{
  "id": "2408.09020",
  "version": "v1",
  "published": "2024-08-16T21:09:18.000Z",
  "updated": "2024-08-16T21:09:18.000Z",
  "title": "On the Edge-Connectivity of the Square of a Graph",
  "authors": [
    "Camino Balbuena",
    "Peter Dankelmann"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected graph. The edge-connectivity of $G$, denoted by $\\lambda(G)$, is the minimum number of edges whose removal renders $G$ disconnected. Let $\\delta(G)$ be the minimum degree of $G$. It is well-known that $\\lambda(G) \\leq \\delta(G)$, and graphs for which equality holds are said to be maximally edge-connected. The square $G^2$ of $G$ is the graph with the same vertex set as $G$, in which two vertices are adjacent if their distance is not more that $2$. In this paper we present results on the edge-connectivity of the square of a graph. We show that if the minimum degree of a connected graph $G$ of order $n$ is at least $\\lfloor \\frac{n+2}{4}\\rfloor$, then $G^2$ is maximally edge-connected, and this result is best possible. We also give lower bounds on $\\lambda(G^2)$ for the case that $G^2$ is not maximally edge-connected: We prove that $\\lambda(G^2) \\geq \\kappa(G)^2 + \\kappa(G)$, where $\\kappa(G)$ denotes the connectivity of $G$, i.e., the minimum number of vertices whose removal renders $G$ disconnected, and this bound is sharp. We further prove that $\\lambda(G^2) \\geq \\frac{1}{2}\\lambda(G)^{3/2} - \\frac{1}{2} \\lambda(G)$, and we construct an infinite family of graphs to show that the exponent $3/2$ of $\\lambda(G)$ in this bound is best possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-16T21:09:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40"
    ],
    "keywords": [
      "edge-connectivity",
      "minimum number",
      "removal renders",
      "minimum degree",
      "connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}