On the Diameters of Friends-and-Strangers Graphs

Let $X$ and $Y$ be $n$-vertex simple graphs. Interpret the vertices of $X$ as positions, and the vertices of $Y$ as people: say two people in the vertex set of $Y$ are friends if they are adjacent and strangers if they are not. Each person picks a position to stand on, yielding a starting configuration. From here, at any point in time, two friends standing on adjacent positions may switch places: we call this operation a friendly swap. From the initial configuration, say the $n$ people have a final configuration in mind, and they know it can be reached from the initial configuration by some sequence of friendly swaps. What is the worst-case (over pairs of starting and final configurations) number of friendly swaps that is necessary in order for the $n$ people to achieve the final configuration from the starting configuration?

We may formalize the problem using the following definition.

See Figure 1 for an illustration of Definition 1.1 on five-vertex graphs. Defant and Kravitz [DK21], in addition to introducing the framework of friends-and-strangers graphs, described the connected components of $\mathsf{FS}(\textsf{{Path}}_{n},Y)$ and $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ in terms of the acyclic orientations of $\overline{Y}$ (the complement of $Y$), and determined both necessary conditions and sufficient conditions for $\mathsf{FS}(X,Y)$ to be connected. In a different paper [Jeo22], we extend their results: [DK21, Corollary 4.14] states that $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is connected if and only if $\overline{Y}$ is a forest with trees of jointly coprime sizes, and we establish that if $X$ is biconnected (i.e., connected and with no cut vertex) and $Y$ is a graph for which $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is connected, then $\mathsf{FS}(X,Y)$ is connected, settling [DK21, Conjecture 7.1]. In [Jeo22], we also initiate the study of the girth of friends-and-strangers graphs. Motivated by [KMS84] and connections to molecular programming as seen in [Bra+19], the framework of friends-and-strangers was later generalized by [Mil23] to permit for multiplicities onto vertices, in which many of the main results of [DK21, Wil74] were also generalized accordingly.

A central objective in the study of friends-and-strangers graphs is to determine necessary and sufficient conditions for their connectivity. Indeed, $\mathsf{FS}(X,Y)$ being connected corresponds exactly to the property that one can go between any two configurations in $\mathsf{FS}(X,Y)$ via some sequence of $(X,Y)$-friendly swaps. Of course, the conditions one may derive will depend upon the assumptions on $X$ and $Y$ under which one works. If one elects to proceed under a regime in which $\mathsf{FS}(X,Y)$ cannot be connected (such as when $X$ and $Y$ are both bipartite; see the discussion around [DK21, Proposition 2.7] and [ADK23, Subsection 2.3] for a parity obstruction which demonstrates why this is the case), one may instead study how small the number of connected components may be under this regime, and the natural question here is to ask for further conditions on $X$ and $Y$ ensuring that $\mathsf{FS}(X,Y)$ achieves the smallest possible number of connected components. As pursued in [DK21, Sections 3 and 4] for (respectively) paths and cycles, one direction of inquiry is to fix (without loss of generality, as we will see in Proposition 2.3(1)) $X$ to be some particular graph, and study the structure of $\mathsf{FS}(X,Y)$ for arbitrary $Y$: see [Def+22, Lee22, WC23, Wil74, Zhu23]. It is also very natural to ask extremal and probabilistic questions concerning the connectivity of friends-and-strangers graphs, such as minimum degree conditions on $X$ and $Y$ which ensure that $\mathsf{FS}(X,Y)$ is connected or for threshold probabilities on Erdős-Rényi random graphs $X,Y$ regarding the connectivity of $\mathsf{FS}(X,Y)$: see [ADK23, Ban22, Jeo23, Mil23, WLC23].

