{
  "abstract": "A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):=\\sum_{\\ell\\in{C}(G)}1/\\ell$ as a measure of the richness of the cycle length spectrum ${C}(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k, n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\\leq n/2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\\geq 2k$ satisfies $s(G)\\geq\\sum_{\\ell=2}^{k}1/(2\\ell)$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26401",
  "author_total": 4,
  "authors": [
    {
      "name": "Richard Montgomery"
    },
    {
      "name": "Aleksa Milojević"
    },
    {
      "name": "Alexey Pokrovskiy"
    },
    {
      "name": "Benny Sudakov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/28w5tsng63k7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26401",
  "fulltext_url": null,
  "id": "28w5tsng63k7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26401",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214092988",
  "pdf_url": "https://arxiv.org/pdf/2609.26401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/28w5tsng63k7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Minimising the harmonic sum of cycle lengths",
  "title_text": "Minimising the harmonic sum of cycle lengths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/28w5tsng63k7s",
  "venue": "arXiv",
  "versions": []
}