Towards a conjecture of Birmelé–Bondy–Reed on the Erdős–Pósa property of long cycles

IF 1 3区数学 Q2 MATHEMATICS Journal of Graph Theory Pub Date : 2022-11-22 DOI:10.1002/jgt.22911

Jie Ma, Chunlei Zu

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Abstract

A conjecture of Birmelé, Bondy, and Reed states that for any integer $ℓ \geq 3$ , every graph $G$ without two vertex-disjoint cycles of length at least $ℓ$ contains a set of at most $ℓ$ vertices which meets all cycles of length at least $ℓ$ . They showed the existence of such a set of at most $2 ℓ + 3$ vertices. This was improved by Meierling, Rautenbach, and Sasse to $5 ℓ ∕ 3 + 29 ∕ 2$ . Here we present a proof showing that at most $3 ℓ ∕ 2 + 7 ∕ 2$ vertices suffice.

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Birmelé-Bondy–Reed关于长周期Erdõs–Pósa性质的猜想

Birmelé、Bondy和Reed的一个猜想指出，对于任何整数ℓ ≥ 3$\ell\ge3$，每个图G$G$至少没有两个长度不相交的顶点循环ℓ $\ell$最多包含一组ℓ $\ell$至少满足所有长度循环的顶点ℓ $\ell$。他们证明了这样一组至多2ℓ + 3$2\ell+3$个顶点。梅尔林、劳滕巴赫和萨斯将比分提高到5ℓ ∕ 3+29∕2$5\ell\unicode{x02215}3+29\unicode{x02215}2$。在这里，我们提出了一个证据，表明最多3ℓ ∕ 2+7∕2$3\ell\unicode{x02215}2+7\unicode{x02215}2$顶点就足够了。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .

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