{"title":"On the inclusion chromatic index of a Halin graph","authors":"","doi":"10.1016/j.disc.2024.114266","DOIUrl":null,"url":null,"abstract":"<div><div>An inclusion-free edge-coloring of a graph <em>G</em> with <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of <em>G</em> is called the <span><math><mi>i</mi><mi>n</mi><mi>c</mi><mi>l</mi><mi>u</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span>-<span><math><mi>f</mi><mi>r</mi><mi>e</mi><mi>e</mi></math></span> <span><math><mi>c</mi><mi>h</mi><mi>r</mi><mi>o</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>c</mi><mspace></mspace><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>x</mi></math></span>, denoted by <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we show that for a Halin graph <em>G</em> with maximum degree <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>, if <em>G</em> is isomorphic to a wheel <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> where Δ is odd, then <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>2</mn></math></span>, otherwise <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span>. We also show a special cubic Halin graph with <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003972","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
An inclusion-free edge-coloring of a graph G with is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of G is called the - , denoted by . In this paper, we show that for a Halin graph G with maximum degree , if G is isomorphic to a wheel where Δ is odd, then , otherwise . We also show a special cubic Halin graph with .
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.