{"title":"Re-1-embeddings of optimal 1-embedded graphs on the projective plane","authors":"Yusuke Suzuki","doi":"10.1016/j.dam.2024.10.005","DOIUrl":null,"url":null,"abstract":"<div><div>It was shown in Schumacher (1986), Suzuki (2010) that every optimal 1-embedded graph on the sphere has at most 8 inequivalent 1-embeddings. In this paper, we prove that the number of inequivalent 1-embeddings of an optimal 1-embedded graph on the projective plane whose quadrangular subgraph is bipartite is at most 24. In the case where such quadrangular subgraphs are nonbipartite, we show an optimal 1-embedded graph having at least <span><math><mi>p</mi></math></span> inequivalent 1-embeddings for any large integer <span><math><mi>p</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 487-496"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004311","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
It was shown in Schumacher (1986), Suzuki (2010) that every optimal 1-embedded graph on the sphere has at most 8 inequivalent 1-embeddings. In this paper, we prove that the number of inequivalent 1-embeddings of an optimal 1-embedded graph on the projective plane whose quadrangular subgraph is bipartite is at most 24. In the case where such quadrangular subgraphs are nonbipartite, we show an optimal 1-embedded graph having at least inequivalent 1-embeddings for any large integer .
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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