不分离循环和5循环双盖

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-04-11 DOI:10.1016/j.disc.2025.114515

Xiaoyu Li , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang

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引用次数: 0

摘要

如果 G-E(C)是连通的，则图 G 中的循环 C 是非分离的。作为攻克众所周知的循环双覆盖猜想及其更强版本：5 循环双覆盖猜想的一种方法，霍夫曼-奥斯坦霍夫（2017）猜想，如果一个 2 边相连的立方图有一个非分离循环 C，那么 G 有一个循环双覆盖。Hoffmann-Ostenhof 等人（European J. Combin.2019）证明，如果一个 2 边相连的图 G 有一个非分离循环 C，使得ϵ(G-E(C))=0 且 w(C)≤3，其中ϵ(G-E(C)) 和 w(C) 分别是 G-E(C) 循环空间的秩和 C 的分量数，那么 G 有一个包含 C 的 5 循环双覆盖，除非 G 可收缩为彼得森图，在这种情况下，G 有一个 6 循环双覆盖。在本文中，我们扩展了他们的结果，证明如果一个 2 边相连的图 G 有一个非分离循环 C，使得⌊ϵ(G-E(C))+52w(C)⌋≤8，那么根据 G 是否可收缩为彼得森图，G 有一个包含 C 的 5 循环双盖或 6 循环双盖。本文还举例说明了主定理的尖锐性。

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Non-separating cycles and 5-cycle double covers

A cycle C in a graph G is non-separating if

G - E (C)

is connected. As an approach to attack the well-known cycle double cover conjecture and its stronger version: the 5-cycle double cover conjecture, it is conjectured by Hoffmann-Ostenhof (2017) that if a 2-edge connected cubic graph has a non-separating cycle C, then G has a cycle double cover. Hoffmann-Ostenhof et al. (European J. Combin. 2019) show that if a 2-edge connected graph G has a non-separating cycle C such that

ϵ (G - E (C)) = 0

and

w (C) \leq 3

, where

ϵ (G - E (C))

and

w (C)

are the rank of the cycle space of

G - E (C)

and the number of the components of C, respectively, then G has a 5-cycle double cover containing C unless G is contractible to the Petersen graph, in which case, G has a 6-cycle double cover. In this paper we extend their result and prove that if a 2-edge connected graph G has a non-separating cycle C such that

⌊ ϵ (G - E (C)) + \frac{5}{2} w (C) ⌋ \leq 8

, then G has a 5-cycle double cover or 6-cycle double cover containing C depending on whether G is contractible to the Petersen graph or not. Examples are also constructed in this paper showing the sharpness of the main theorem.

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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.