The setup proposed by Definition 1.1 is quite general. Indeed, friends-and-strangers graphs serve both as a common natural generalization of many classical combinatorial objects and as a framework which embodies many important problems in discrete mathematics and theoretical computer science. We illustrate this claim with a non-exhaustive listing of relevant examples. The graph $\mathsf{FS}(X,K_{n})$ is the Cayley graph of the symmetric group on the vertex set of $X$ generated by the transpositions corresponding to the edges of $X$; we refer the reader to [DK21] and the references therein for a comprehensive discussion regarding the relevance of friends-and-strangers graphs within algebraic combinatorics. Letting $X$ be the $4$-by-$4$ grid and $Y$ a star graph, studying $\mathsf{FS}(X,Y)$ is equivalent to studying the configurations and moves that can be performed on the famous $15$-puzzle (with the central vertex of the star graph corresponding to the empty tile); see [BK23, DR18, Par15, Yan11] for similar inquiries of a recreational flavor. The works [Naa00, Sta08] both study the structure of the graph $\mathsf{FS}(\textsf{{Path}}_{n},Y)$ under certain restrictions on $Y$, while the works [BR99, Rei98] utilize $\mathsf{FS}(\textsf{{Path}}_{n},Y)$ to investigate the acyclic orientations of $\overline{Y}$. Asking if $X$ and $Y$ pack [BE78, KO09, SS78, Yap88, Yus07] in the graph packing literature is equivalent to asking if there exists an isolated vertex in $\mathsf{FS}(X,Y)$. Studying the token swapping problem [Aic+22, Bin+23, BMR18, Mil+16, Yam+15] on the graph $X$ is equivalent to studying distances between configurations in $\mathsf{FS}(X,K_{n})$. Finally, as we will briefly touch upon in Subsection 4.4, the interchange process on the graph $X$ [AF02, AK13, Ang03, BD06, CLR10, ES23, Ham15, HS21, Sch05] can be phrased in terms of (continuous-time) random walks on $\mathsf{FS}(X,K_{n})$.

Unlike the existing body of work that studies the connectivity of friends-and-strangers graphs, the present paper initiates the study of their diameters, corresponding to the length of the “longest shortest path,” with lengths of shortest paths evaluated over all pairs of vertices. Indeed, the diameter of a connected component of $\mathsf{FS}(X,Y)$ corresponds to the largest number of $(X,Y)$-friendly swaps necessary to achieve one configuration in the component from another. In a more recreational tone, if we think of $\mathsf{FS}(X,Y)$ as a generalized $15$-puzzle, we are asking for the longest solution length for any solvable puzzle involving “board $X$ and rules $Y$.” The works [ADK23, DK21] both posed the following question, which asks whether the distance between any two configurations in $\mathsf{FS}(X,Y)$ is polynomial in the size of $X$ and $Y$.

In Section 2, we introduce some background that we shall need later in the work. Before tackling the more global Question 1.2, in Section 3, we fix (without loss of generality) $X$ to be a complete, path, or cycle graph, and derive upper bounds on the maximum diameter of a component of $\mathsf{FS}(X,Y)$ in each setting. Our results on paths and cycles address an open problem posed in [DK21, Subsection 7.3]. Furthermore, the discussion therein suggests that one must restrict their attention to rather contrived choices of graphs $X$ and $Y$ in order for $\mathsf{FS}(X,Y)$ to have a component with diameter that is superpolynomial in the size of $X$ and $Y$, suggesting that Question 1.2 may be challenging to settle via constructive means if it holds in the negative.

In Section 4, we establish the main result of this article, Theorem 1.3, which answers Question 1.2 in the negative. We prove this theorem by constructing, for all integers $L\geq 1$, graphs $X_{L}$ and $Y_{L}$ on the same number of vertices: see Figure 2 for a schematic diagram of the construction for $L=3$. The construction is such that the number of vertices of $X_{L}$ and $Y_{L}$ is $\Theta(L)$, and there exist two configurations $\sigma_{s},\sigma_{f}\in V(\mathsf{FS}(X_{L},Y_{L}))$ which lie in the same connected component $\mathscr{C}$ of $\mathsf{FS}(X_{L},Y_{L})$ and for which the distance between $\sigma_{s}$ and $\sigma_{f}$ is $e^{\Omega(n)}$.

At the end of Section 4, we briefly discuss implications of Theorem 1.3 to the study of random walks on friends-and-strangers graphs, and deduce what might be thought of as the natural stochastic analogue of Theorem 1.3. Random walks on friends-and-strangers graphs model a variant of the interchange process where we may prohibit certain pairs of particles from swapping positions. This result contrasts many classical theorems regarding rapidly mixing Markov chains, all of which may be readily rewritten using the language of friends-and-strangers graphs, and makes progress on another open problem posed in [ADK23, Section 7]. We conclude the work with Section 5, which suggests several open problems and directions for future research.

In this article, unless stated otherwise, we assume that all graphs are simple. We employ standard asymptotic notation in this paper. Unless stated otherwise, all asymptotic notation in this paper will be with respect to $n$. Purely for the sake of completeness, we state the following standard notation.

• •

  The vertex and edge sets of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively.

• •

  The complement of the graph $G$ is denoted $\overline{G}$.

• •

  The statement that $G$ and $H$ are isomorphic is written as $G\cong H$.

