{"title":"The conflict-free connection number and the minimum degree-sum of graphs","authors":"","doi":"10.1016/j.amc.2024.128981","DOIUrl":null,"url":null,"abstract":"<div><p>In the context of an edge-coloured graph <em>G</em>, a path within the graph is deemed <em>conflict-free</em> when a colour is exclusively applied to one of its edges. The presence of a conflict-free path connecting any two unique vertices of an edge-coloured graph is what defines it as <em>conflict-free connected</em>. The <em>conflict-free connection number</em>, indicated by <span><math><mi>c</mi><mi>f</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the fewest number of colours necessary to make <em>G</em> conflict-free connected. Consider the subgraph <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a connected graph <em>G</em>, which is constructed from the cut-edges of <em>G</em>. Let <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the minimum degree-sum of any 3 independent vertices in <em>G</em>. In this study, we establish that for a connected graph <em>G</em> with an order of <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span> and <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, the following conditions hold: (1) <span><math><mi>c</mi><mi>f</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>3</mn></math></span> when <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span>; (2) <span><math><mi>c</mi><mi>f</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span> when <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> forms a linear forest. Moreover, we will now demonstrate that if <em>G</em> is a connected, non-complete graph with <em>n</em> vertices, where <span><math><mi>n</mi><mo>≥</mo><mn>43</mn></math></span>, <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is a linear forest, <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>3</mn></math></span>, and <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>14</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>, then <span><math><mi>c</mi><mi>f</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>. Moreover, we also determine the upper bound of the number of cut-edges of a connected graph depending on the degree-sum of any three independent vertices.</p></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300324004429","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In the context of an edge-coloured graph G, a path within the graph is deemed conflict-free when a colour is exclusively applied to one of its edges. The presence of a conflict-free path connecting any two unique vertices of an edge-coloured graph is what defines it as conflict-free connected. The conflict-free connection number, indicated by , is the fewest number of colours necessary to make G conflict-free connected. Consider the subgraph of a connected graph G, which is constructed from the cut-edges of G. Let be the minimum degree-sum of any 3 independent vertices in G. In this study, we establish that for a connected graph G with an order of and , the following conditions hold: (1) when ; (2) when forms a linear forest. Moreover, we will now demonstrate that if G is a connected, non-complete graph with n vertices, where , is a linear forest, , and , then . Moreover, we also determine the upper bound of the number of cut-edges of a connected graph depending on the degree-sum of any three independent vertices.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.