{
  "id": "2307.08121",
  "version": "v1",
  "published": "2023-07-16T18:09:06.000Z",
  "updated": "2023-07-16T18:09:06.000Z",
  "title": "Evacuating \"O''- and \"Y''-shaped houses on fire: the connectivity of friends-and-strangers graphs on complete multipartite graphs",
  "authors": [
    "Honglin Zhu"
  ],
  "comment": "22 pages, 23 figures",
  "categories": [
    "math.CO"
  ],
  "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 $\\sigma: V(X) \\to V(Y)$, where two bijections $\\sigma$ and $\\sigma'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \\in V(X)$ such that $\\sigma(a), \\sigma(b) \\in V(Y)$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-16T18:09:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "complete multipartite graph",
      "friends-and-strangers graph",
      "complete bipartite graph",
      "connectivity",
      "vertex set consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}