{
  "id": "2110.12424",
  "version": "v2",
  "published": "2021-10-24T12:22:07.000Z",
  "updated": "2022-08-18T11:18:17.000Z",
  "title": "A Note on Minimum Degree Condition for Hamiltonian $(a,b)$-Cycles in Hypergraphs",
  "authors": [
    "Jian Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\\cup B_i$ and $B_i\\cup A_{i+1}$ (subscripts module $t$) are edges for all $i=0,1,\\ldots,t-1$. Let $\\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\\geq 5k$ and $n$ divisible by $k$. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists $\\alpha\\in (0,1)$ such that $\\delta_a(\\mathcal{H})\\geq (\\alpha +o(1))\\binom{n-a}{b}$ and $\\delta_b(\\mathcal{H})\\geq (1-\\alpha +o(1))\\binom{n-b}{a}$, then $\\mathcal{H}$ contains a Hamilton $(a,b)$-cycle. As a corollary, we prove that if $\\delta_{\\ell}(\\mathcal{H})\\geq (1/2 +o(1))\\binom{n-\\ell}{k-\\ell}$ for some $\\ell \\geq k/2$, then $\\mathcal{H}$ contains a Hamilton $(k-\\ell,\\ell)$-cycle and this is asymptotically best possible.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-18T11:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum degree condition",
      "hamiltonian",
      "uniform hypergraph",
      "vertex set",
      "subscripts module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}