{"title":"小阶广义彼得森图的定位色度数","authors":"Redha Sakri , Moncef Abbas","doi":"10.1016/j.exco.2024.100141","DOIUrl":null,"url":null,"abstract":"<div><p>It was conjectured by Asmiati (2018) that the generalized Petersen graph <span><math><mrow><mi>P</mi><mfenced><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></mfenced></mrow></math></span> has a locating chromatic number 4 if and only if <span><math><mrow><mo>(</mo><mi>n</mi><mspace></mspace><mi>o</mi><mi>d</mi><mi>d</mi><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></math></span> or <span><math><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>4</mn><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></math></span>. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if <span><math><mrow><mo>(</mo><mn>5</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>12</mn><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>)</mo></mrow></math></span> and <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow><mo>≠</mo><mrow><mo>(</mo><mn>12</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>L</mi></mrow></msub></mrow></msub><mfenced><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mfenced><mo>=</mo><mn>4</mn></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100141"},"PeriodicalIF":0.0000,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000077/pdfft?md5=0c1ce0bbc9c76ab3ef2eb212405914a8&pid=1-s2.0-S2666657X24000077-main.pdf","citationCount":"0","resultStr":"{\"title\":\"The locating chromatic number of generalized Petersen graphs with small order\",\"authors\":\"Redha Sakri , Moncef Abbas\",\"doi\":\"10.1016/j.exco.2024.100141\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>It was conjectured by Asmiati (2018) that the generalized Petersen graph <span><math><mrow><mi>P</mi><mfenced><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></mfenced></mrow></math></span> has a locating chromatic number 4 if and only if <span><math><mrow><mo>(</mo><mi>n</mi><mspace></mspace><mi>o</mi><mi>d</mi><mi>d</mi><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></math></span> or <span><math><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>4</mn><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></math></span>. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if <span><math><mrow><mo>(</mo><mn>5</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>12</mn><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>)</mo></mrow></math></span> and <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow><mo>≠</mo><mrow><mo>(</mo><mn>12</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>L</mi></mrow></msub></mrow></msub><mfenced><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mfenced><mo>=</mo><mn>4</mn></mrow></math></span>.</p></div>\",\"PeriodicalId\":100517,\"journal\":{\"name\":\"Examples and Counterexamples\",\"volume\":\"5 \",\"pages\":\"Article 100141\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-02-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S2666657X24000077/pdfft?md5=0c1ce0bbc9c76ab3ef2eb212405914a8&pid=1-s2.0-S2666657X24000077-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Examples and Counterexamples\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666657X24000077\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Examples and Counterexamples","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666657X24000077","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The locating chromatic number of generalized Petersen graphs with small order
It was conjectured by Asmiati (2018) that the generalized Petersen graph has a locating chromatic number 4 if and only if or . In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if and and , then .
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