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{
    "pk": 65022,
    "title": "A tower lower bound for the degree relaxation of the Regularity Lemma",
    "subtitle": null,
    "abstract": "It is well-known that if \\((A,B)\\) is an \\(\\tfrac{\\varepsilon}{2}\\)-regular pair (in the sense of Szemerédi) then there exist sets \\(A'\\subset A\\) and \\(B'\\subset B\\) with \\(|A'|\\le \\varepsilon|A|\\) and \\(|B'|\\le \\varepsilon|B|\\) so that the degrees of all vertices in \\(A\\setminus A'\\) differ by at most \\(\\varepsilon|B|\\) and the degrees of all vertices in \\(B\\setminus B'\\) differ by at most \\(\\varepsilon|A|\\). We call such a property \\(\\varepsilon\\)-degularity. This leads to the notion of an \\(\\varepsilon\\)-degular partition of a graph in the same way as the definition of \\(\\varepsilon\\)-regular pairs leads to the notion of \\(\\varepsilon\\)-regular partitions.\nWe show that there exist graphs in which any \\(\\varepsilon\\)-degular partition requires the number of clusters to be \\(\\mathrm{tower}(\\Theta(\\varepsilon^{-1/3}))\\). That is, even though degularity is a substantial relaxation of regularity, in general one cannot improve much on the bounds that come with Szemerédi's regularity lemma.\n \nMathematics Subject Classifications: 05C35\n \nKeywords: Szemerédi's regularity lemma, degree",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Szemerédi's regularity lemma"
        },
        {
            "word": "degree"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/46m655nk",
    "frozenauthors": [
        {
            "first_name": "Frederik",
            "middle_name": "",
            "last_name": "Garbe",
            "name_suffix": "",
            "institution": "Heidelberg University, Faculty of Mathematics and Computer Science, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany",
            "department": ""
        },
        {
            "first_name": "Jan",
            "middle_name": "",
            "last_name": "Hladký",
            "name_suffix": "",
            "institution": "Institute of Computer Science of the Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07 Prague, Czechia",
            "department": ""
        }
    ],
    "date_submitted": "2026-02-02T09:39:55Z",
    "date_accepted": "2026-02-02T09:39:55Z",
    "date_published": "2025-12-20T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65022/galley/49832/download/"
        }
    ]
}