{"title":"2-Distance (Δ + 1)-coloring of sparse graphs using the potential method","authors":"","doi":"10.1016/j.disc.2024.114292","DOIUrl":null,"url":null,"abstract":"<div><div>A 2-distance <em>k</em>-coloring of a graph is a proper <em>k</em>-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (<span><math><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span>)-coloring for graphs with maximum average degree less than <span><math><mfrac><mrow><mn>18</mn></mrow><mrow><mn>7</mn></mrow></mfrac></math></span> and maximum degree <span><math><mi>Δ</mi><mo>≥</mo><mn>7</mn></math></span>. As a corollary, every planar graph with girth at least 9 and <span><math><mi>Δ</mi><mo>≥</mo><mn>7</mn></math></span> admits a 2-distance <span><math><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-21","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/S0012365X24004230","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
A 2-distance k-coloring of a graph is a proper k-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance ()-coloring for graphs with maximum average degree less than and maximum degree . As a corollary, every planar graph with girth at least 9 and admits a 2-distance -coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.
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.