Hoffmann-Ostenhof's 3-Decomposition Conjecture

IF 0.7 3区数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2025-07-01 Epub Date: 2025-02-20 DOI:10.1016/j.disc.2025.114454

Genghua Fan, Chuixiang Zhou

引用次数: 0

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

\frac{n - 4}{6}

. Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture.

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Hoffmann-Ostenhof的3分解猜想

3-分解猜想指出，每一个连通的三次图都可以分解成一个生成树、一组环和一个匹配。不同的人已经独立地证明了每一个连通三次图都可以分解成一棵生成树、一组环和一组至多两条边的顶点不相交路径。本文建立了两条边的路径数的一个界，证明了每一个有n个顶点的连通三次图都可以分解为至多两条边的生成树、一组环和一组顶点不相交的路径，使得两条边的路径数至多为n−46。我们的证明是基于结构分析的，这可能为攻击3分解猜想提供一种新的方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

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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