Bipartite Friends and Strangers Walking on Bipartite Graphs
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Given n-vertex simple graphs X and Y, the friends-and-strangers graph 𝖥𝖲(X, Y) has as its vertices all n! bijections from V(X) to V(Y), where two bijections are adjacent if and only if they differ on two adjacent elements of V(X) whose mappings are adjacent in Y. We consider the setting where X and Y are both edge-subgraphs of K_r,r: due to a parity obstruction, 𝖥𝖲(X,Y) is always disconnected in this setting. Sharpening a result of Bangachev, we show that if X and Y respectively have minimum degrees δ(X) and δ(Y) and they satisfy δ(X) + δ(Y) ≥⌊ 3r/2 ⌋ + 1, then 𝖥𝖲(X,Y) has exactly two connected components. This proves that the cutoff for 𝖥𝖲(X,Y) to avoid isolated vertices is equal to the cutoff for 𝖥𝖲(X,Y) to have exactly two connected components. We also consider a probabilistic setup in which we fix Y to be K_r,r, but randomly generate X by including each edge in K_r,r independently with probability p. Invoking a result of Zhu, we exhibit a phase transition phenomenon with threshold function (log r)/r: below the threshold, 𝖥𝖲(X,Y) has more than two connected components with high probability, while above the threshold, 𝖥𝖲(X,Y) has exactly two connected components with high probability. Altogether, our results settle a conjecture and completely answer two problems of Alon, Defant, and Kravitz.
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