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{
    "pk": 64861,
    "title": "Minimizing cycles in tournaments and normalized \\(q\\)-norms",
    "subtitle": null,
    "abstract": "Akin to the Erdős-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any \\(d\\in (0,1]\\), among all \\(n\\)-vertex tournaments with \\(d\\binom{n}{3}\\) many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Král' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for \\(d\\geq 1/36\\).  In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers \\(\\ell\\not\\equiv 2\\mod 4\\), there exists some constant \\(c_\\ell>0\\) such that if \\(d\\geq 1-c_\\ell\\), then the number of \\(\\ell\\)-cycles is also asymptotically minimized by the same extremal examples. In doing so, we answer a question of Linial and Morgenstern about minimizing the \\(q\\)-norm of a probabilistic vector with given \\(p\\)-norm for integers \\(q>p>1\\). For integers \\(\\ell\\equiv 2\\mod 4\\), however the same phenomena do not hold for \\(\\ell\\)-cycles, for which we can construct an explicit family of tournaments containing fewer \\(\\ell\\)-cycles for any given number of \\(3\\)-cycles. We propose two conjectures concerning the minimization problem for general cycles.\n \nMathematics Subject Classifications: 05C20, 05C35, 05C38\n \nKeywords: Tournaments, cycles, spectrum",
    "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": "Tournaments"
        },
        {
            "word": "cycles"
        },
        {
            "word": "spectrum"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/2rm2x1nm",
    "frozenauthors": [
        {
            "first_name": "Jie",
            "middle_name": "",
            "last_name": "Ma",
            "name_suffix": "",
            "institution": "School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China",
            "department": ""
        },
        {
            "first_name": "Tianyun",
            "middle_name": "",
            "last_name": "Tang",
            "name_suffix": "",
            "institution": "Department of Mathematics, National University of Singapore, Singapore",
            "department": ""
        }
    ],
    "date_submitted": "2022-10-11T15:58:45Z",
    "date_accepted": "2022-10-11T15:58:45Z",
    "date_published": "2022-10-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64861/galley/49671/download/"
        }
    ]
}