Journal of Combinatorial Theory Series B最新文献

英文中文

Weak diameter choosability of graphs with an excluded minor 具有排除次要项的图的弱直径可选择性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.005

Joshua Crouch, Chun-Hung Liu

Weak diameter coloring of graphs recently attracted attention, partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph H, H-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring H-minor free graphs with bounded weak diameter. As a corollary, H-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.

When H is planar, we prove a much stronger result: for every 2-list-assignment L of an H-minor free graph, every precoloring with bounded weak diameter can be extended to an L-coloring with bounded weak diameter. It is a common generalization of earlier results for non-list-coloring with bounded weak diameter and for list-coloring with bounded clustering without allowing precolorings. As a corollary, for any planar graph H and H-minor free graph G, there are exponentially many list-colorings of G with bounded weak diameter (and with bounded clustering if G also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.

We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree Δ are k-choosable with bounded weak diameter only when

k = Ω (\log Δ / \log \log Δ)

. On the other hand, we show that odd H-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd H-minor free graphs with bounded maximum degree.

图的弱直径着色近年来引起人们的关注，部分原因是它与度量空间的渐近维数有关。本文研究了图的弱直径表着色问题。Dvořák和Norin证明了具有有界欧拉属的图是具有有界弱直径的3-可选图。在本文中，我们推广了它们的结果，证明了对于每一个图H， H次自由图都是具有有界弱直径的3-可选图。上界3是最优的，它加强了之前关于弱直径有界的非列表着色h次自由图的结果。作为一个推论，具有有界最大度的h次自由图在有界聚类中是3-可选的，加强了之前关于非列表着色的结果。当H是平面时，我们证明了一个更强的结果：对于H次自由图的每一个2-列表赋值L，每一个弱直径有界的预着色都可以推广到弱直径有界的L着色。对于有界弱直径的非列表着色和不允许预着色的有界聚类的列表着色，这是早期结果的一般推广。作为推论，对于任意平面图H和H次自由图G， G的弱直径有界（如果G的最大度也有界，则G的聚类有界）存在指数多列着色；并且每一个层树宽度有界、最大度有界的图都有指数次的有界聚类的3色。通过证明一些最大度为Δ的二部图只有在k=Ω（log （Δ） /log (log)）时才具有弱直径有界的k-可选性，我们还证明了上述关于列表着色的结果不能推广到奇次自由图。另一方面，我们证明了奇h小图是弱直径有界的3色图，暗示了关于最大度有界的奇h小自由图的聚类着色的早期结果。

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引用次数: 0

Induced C4-free subgraphs with large average degree 具有较大平均度的无 C4 子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.002

Xiying Du , António Girão , Zach Hunter , Rose McCarty , Alex Scott

We prove that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subgraph which is

C_{4}

-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.

We show that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subdivision of

K_{k}

. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both k and s.

We also show that for any hereditary degree-bounded class

F

, there exists a constant

C = C_{F}

so that

C^{s^{3}}

is a degree-bounding function for

F

. This is the first bound of any type on the rate of growth of such functions.

我们证明了存在一个常数C，使得对于所有s，k∈N，如果G的平均度至少为kCs3且不包含Ks，s作为子图，则它包含一个不含c4且平均度至少为k的诱导子图。已知s和k的某个函数是足够的，但这是第一个显式界。我们给出了这个结果的几个应用，包括以下两个推论的简短和简化的证明。我们证明了存在一个常数C，使得对于所有s，k∈N，如果G的平均度至少为kCs3，并且不包含Ks，s作为子图，则它包含Kk的诱导子图。这是对k hn和Osthus的一个著名定理的第一个定量改进；他们的证明给出了一个在k和s上都是三指数的界。我们还证明了对于任何遗传度有界类F，存在一个常数C=CF，使得Cs3是F的一个度有界函数。这是关于这类函数增长率的任何类型的第一个界。

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引用次数: 0

A matrix realization of spectral bounds 谱界的矩阵实现

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.006

Yen-Jen Cheng , Chih-wen Weng

We give a unified and systematic way to find bounds for the largest real eigenvalue of a nonnegative matrix by considering its modified quotient matrix. We leverage this insight to identify the unique matrix whose largest real eigenvalue is maximum among all

(0, 1)

-matrices with a specified number of ones. This result resolves a problem that was posed independently by R. Brualdi and A. Hoffman, as well as F. Friedland, back in 1985.

