{
  "annotation_id": "e1eb8841-6162-4c9e-a425-a05ed5a70c37",
  "date_created": "2026-02-17T05:42:09.935000Z",
  "date_modified": "2026-02-17T05:42:09.935000Z",
  "file_hash": "7c68d5ae105ba87c41cb3053e5a97023c1464399ad488016804dc286ff660473",
  "private": false,
  "record": {
    "abstract": "For a simple graph $G$, let $n$ denote its number of vertices, and let $N(G,K_t)$ denote the number of copies of $K_t$ in $G$. Zykov\u0027s theorem (1949) asserts that for any $K_{r+1}$-free graph and $t \\ge 2$, \\[ N(G,K_t) \\le {r \\choose t}\\left(\\frac{n}{r}\\right)^t \\] We generalize Zykov\u0027s bound within a vertex-based localization framework.\n  For each vertex $v \\in V(G)$, let $c(v)$ denote the order of the largest clique containing $v$. In this paper, we show that \\[ N(G,K_t) \\le n^{t-1} \\sum_{v \\in V(G)} \\frac{1}{c(v)^t} {c(v) \\choose t} \\] We further show that equality holds if and only if $G$ is a regular complete multipartite graph. \\newline Note that if we impose the condition that, $G$ is $K_{r+1}$-free, then $c(v) \\leq r$ for all $v \\in V(G)$. Thus, plugging $c(v) = r$ for all $v \\in V(G)$, we retrieve Zykov\u0027s bound.",
    "arxiv_id": "2512.02958",
    "authors": [
      "Rajat Adak",
      "L. Sunil Chandran"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Generalized Zykov\u0027s Theorem",
    "url": "https://arxiv.org/abs/2512.02958",
    "version": "v2"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}