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{
    "pk": 64870,
    "title": "Linear-sized independent sets in random cographs and increasing subsequences in separable permutations",
    "subtitle": null,
    "abstract": "This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations.  First, we prove that, with high probability as \\(n\\) gets large, the largest independent set in a uniform random cograph with \\(n\\) vertices has size \\(o(n)\\). This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit.  Second, and unexpectedly given the above results, we show that for \\(\\beta >0\\) sufficiently small, the expected number of independent sets of size \\(\\beta n\\) in a uniform random cograph with \\(n\\) vertices grows exponentially fast with \\(n\\). We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions.\n \nMathematics Subject Classifications: 60C05, 05C80, 05C69, 05A05\n \nKeywords: Combinatorial graph theory, combinatorial probability, cographs, random  graphs, graphons, self-similarity",
    "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": "Combinatorial graph theory"
        },
        {
            "word": "combinatorial probability"
        },
        {
            "word": "cographs"
        },
        {
            "word": "random  graphs"
        },
        {
            "word": "graphons"
        },
        {
            "word": "self-similarity"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/23340676",
    "frozenauthors": [
        {
            "first_name": "Frédérique",
            "middle_name": "",
            "last_name": "Bassino",
            "name_suffix": "",
            "institution": "Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430 Villetaneuse, France",
            "department": ""
        },
        {
            "first_name": "Mathilde",
            "middle_name": "",
            "last_name": "Bouvel",
            "name_suffix": "",
            "institution": "Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland and Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France",
            "department": ""
        },
        {
            "first_name": "Michael",
            "middle_name": "",
            "last_name": "Drmota",
            "name_suffix": "",
            "institution": "Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria",
            "department": ""
        },
        {
            "first_name": "Valentin",
            "middle_name": "",
            "last_name": "Féray",
            "name_suffix": "",
            "institution": "Université de Lorraine, CNRS, IECL, F-54000 Nancy, France",
            "department": ""
        },
        {
            "first_name": "Lucas",
            "middle_name": "",
            "last_name": "Gerin",
            "name_suffix": "",
            "institution": "CMAP, École polytechnique, CNRS, I.P. Paris, 91128 Palaiseau, France",
            "department": ""
        },
        {
            "first_name": "Mickaël",
            "middle_name": "",
            "last_name": "Maazoun",
            "name_suffix": "",
            "institution": "Department of Statistics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, U.K.",
            "department": ""
        },
        {
            "first_name": "Adeline",
            "middle_name": "",
            "last_name": "Pierrot",
            "name_suffix": "",
            "institution": "Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France",
            "department": ""
        }
    ],
    "date_submitted": "2022-10-13T06:30:56Z",
    "date_accepted": "2022-10-13T06:30:56Z",
    "date_published": "2022-10-15T07:00:00Z",
    "render_galley": null,
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            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64870/galley/49680/download/"
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    ]
}