma (any connected graph has a vertex whose removal leaves the graph connected) to iteratively “fix” vertices. At each iteration a vertex v not yet fixed is chosen so that the remaining subgraph stays connected; a short path inside the current unfixed subgraph moves the correct person to v, after which v is never touched again. The number of swaps per iteration is bounded by the size of the unfixed set, yielding the stated total bounds. The stronger degree sum condition in Theorem 1.4 allows a more aggressive reduction, leading to the quadratic bound.

2. Exact diameter for the complete‑friend graph.
   For FS(Kₙ,Y) the authors improve Jeong’s earlier bound 2n²−5n+3 to a tight bound ⌊n²/2⌋. The algorithm is essentially the same leaf‑fixing process specialized to the complete friendship graph, where any two people may swap provided they occupy adjacent locations. They also show that when Y is disconnected, the configuration space decomposes according to the connected components of Y, and each component’s diameter is again bounded by ⌊n²/2⌋.

3. Random‑graph setting.
   Let X∼G(n,p) and Y∼G(n,q) be independent Erdős‑Rényi graphs. If the product of edge probabilities satisfies pq ≥ 100·log n / n, then with probability at least 1−o(n⁻²) the distance between any two fixed permutations τ and ω in FS(X,Y) is O(n⁵). The proof leverages the fact that dense random graphs have short paths between almost all vertex pairs, together with the O(n³) diameter bound for FS(Starₙ,Y) (a result of Kornhauser‑Miller‑Spirakis). By first moving τ to a “star” configuration and then to ω, the total number of swaps stays polynomial. This result improves on earlier work that required a larger product pq (≈exp(log log n)/n) to guarantee connectivity; here only a single pair of configurations is considered, which allows a weaker condition.

The paper also revisits classic results: the Kornhauser‑Miller‑Spirakis O(n³) bound for sliding puzzles on arbitrary graphs, Wilson’s connectivity theorem for FS(Starₙ,Y), and token‑swapping bounds (FS(Kₙ,Y) has diameter O(n²)). By extending these ideas, the authors obtain the first general polynomial diameter bounds for a broad class of friends‑and‑strangers graphs.

Finally, the authors conjecture that whenever FS(X,Y) is connected, its diameter is polynomial in n. Their theorems cover most of the parameter space identified by Bangachev, leaving only a small “green dotted” region where connectivity is known but a polynomial bound is still open. They suggest further work on lowering the exponent in the O(n⁵) bound, tightening the random‑graph thresholds, and exploring the diameter behavior when minimum degree conditions are relaxed.