• •

  For a subset $S\subset V(G)$, we let $G|_{S}$ denote the induced subgraph of $G$ with vertex set $S$.

• •

  The open neighborhood of $v\in V(G)$, which is the collection of all neighbors of $v$, is denoted by $N_{G}(v)$. The closed neighborhood of $v\in V(G)$ is denoted $N_{G}[v]=N_{G}(v)\cup\{v\}$. For a subset of vertices $S\subseteq V(G)$, we let

  $\displaystyle N_{G}(S)=\bigcup_{v\in S}N_{G}(v),$

  $\displaystyle N_{G}[S]=\bigcup_{v\in S}N_{G}[v].$

• •

  The disjoint of a collection of graphs $\{G_{i}\}_{i\in I}$, notated $\bigoplus_{i\in I}G_{i}$, is the graph with vertex set $\bigsqcup_{i\in I}V(G_{i})$ and edge set $\bigsqcup_{i\in I}E(G_{i})$. This readily extends to expressing a graph as the disjoint union of its connected components.

• •

  The distance $d(v,w)$ between $v,w\in V(G)$ is the length of the shortest path from $v$ to $w$. The diameter of a component $\mathscr{C}$ of $G$ is $\max_{v,w\in V(\mathscr{C})}d(v,w)$.

Graphs with vertex set $[n]\mathrel{\mathop{\ordinarycolon}}=\{1,\dots,n\}$ that will be relevant later are

• •

  the complete graph $K_{n}$, with $E(K_{n})\mathrel{\mathop{\ordinarycolon}}=\{\{i,j\}\mathrel{\mathop{\ordinarycolon}}\{i,j\in[n],\ i\neq j\}\}$;

• •

  the complete bipartite graph $K_{i,j}$, with $E(K_{i,j})\mathrel{\mathop{\ordinarycolon}}=\{\{v_{1},v_{2}\}\mathrel{\mathop{\ordinarycolon}}v_{1}\in[i],v_{2}\in\{i+1,\dots,i+j\}\}$, which naturally partitions $V(K_{i,j})$ into two sets (henceforth called partite sets);

• •

  the path graph $\textsf{{Path}}_{n}$, with $E(\textsf{{Path}}_{n})\mathrel{\mathop{\ordinarycolon}}=\{\{i,i+1\}\mathrel{\mathop{\ordinarycolon}}i\in[n-1]\}$;

• •

  the cycle graph $\textsf{{Cycle}}_{n}$, with $E(\textsf{{Cycle}}_{n})\mathrel{\mathop{\ordinarycolon}}=\{\{i,i+1\}\mathrel{\mathop{\ordinarycolon}}i\in[n-1]\}\cup\{\{n,1\}\}$;

• •

  the star graph $\textsf{{Star}}_{n}\mathrel{\mathop{\ordinarycolon}}=K_{1,n-1}$.

An orientation of a graph $G$ is an assignment of a direction to every edge of $G$, and an acyclic orientation of $G$ is an orientation with no directed cycles. Denote the set of all acyclic orientations of $G$ by $\textsf{{Acyc}}(G)$. We will be interested in operations on acyclic orientations of $G$ called flips and double-flips, as defined in [DK21]. Notably, it was shown in [DK21, Theorem 4.7] that double-flips on acyclic orientations in $\textsf{{Acyc}}(\overline{Y})$ are paramount in describing the connected components of $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$.

Letting $\alpha\in\textsf{{Acyc}}(G)$, converting a source of $\alpha$ into a sink or a sink of $\alpha$ into a source by reversing the directions of all its incident edges results in another acyclic orientation $\alpha^{\prime}$ of $G$. We call such an operation a flip, and we say that $\alpha$ and $\alpha^{\prime}$ are flip equivalent, denoted $\alpha\sim\alpha^{\prime}$. In the literature, the equivalence classes in $\textsf{{Acyc}}(G)/\!\!\sim$ are called toric acyclic orientations; we refer the interested reader to [Che10, DMR16, MM11, Pre86, Spe09] for related reading. We will further say that we perform an inflip on $\alpha$ if we convert a source into a sink (the direction of all incident edges “go into” the new sink), and an outflip if we convert a sink into a source.¹¹1In particular, we may apply these operations to isolated vertices.

