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{
    "pk": 64867,
    "title": "Period collapse in Ehrhart quasi-polynomials of \\(\\{1,3\\}\\)-graphs",
    "subtitle": null,
    "abstract": "A graph whose nodes have degree \\(1\\) or \\(3\\) is called a \\(\\{1,3\\}\\)-graph. Liu and Osserman associated a polytope to each \\(\\{1,3\\}\\)-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set \\(\\{0,\\frac14,\\frac12,1\\}\\), which implies that the period of their Ehrhart quasi-polynomials is either \\(1, 2\\), or \\(4\\). We show that the period of the Ehrhart quasi-polynomial of these polytopes is 2 if the graph is a tree, the period is at most 2 if the graph is cubic, and the period is \\(4\\) otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.\n \nMathematics Subject Classifications: 05C30, 05C76, 52B20\n \nKeywords: Ehrhart polynomials, period collapse",
    "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": "Ehrhart polynomials"
        },
        {
            "word": "period collapse"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8n44x9vm",
    "frozenauthors": [
        {
            "first_name": "Cristina",
            "middle_name": "G.",
            "last_name": "Fernandes",
            "name_suffix": "",
            "institution": "Instituto de Matemática e Estatística, Universidade de S ao Paulo, 05508-090 S ao Paulo, Brazil",
            "department": ""
        },
        {
            "first_name": "José",
            "middle_name": "C.",
            "last_name": "de Pina",
            "name_suffix": "",
            "institution": "Instituto de Matemática e Estatística, Universidade de S ao Paulo, 05508-090 S ao Paulo, Brazil",
            "department": ""
        },
        {
            "first_name": "Jorge",
            "middle_name": "Luis",
            "last_name": "Ramírez Alfonsín",
            "name_suffix": "",
            "institution": "IMAG, Univ. Montpellier, CNRS, Montpellier, France and UMI2924 - Jean-Christophe Yoccoz, CNRS-IMPA",
            "department": ""
        },
        {
            "first_name": "Sinai",
            "middle_name": "",
            "last_name": "Robins",
            "name_suffix": "",
            "institution": "Instituto de Matemática e Estatística, Universidade de S ao Paulo, 05508-090 S ao Paulo, Brazil",
            "department": ""
        }
    ],
    "date_submitted": "2022-10-12T14:30:44Z",
    "date_accepted": "2022-10-12T14:30:44Z",
    "date_published": "2022-10-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64867/galley/49677/download/"
        }
    ]
}