{
  "id": "2209.12126",
  "version": "v1",
  "published": "2022-09-25T02:52:21.000Z",
  "updated": "2022-09-25T02:52:21.000Z",
  "title": "Edge-fault-tolerance about the SM-λ property of hypercube-like networks",
  "authors": [
    "Dong Liu. Pingshan Li",
    "Bicheng Zhang"
  ],
  "comment": "9 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The edge-fault-tolerance of networks is of great significance to the design and maintenance of networks. For any pair of vertices $u$ and $v$ of the connected graph $G$, if they are connected by $\\min \\{ \\deg_G(u),\\deg_G(v)\\}$ edge-disjoint paths, then $G$ is strong Menger edge connected (SM-$\\lambda$ for short). The conditional edge-fault-tolerance about the SM-$ \\lambda$ property of $G$, written $sm_\\lambda^r(G)$, is the maximum value of $m$ such that $G-F$ is still SM-$\\lambda$ for any edge subset $F$ with $|F|\\leq m$ and $\\delta(G-F)\\geq r$, where $\\delta(G-F)$ is the minimum degree of $G-F$. Previously, most of the exact value for $sm_\\lambda^r(G)$ is aimed at some well-known networks when $r\\leq 2$, and a few of the lower bounds on some well-known networks for $r\\geq 3$. In this paper, we firstly determine the exact value of $sm_\\lambda^r(G)$ on class of hypercube-like networks (HL-networks for short, including hypercubes, twisted cubes, crossed cubes etc.) for a general $r$, that is, $sm_\\lambda^r(G_n)=2^r(n-r)-n$ for every $G_n\\in HL_n$, where $n\\geq 3$ and $1\\leq r \\leq n-2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-25T02:52:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hypercube-like networks",
      "exact value",
      "well-known networks",
      "strong menger edge",
      "great significance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}