{"title":"Hoffmann-Ostenhof's 3-Decomposition Conjecture","authors":"Genghua Fan, Chuixiang Zhou","doi":"10.1016/j.disc.2025.114454","DOIUrl":null,"url":null,"abstract":"<div><div>The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a matching. It has been proved independently by different groups of people that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges. In this paper, we establish a bound on the number of paths of two edges, proving that every connected cubic graph on <em>n</em> vertices can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges such that the number of paths of two edges is at most <span><math><mfrac><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></span>. Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114454"},"PeriodicalIF":0.7000,"publicationDate":"2025-02-20","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/S0012365X25000627","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
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a matching. It has been proved independently by different groups of people that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges. In this paper, we establish a bound on the number of paths of two edges, proving that every connected cubic graph on n vertices can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges such that the number of paths of two edges is at most . Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture.
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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