{
  "id": "2505.08951",
  "version": "v1",
  "published": "2025-05-13T20:28:55.000Z",
  "updated": "2025-05-13T20:28:55.000Z",
  "title": "Sensitivity and Hamming graphs",
  "authors": [
    "Sara Asensio",
    "Yuval Filmus",
    "Ignacio García-Marco",
    "Kolja Knauer"
  ],
  "comment": "8 pages, 1 figure",
  "categories": [
    "math.CO",
    "cs.CC"
  ],
  "abstract": "For any $m\\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, Garc\\'ia-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-13T20:28:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamming graph",
      "ary sensitivity conjecture",
      "low maximum degree",
      "ary function",
      "polynomial expression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}