Similarly, flipping a nonadjacent source and sink of $\alpha$ into (respectively) a sink and a source results in another acyclic orientation $\alpha^{\prime\prime}$ of $G$: we call such an operation a double-flip, and we say $\alpha$ and $\alpha^{\prime\prime}$ are double-flip equivalent, denoted $\alpha\approx\alpha^{\prime\prime}$. It is easy to show that $\sim$ and $\approx$ are equivalence relations on $\textsf{{Acyc}}(G)$. We denote the set of double-flip equivalence classes of $\textsf{{Acyc}}(G)$ by $\textsf{{Acyc}}(G)/\!\!\approx$, and denote the double-flip equivalence class for which $\alpha$ is a representative by $[\alpha]_{\approx}$.

Assume $V(G)=[n]$, and take $\alpha\in\textsf{{Acyc}}(G)$. Associated to the acyclic orientation $\alpha$ is a poset $P_{\alpha}=([n],\leq_{\alpha})$, where $i\leq_{\alpha}j$ if and only if there exists a directed path from $i$ to $j$ in $\alpha$. We define a linear extension of $P_{\alpha}$ to be any permutation $\sigma\in\mathfrak{S}_{n}$ such that $\sigma^{-1}(i)\leq\sigma^{-1}(j)$ whenever $i\leq_{\alpha}j$. We let $\mathcal{L}(\alpha)$ denote the collection of linear extensions of $P_{\alpha}$. For any $\sigma\in\mathfrak{S}_{n}$, it is not hard to see that there exists a unique acyclic orientation $\alpha_{G}(\sigma)\in\textsf{{Acyc}}(G)$ for which $\sigma\in\mathcal{L}(\alpha_{G}(\sigma))$, and that this acyclic orientation is the result of directing each edge $\{i,j\}\in E(G)$ from $i$ to $j$ if and only if $\sigma^{-1}(i)<\sigma^{-1}(j)$. It is also not hard to see that the poset $P_{\alpha}$ associated to $\alpha\in\textsf{{Acyc}}(G)$ has a linear extension (e.g., for $i\in[n]$, we can construct a linear extension $\sigma$ by setting $\sigma^{-1}(i)$ to be a source of $\alpha$, then removing the source and all incident edges from $\alpha$; in an abuse of notation,²²2We will commit similar abuses of notation in Section 3. They should not raise any confusion when invoked. we understand $\alpha$ here as being mutated over the course of this greedy algorithm). We write

We refer the reader to [DK21, Section 4] for a more comprehensive discussion regarding why these notions are of importance in the study of friends-and-strangers graphs (though this is illuminated in passing in Subsection 2.2 and in the arguments of Section 3).

For a graph $G$ and acyclic orientation $\alpha\in\textsf{{Acyc}}(G)$, we can partition the directed edges of any cycle subgraph $\mathcal{C}$ of $G$ into $\mathcal{C}_{\alpha}^{-}$ and $\mathcal{C}_{\alpha}^{+}$, corresponding to edges directed in one of two possible directions under $\alpha$ in $\mathcal{C}$. The article [Pre86] studied precisely when an acyclic orientation could be reached from another by a sequence of inflips or outflips, while [Pro21] extends this result by providing an upper bound on the number of inflips or outflips necessary to reach $\alpha$ from $\alpha^{\prime}$ whenever $\alpha\sim\alpha^{\prime}$.

We build on Lemma 2.1. The following proposition establishes that we could have defined flip equivalence strictly with respect to inflips or outflips, as this would have resulted in the same notion.

We mention those general properties of friends-and-strangers graphs that we will need later in the article. We refer the reader to [DK21, Section 2] for a thorough treatment of the general properties of friends-and-strangers graphs.

The definitions concerning acyclic orientations that were introduced in Subsection 2.1 were observed in [DK21] to be central in describing the structure of the connected components of $\mathsf{FS}(\textsf{{Path}}_{n},Y)$ and $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$, which are the graphs we will be interested in during Section 3. Specifically, we have the following theorems.

Defant and Kravitz [DK21] also determined precisely when $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is connected. The coprimality condition on the sizes of the components of $\overline{Y}$ in Theorem 2.6 may seem surprising at first glance. We refer the reader to the discussion around [DK21, Corollary 4.12] and [DK21, Corollary 4.14] to see where this condition emerges and why it is a natural one.

We begin by setting $Y=K_{n}$. Take any two configurations $\sigma,\tau\in V(\mathsf{FS}(X,K_{n}))$ that lie in the same connected component. Consider the following iterative algorithm, applied starting from $\sigma$ and proceeding sequentially on $i\in[n]$. In an abuse of notation, $\sigma$ is understood to be mutated over the course of this algorithm as we perform $(X,K_{n})$-friendly swaps to modify its mappings.

1. (1)

   If $\sigma(i)=\tau(i)$, do nothing.