利用非负矩阵的修正商矩阵，给出了求非负矩阵最大实特征值界的统一、系统的方法。我们利用这一见解来识别唯一矩阵，其最大实特征值在所有(0,1)-具有指定数量的矩阵中是最大的。这一结果解决了R. Brualdi和a . Hoffman以及F. Friedland在1985年独立提出的一个问题。

引用次数: 0

Haar graphical representations of finite groups and an application to poset representations 有限群的图解表示及其在偏置集表示中的应用

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.001

Joy Morris , Pablo Spiga

Let R be a group and let S be a subset of R. The Haar graph

Haar (R, S)

of R with connection set S is the graph having vertex set

R \times {- 1, 1}

, where two distinct vertices

(x, - 1)

and

(y, 1)

are declared to be adjacent if and only if

y x^{- 1} \in S

. The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs.

For every

g \in R

, the mapping

ρ_{g} : (x, ε) \mapsto (x g, ε)

\forall (x, ε) \in R \times {- 1, 1}

, is an automorphism of

Haar (R, S)

. In particular, the set

\hat{R} : = {ρ_{g} | g \in R}

is a subgroup of the automorphism group of

Haar (R, S)

isomorphic to R. In the case that the automorphism group of

Haar (R, S)

equals

\hat{R}

, the Haar graph

Haar (R, S)

is said to be a Haar graphical representation of the group R.

Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions.

Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.

设R是一个群，S是R的一个子集。具有连接集S的R的Haar图Haar（R，S）是具有顶点集rx{- 1,1}的图，其中两个不同的顶点（x, - 1）和（y,1）被声明为相邻当且仅当x - 1∈S。Haar图这个名字是由tomajov Pisanski在对这类图的第一次研究中创造的。每g∈R映射ρg: (x,ε)↦(xgε)∀(x,ε)∈R×{−1,1},是哈尔(R, S)的自同构。特别地，集合R:={ρg|g∈R}是与R同构的Haar（R，S）的自同构群的一子群。在Haar（R，S）的自同构群等于R的情况下，我们称Haar图Haar（R，S）是群R的一个Haar图表示。具体地说，我们证明了除了阿贝尔群和其他10个小群外，每个有限群都有一个Haar图表示。我们在Haar图上的工作使我们能够改进Babai在1980年关于群在偏置集上表示的结果，在这个方向上取得了最好的结果。对Babai关于群在分布格上表示的相关结果进行了改进。

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引用次数: 0

Pivot-minors and the Erdős-Hajnal conjecture 支点小调和Erdős-Hajnal猜想

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.004

James Davies

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph H, there exists

ϵ > 0

such that every n-vertex graph with no pivot-minor isomorphic to H contains two sets

A, B

of vertices such that

| A |, | B | ⩾ ϵ n

and A is complete or anticomplete to B.

证明了Kim和Oum的一个猜想，即图的每一个适当的支点-次闭类都具有强Erdős-Hajnal性质。更准确地说，对于每个图H，存在ϵ>；0，使得每个n顶点图与H没有轴心次同构包含两个顶点集A，B，使得|A|，|B|小于ϵn和A对于B是完全的或反完全的。

引用次数: 0

Optimal bounds for zero-sum cycles. I. Odd order 零和循环的最优边界。I. 奇数阶

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.003

Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner

For a finite (not necessarily abelian) group

(Γ, \cdot)

, let

n (Γ)

denote the smallest positive integer n such that for each labelling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich [2] initiated the study of the parameter

n (\cdot)

on cyclic groups and proved

n (Z_{q}) = O (q \log q)

. This was later improved to a linear bound of

n (Γ) \leq 8 | Γ |

for every finite abelian group by Mészáros and the last author [8], and then further to

n (Γ) \leq 2 | Γ | - 1

for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma [3] as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta [1].

In this series of two papers we conclude this line of research by proving that

n (Γ) \leq | Γ | + 1

for every finite group

(Γ, \cdot)

, which is the best possible such bound in terms of the group order and precisely determines the value of

n (Γ)

for all cyclic groups as

n (Z_{q}) = q + 1

In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.

