{"title":"广义旋转星痕的完美匹配覆盖指数","authors":"Wenjuan Zhou, Rong-Xia Hao, Yilun Luo","doi":"10.1016/j.amc.2024.129101","DOIUrl":null,"url":null,"abstract":"<div><div>Let Γ be a 3-regular bridgeless graph and <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> be the perfect matching cover index of Γ. It is conjectured by Berge that <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>5</mn></math></span>. Esperet and Mazzuoccolo (2013) <span><span>[4]</span></span> proved that deciding whether Γ satisfies <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>4</mn></math></span> is NP-complete. Máčajová and Škoviera (2021) <span><span>[13]</span></span> gave a family <span><math><mi>R</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><mi>k</mi><mo>≥</mo><mn>4</mn><mo>}</mo></math></span> of rotation snarks. We construct a family <span><math><mrow><mi>FR</mi></mrow><mo>=</mo><mo>{</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span> of generalized rotation snarks. In this paper, we show that each <span><math><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>. As a corollary, each <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Perfect matching cover indices of generalized rotation snarks\",\"authors\":\"Wenjuan Zhou, Rong-Xia Hao, Yilun Luo\",\"doi\":\"10.1016/j.amc.2024.129101\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let Γ be a 3-regular bridgeless graph and <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> be the perfect matching cover index of Γ. It is conjectured by Berge that <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>5</mn></math></span>. Esperet and Mazzuoccolo (2013) <span><span>[4]</span></span> proved that deciding whether Γ satisfies <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>4</mn></math></span> is NP-complete. Máčajová and Škoviera (2021) <span><span>[13]</span></span> gave a family <span><math><mi>R</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><mi>k</mi><mo>≥</mo><mn>4</mn><mo>}</mo></math></span> of rotation snarks. We construct a family <span><math><mrow><mi>FR</mi></mrow><mo>=</mo><mo>{</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span> of generalized rotation snarks. In this paper, we show that each <span><math><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>. As a corollary, each <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>.</div></div>\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2024-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0096300324005629\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300324005629","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
Perfect matching cover indices of generalized rotation snarks
Let Γ be a 3-regular bridgeless graph and be the perfect matching cover index of Γ. It is conjectured by Berge that . Esperet and Mazzuoccolo (2013) [4] proved that deciding whether Γ satisfies is NP-complete. Máčajová and Škoviera (2021) [13] gave a family of rotation snarks. We construct a family of generalized rotation snarks. In this paper, we show that each satisfies . As a corollary, each satisfies .
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