Packing directed cycles of specified odd length into digraphs and alternating cycles into bipartite graphs

IF 0.7 3区数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-11-06 DOI:10.1016/j.disc.2024.114306

Shuya Chiba , Koshin Yoshida

引用次数: 0

Abstract

In this paper, we prove the following result. For given integers

k \geq 1, t \geq 0

with

k \geq t

and an odd integer

ℓ \geq 3

, there exists an integer

n_{0} = n_{0} (k, t, ℓ)

satisfying the following: If D is a digraph of order

n \geq n_{0}

, and if

d_{D}^{+} (u) + d_{D}^{-} (v) \geq n + t

for every two distinct vertices u and v with

(u, v) \notin A (D)

, then D contains k vertex-disjoint directed cycles of length ℓ or

ℓ + 1

such that at least t of them are of length ℓ. This is a common extension of the results obtained by Brandt et al. (1997) and, Chiba and Yamashita (2018). We also discuss the relation between our result and problems on packing alternating cycles into bipartite graphs.

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将指定奇数长度的有向循环打包到数图中，将交替循环打包到二方图中

本文将证明以下结果。对于给定整数 k≥1,t≥0，且 k≥t 和奇整数 ℓ≥3，存在满足以下条件的整数 n0=n0(k,t,ℓ)：如果 D 是阶数 n≥n0 的数图，且对于每两个不同的顶点 u 和 v，且 (u,v)∉A(D) 的 dD+(u)+dD-(v)≥n+t ，那么 D 包含长度为 ℓ 或 ℓ+1 的 k 个顶点相交的有向循环，且其中至少有 t 个循环的长度为 ℓ。这是对 Brandt 等人（1997）以及 Chiba 和 Yamashita（2018）所获结果的常见扩展。我们还讨论了我们的结果与将交替循环打包进双方形图问题之间的关系。

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.