2. (2)

   If $\sigma(i)\neq\tau(i)$, swap $\tau(i)$ onto $i$ along a simple path, then swap $\sigma(i)$ back along the simple path that $\tau(i)$ traversed.

It is straightforward to prove via induction that at the beginning of any iteration $i\in[n]$, $\sigma(i)$ and $\tau(i)$ lie upon the same connected component of $X$ (so that the algorithm may always proceed), and that $\sigma(j)=\tau(j)$ for all $j<i$. Thus, $\sigma=\tau$ when the algorithm terminates after $n-1$ iterations (it must be that $\sigma(n)=\tau(n)$ at the beginning of the $n$^th iteration). For any iteration $i\in[n]$, step (2) requires at most $n-1$ $(X,K_{n})$-friendly swaps to move $\tau(i)$ onto $i$, and at most $n-2$ $(X,K_{n})$-friendly swaps to move $\sigma(i)$ back. This establishes that the diameter of any component of $\mathsf{FS}(X,K_{n})$ is therefore at most $(n-1)((n-1)+(n-2))=2n^{2}-5n+3=O(n^{2})$.

Finding the exact distance between two configurations in $\mathsf{FS}(X,K_{n})$ is known as the token swapping problem on $X$ in the theoretical computer science literature. The $O(n^{2})$ bound on the diameter of any component of $\mathsf{FS}(X,K_{n})$ is well known, and we also have a bound of $\Omega(n^{2})$ on the diameter of any component of $\mathsf{FS}(X,K_{n})$ for particular choices of $X$ (e.g., see Remark 3.4). In general, computing exact distances between two configurations in $\mathsf{FS}(X,K_{n})$, as well as the diameters of its connected components, is challenging, even when imposing additional assumptions on $X$ (e.g., see [Bin+23, Yam+15]). There do exist, however, exact polynomial-time algorithms which solve the token swapping problem for a number of choices of $X$, including cliques [Cay49], paths [Jer85], stars [PV90], cycles [KSY19, van+16], and complete bipartite graphs [Yam+15]. See Subsection 5.5 for additional discussion regarding matters of hardness and approximation.

In this subsection, we fix $X=\textsf{{Path}}_{n}$. We begin by introducing a notion which will serve as a monovariant in the proof of Proposition 3.2.

In other words, the ordered pair $(i,j)$ is a $(\sigma,\tau)$-inversion if the relative ordering of the inverse images of $i,j$ under $\sigma$ is opposite that of $\tau$. If (without loss of generality) $\tau$ is the identity permutation, then $\textup{inv}(\sigma,\tau)=\textup{inv}(\sigma)$, the number of inversions of $\sigma$. It also follows immediately that $\textup{inv}(\sigma,\tau)=0$ if and only if $\sigma=\tau$.

Certainly, the two vertices incident to an edge of $\overline{Y}$ are comparable in the poset $P_{\alpha}=([n],\leq_{\alpha})$ for any $\alpha\in\textsf{{Acyc}}(\overline{Y})$. This yields the following statement, as $\binom{n}{2}-p_{\alpha}\leq\binom{n}{2}-|E(\overline{Y})|=|E(Y)|$. For simplicity, we appeal to Theorem 3.3, rather than Proposition 3.2, in forthcoming arguments.

In this subsection, we fix $X=\textsf{{Cycle}}_{n}$. The setting $Y=K_{n}$ has been studied in the context of circular permutations [Kim16, van+16]. In particular, [Kim16, Procedure 3.6] provides an algorithm that achieves the minimal number of $(\textsf{{Cycle}}_{n},K_{n})$-friendly swaps between any two permutations in $\mathfrak{S}_{n}$. Extracting these results yields that the diameter of $\mathsf{FS}(\textsf{{Cycle}}_{n},K_{n})$ is $\lfloor n^{2}/4\rfloor$. In the spirit of Remark 3.4, it follows that there exist families of $n$-vertex graphs $Y$ for which $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ has diameter $\Theta(n^{2})$, and it is worth asking what conditions on $Y$ yield that the maximum diameter of a connected component of $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is at most quadratic in $n$. In this direction, we have the following proposition.

We were unable to extend the $O(n^{2})$ bound from Proposition 3.6 to general $Y$, although we suspect that this is the truth (see Subsection 5.2). However, the existence of a universal constant $C>0$ such that the maximum diameter of a component of $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is $O(n^{C})$ remains highly desirable. In conjunction with Theorem 2.6, the following theorem yields such a result whenever $\mathsf{FS}(\textsf{{Cycle}}_{n},Y)$ is connected.