{"title":"A note on clique immersion of strong product graphs","authors":"","doi":"10.1016/j.disc.2024.114237","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>G</mi><mo>,</mo><mi>H</mi></math></span> be graphs, and <span><math><mi>G</mi><mo>⁎</mo><mi>H</mi></math></span> represent a specific graph product of <em>G</em> and <em>H</em>. Define <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> as the largest <em>t</em> for which <em>G</em> contains a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-immersion. Collins, Heenehan, and McDonald posed the question: given <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>t</mi></math></span> and <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi></math></span>, how large can <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo></math></span> be? They conjectured <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mi>t</mi><mi>r</mi></math></span> when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when <em>H</em> contains <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003686/pdfft?md5=2175b8b68439085105021d9c5e79d193&pid=1-s2.0-S0012365X24003686-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003686","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let be graphs, and represent a specific graph product of G and H. Define as the largest t for which G contains a -immersion. Collins, Heenehan, and McDonald posed the question: given and , how large can be? They conjectured when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when H contains as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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