On the Asymmetric Generalizations of Two Extremal Questions on Friends-and-Strangers Graphs

TL;DR

This paper investigates the connectivity properties of friends-and-strangers graphs derived from pairs of graphs with certain degree conditions, providing new thresholds and characterizations for their number of connected components.

Contribution

It establishes new degree-based conditions ensuring the connectivity or bipartite component structure of friends-and-strangers graphs, resolving several recent conjectures and questions.

Findings

Connectedness when minimum degrees exceed specific thresholds.

Exactly two components in bipartite cases under certain degree sums.

Precise bounds for minimum degrees in bipartite graphs for two-component structure.

Abstract

For two graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality $n,$ the friends-and-strangers graph $\mathsf{FS}(X,Y)$ was recently defined by Defant and Kravitz. The vertices of $\mathsf{FS}(X,Y)$ are the bijections from $V(X)$ to $V(Y),$ and two bijections $\sigma$ and $\tau$ are adjacent if they agree everywhere except at two vertices $a,b\in V(X)$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y.$ We study generalized versions of two problems by Alon, Defant, and Kravitz. First, we show that if $X$ and $Y$ have minimum degrees $\delta(X)$ and $\delta(Y)$ that satisfy $\delta(X)> n/2, \delta(Y)>n/2,$ and $2\min(\delta(X), \delta(Y))+3\max(\delta(X), \delta(Y))\ge 3n,$ then $\mathsf{FS}(X,Y)$ is connected. As a corollary, we settle a recent conjecture by Alon, Defant, and Kravitz stating that there exists a number $d_n = 3n/5 + O(1)$ such that if both $X$ and $Y$ have minimum degrees at least $d_n,$ the graph $\mathsf{FS}(X,Y)$ is connected. When $X$ and $Y$ are bipartite, a parity obstruction prevents $\mathsf{FS}(X,Y)$ from being connected. We show that if $X$ and $Y$ are edge-subgraphs of $K_{r,r}$ that satisfy $\delta(X)+\delta(Y)\ge 3r/2+1,$ then the graph $\mathsf{FS}(X,Y)$ has exactly two connected components. As a corollary, we provide an almost complete answer to another recent question of Alon, Defant, and Kravitz asking for the minimum number $d^*_{r,r}$ such that for any edge-subgraph $X$ of $K_{r,r}$ satisfying $\delta(X)\ge d^*_{r,r},$ the graph $\mathsf{FS}(X,K_{r,r})$ has exactly two connected components. We show that $d^*_{r,r} = r/2+1$ when $r$ is even and $d^*_{r,r}\in \{\lceil r/2\rceil, \lceil r/2\rceil+1\}$ when $r$ is odd.