对于有限（不一定是阿贝尔）群（Γ，⋅），设n（Γ）表示最小的正整数n，使得对于使用Γ中的元素标记n阶的完全有向图的每个弧，存在一个有向循环，使得沿循环的弧标记乘以单位。Alon和Krivelevich[2]率先研究了n（⋅）在循环群上的参数，并证明了n(Zq)=O（qlog）。后来由Mészáros和最后一位作者[8]对每个有限阿贝群改进为n（Γ）≤8|Γ|的线性界，然后由Berendsohn， Boyadzhiyska和Kozma[3]以及Akrami， Alon, Chaudhury, Garg， Mehlhorn和Mehta[1]进一步改进为n（Γ）≤2|Γ|−1对每个非平凡有限群独立的线性界。在本系列的两篇论文中，我们通过证明n（Γ）≤|Γ|+1对于每一个有限群（Γ，⋅），这是群序的最佳可能界，并精确地决定了n（Γ）对于所有循环群的值为n(Zq)=q+1，从而总结了这条研究路线。本文对所有奇阶群证明了上述结果。偶序群的证明需要克服大量额外的障碍，这将在本系列的第二部分中介绍。

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引用次数: 0

Embedding connected factorizations II 嵌入连通分解II

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-04 DOI: 10.1016/j.jctb.2025.03.003

Amin Bahmanian , Anna Johnsen-Yu

Let

λ K_{n}^{h}

be the complete h-uniform n-vertex hypergraph in which each edge is repeated λ times. For

r : = (r_{1}, \dots, r_{k})

, a (partial) r-factorization of

λ K_{n}^{h}

is a partition of the edges of

λ K_{n}^{h}

into factors

F_{1}, \dots, F_{k}

such that each factor is spanning and the degree of all vertices in each

F_{i}

is (at most)

r_{i}

. Suppose that

n \geq (h - 1) (2 m - 1)

. We establish necessary and sufficient conditions that ensure a partial r-factorization of

λ K_{m}^{h}

can be embedded in a connected r-factorization of

λ K_{n}^{h}

. Moreover, we prove a general result which leads to a complete characterization of partial

(s_{1}, \dots, s_{q})

-factorizations of any sub-hypergraph of

λ K_{m}^{h}

in connected r-factorizations of

λ K_{n}^{h}

so long as q meets a natural upper bound. These results can be seen as unified generalizations of many classical combinatorial results, and can also be restated as results on embedding partial symmetric latin hypercubes.

设λ knh为每条边重复λ次的完全h-均匀n顶点超图。对于r:=(r1，…，rk)， λKnh的（部分）r-分解是将λKnh的边划分为因子F1，…，Fk，使得每个因子都是生成的，并且每个Fi中所有顶点的度数（最多）为ri。设n≥(h−1)（2m−1）。建立了λKmh的部分r因子分解可以嵌入到λ kh的连通r因子分解中的充分必要条件。此外，我们证明了一个一般结果，该结果使得λKmh的任何子超图在λKnh的连通r-分解中，只要q满足自然上界，就可以得到部分（s1，…，sq）分解的完全刻画。这些结果可以看作是许多经典组合结果的统一推广，也可以重述为嵌入偏对称拉丁超立方体的结果。

引用次数: 0

A splitter theorem on 3-connected binary matroids and inner fans 3连通二元拟阵和内扇的一个分岔定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-01 DOI: 10.1016/j.jctb.2025.03.004

João Paulo Costalonga

We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids

N < M

, there is a disjoint family

{X_{1}, \dots, X_{n}} \subseteq 2^{E (M)}

such that

r (X_{1}) + \dots + r (X_{n}) = r (X_{1} \cup \dots \cup X_{n}) \geq r (M) - r (N)

, each

si (M / X_{i})

is 3-connected with an N-minor, and either

| X_{i} | = 1

or X is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.

我们建立了关于元素的3连通二元拟阵的分裂型定理，这些元素的收缩保留了固定的3连通次元和垂直的3连通。我们建立了对于3连通的简单二元拟阵N<；M，存在一个不相交的族{X1，…，Xn}，使得r(X1)+⋯+r(Xn)=r(X1∪⋯∪Xn)≥r(M) - r(N)，每个si（M/Xi）与一个N次元3连通，且|Xi|=1或X是一个特殊类型的扇。我们还在特定的假设下建立了一个更强的版本。这些结果有几个结论，包括对3连通图中可缩边的一些结果在二元拟阵上的推广，以及图和二元拟阵的其他一些结构结果。

引用次数: 0

Improved bounds for zero-sum cycles in Zpd 改进了Zpd中零和循环的边界

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-03-21 DOI: 10.1016/j.jctb.2025.03.001

Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner

<div><div>For a finite abelian group <math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>+</mo><mo>)</mo></math>, let <math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math> denote the smallest positive integer n such that for each labeling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter <math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math> on cyclic groups and proved that <math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></math>. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form <math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math> was recently announced by Campbell, Gollin, Hendrey and the last author. While this bound is tight when the group Γ is cyclic, in cases when Γ is far from being cyclic, significant improvements on the bound can be made. In this direction, studying the prototypical case when <math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> is a power of a cyclic group of prime order, Letzter and Morrison [Journal of Combinatorial Theory Series B, 2024] showed that <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> and that <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo>⁡</mo><mi>d</mi><mo>)</mo></math>. They then posed the problem of proving an (asymptotically optimal) upper bound of <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><mo>)</mo></math> for all primes p and <math><mi>d</mi><mo>∈</mo><mi>N</mi></math>. In this paper, we solve this problem for <math><mi>p</mi><mo>=</mo><mn>2</mn></math> and improve their bound for all primes <math><mi>p</mi><mo>≥</mo><mn>3</mn></math> by proving <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mn>5</mn><mi>d</mi></math> and <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow>

对于有限阿贝尔群（Γ，+），设n（Γ）表示最小的正整数n，使得对于使用Γ中的元素标记n阶的完全有向图的每个弧，存在一个有向循环，使得沿循环的弧标记的总和等于0。Alon和Krivelevich开创了循环群上n（⋅）参数的研究，证明了n(Zq)=O（qlog）。此后，对这一界进行了若干改进和推广，最近由Campbell， Gollin， hendry和最后一位作者提出了n（Γ）≤|Γ|+1的群阶最优界。当组Γ是循环的时候，这个边界是紧的，而当Γ远不是循环的时候，可以对边界进行重大改进。在这个方向上，Letzter和Morrison [Journal of Combinatorial Theory Series B， 2024]研究了Γ=Zpd是一个素阶循环群幂的典型情况，证明了n(Zpd)≤O(pd(log d)2), n(Z2d)≤O（log d）。然后，他们提出了证明对于所有素数p和d∈n， n(Zpd)≤O（pd）的（渐近最优）上界的问题。本文通过证明n(Z2d)≤5d和n(Zpd)≤O(pdlog (d))，解决了p=2时的这一问题，并改进了p≥3时所有素数的界。当第一个边界决定n（Z2d）到5的乘法误差时，第二个边界紧到一个log (d)因子。此外，我们的结果表明，对于任意p和d， n(Zpd)=Θ（pd）的紧界是由著名的Jaeger， Linial， Payan和Tarsi在Zpd上的加性基猜想的一个（强形式）推导出来的。在证明这些结果的过程中，我们建立了Haxell在矩阵环境下的超图匹配结果的推广。具体地说，我们得到了超图中超边由一个拟阵的元素标记的匹配存在的充分条件，并得到了该超图中匹配中的边可以引出该拟阵的一组基。我们认为，这些声明具有独立的利益。

引用次数: 0

Sparse induced subgraphs of large treewidth 大树宽的稀疏诱导子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-03-20 DOI: 10.1016/j.jctb.2025.03.002

Édouard Bonnet

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) [22] or by a classic result of Chekuri and Chuzhoy (2015) [5], we show that for any natural numbers t and w, and real

ε > 0

, there is an integer

W : = W (t, w, ε)

such that every graph with treewidth at least W and no

K_{t, t}

subgraph admits a 2-connected n-vertex induced subgraph with treewidth at least w and at most

(1 + ε) n

edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weißauer (2019) [25] that graphs of large treewidth have a large biclique subgraph or a long induced cycle.

受著名的Robertson和Seymour（1986）的网格小定理（Grid Minor theorem）提供的树宽稀疏子图（即保持树宽较大的稀疏子图）的诱导对偶，或Chekuri和chuchoy（2015）的经典结果（[5]）的激励，我们表明，对于任何自然数t和w，以及实数ε>；0，存在一个整数W:=W(t， W,ε)，使得每个树宽至少W且没有Kt，t子图的图都存在一个树宽至少W且最多（1+ε）n条边的2连通n顶点诱导子图。诱导子图可以是细分壁面，也可以是细分壁面的线形图，也可以是细分壁面的生成超图。这特别扩展了Weißauer(2019)[25]的结果，即大树宽的图有一个大的双曲线子图或一个长诱导周期。

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引用次数: 0

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