Journal of Combinatorial Theory Series B最新文献

英文中文

On a conjecture of Tokushige for cross-t-intersecting families 关于交叉族的Tokushige猜想

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-12-06 DOI: 10.1016/j.jctb.2024.11.005

Huajun Zhang , Biao Wu

Two families of sets

A

and

B

are called cross-t-intersecting if

| A \cap B | \geq t

for all

A \in A

B \in B

. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting families. This incorporates the classical Erdős–Ko–Rado (EKR) problem. In the present paper, we prove that if

A \subseteq (\begin{matrix} [n] \\ k \end{matrix})

and

B \subseteq (\begin{matrix} [n] \\ k \end{matrix})

are cross-t-intersecting with

k \geq t \geq 3

and

n \geq (t + 1) (k - t + 1)

, then

| A | | B | \leq {(\begin{matrix} n - t \\ k - t \end{matrix})}^{2}

. Moreover, equality holds if and only if

A = B

is a maximum t-intersecting subfamily of

(\begin{matrix} [n] \\ k \end{matrix})

. This confirms a conjecture of Tokushige for

t \geq 3

如果对于所有A∈A， B∈B, |A∩B|≥t，则集合A和B的两个族称为交叉t相交。极值集理论中的一个活跃问题是确定交叉族大小的最大积。这包含了经典的Erdős-Ko-Rado （EKR）问题。在本文中，我们证明了如果A、B两种面包车（[n]k）在k≥t≥3、n≥(t+1)（k−t+1）时呈t相交，则|A||B|≤(n−tk−t)2。而且，当且仅当A=B是（[n]k）的最大t相交子族时，等式成立。这证实了t≥3时Tokushige的一个猜想。

引用次数: 0

Linear three-uniform hypergraphs with no Berge path of given length 没有给定长度的Berge路径的线性三均匀超图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-12-05 DOI: 10.1016/j.jctb.2024.11.003

Ervin Győri , Nika Salia

Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an n-vertex 3-uniform linear hypergraph, without a Berge path of length k as a subgraph is at most

\frac{(k - 1)}{6} n

for

k \geq 4

. The bound is sharp for infinitely many k and n.

研究了Erdős-Gallai定理在一般超图中的推广。本文证明了线性超图Erdős-Gallai定理的推广。特别地，我们证明了在一个n顶点3-一致线性超图中，当k≥4时，不存在长度为k的Berge路径作为子图时，超边的数目最多为（k−1）6n。对于无穷多个k和n，边界很明显。

引用次数: 0

Note on disjoint faces in simple topological graphs 注意简单拓扑图中的不相交面

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-11-28 DOI: 10.1016/j.jctb.2024.11.002

Ji Zeng

We prove that every n-vertex complete simple topological graph generates at least

Ω (n)

pairwise disjoint 4-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every n-vertex complete simple topological graph drawn in the unit square generates a 4-face with area at most

O (1 / n)

. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for k-faces with arbitrary

k \geq 3

证明了每个n顶点完备简单拓扑图至少生成Ω(n)对不相交的4面。这比Hubard和Suk最近的研究结果有所改进。作为直接推论，在单位方格中绘制的每一个n顶点完全简单拓扑图都会生成一个面积不超过O（1/n）的4面。这可以看作是四边形的Heilbronn问题的拓扑变体。我们构造了一些例子来证明我们的结果是渐近紧的。我们还讨论了任意k≥3的k面的类似问题。

引用次数: 0

A characterization of the Grassmann graphs 格拉斯曼图的表征

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-11-14 DOI: 10.1016/j.jctb.2024.11.001

Alexander L. Gavrilyuk , Jack H. Koolen

The Grassmann graph

J_{q} (n, D)

is a graph on the D-dimensional subspaces of

F_{q}^{n}

with two subspaces being adjacent if their intersection has dimension

D - 1

. Characterizing these graphs by their intersection numbers is an important step towards a solution of the classification problem for

(P and Q)

-polynomial association schemes, posed by Bannai and Ito in their monograph “Algebraic Combinatorics I” (1984).

Metsch (1995) [37] showed that the Grassmann graph

J_{q} (n, D)

with

D \geq 3

is characterized by its intersection numbers except for the following two principal open cases:

n = 2 D

n = 2 D + 1

. Van Dam and Koolen (2005) [57] constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs

J_{q} (2 D + 1, D)

(for any prime power q and

D \geq 2

), but not isomorphic to the latter ones. This shows that characterizing the graphs in the remaining cases would require a conceptually new approach.

We prove that the Grassmann graph

J_{q} (2 D, D)

is characterized by its intersection numbers provided that D is large enough.

格拉斯曼图 Jq(n,D) 是 Fqn 的 D 维子空间上的图，如果两个子空间的相交维数为 D-1，则这两个子空间相邻。Metsch (1995) [37]指出，D≥3的格拉斯曼图 Jq(n,D)由其交点数表征，但以下两种主要开放情况除外：n=2D 或 n=2D+1。Van Dam 和 Koolen（2005）[57] 构建的扭曲格拉斯曼图与格拉斯曼图 Jq(2D+1,D)（对于任意质幂 q 和 D≥2）具有相同的交点数，但与后者不同构。我们证明，只要 D 足够大，格拉斯曼图 Jq(2D,D) 的交点数就是它的特征。

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引用次数: 0

Counting cycles in planar triangulations 平面三角形中的循环计数

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-11-05 DOI: 10.1016/j.jctb.2024.10.002

On-Hei Solomon Lo , Carol T. Zamfirescu

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is

Ω (n)

for any cycle length at most

3 + \max {rad (G^{⁎}), ⌈ {(\frac{n - 3}{2})}^{\log_{3} 2} ⌉}

, where

rad (G^{⁎})

denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian n-vertex triangulations containing

O (n)

many k-cycles for any

k \in {⌈ n - \sqrt[5]{n} ⌉, \dots, n}

. Furthermore, we prove that planar 4-connected n-vertex triangulations contain

Ω (n)

many k-cycles for every

k \in {3, \dots, n}

, and that, under certain additional conditions, they contain

Ω (n^{2})

k-cycles for many values of k, including n.

我们研究了平面 n 顶点三角剖分 G 中指定长度循环的最小数目。我们证明，对于循环长度最多为 3+max{rad(G⁎),⌈(n-32)log32⌉} 的任意循环，该数目为 Ω(n)，其中 rad(G⁎) 表示三角剖分的对偶半径，它至少是对数，但可以是三角剖分顺序的线性。我们还证明，对于任意 k∈{⌈n-n5⌉,...,n}，存在包含 O(n) 个 k 循环的平面哈密顿 n 顶点三角剖分。此外，我们还证明了平面四连 n 顶点三角形在任何 k∈{3,...,n} 条件下都包含 Ω(n) 个 k 循环，而且在某些附加条件下，它们在包括 n 在内的许多 k 值上都包含 Ω(n2) 个 k 循环。

引用次数: 0

Trees with many leaves in tournaments 锦标赛中树叶繁茂的树木

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-10-15 DOI: 10.1016/j.jctb.2024.10.001

Alistair Benford , Richard Montgomery

Sumner's universal tournament conjecture states that every

(2 n - 2)

-vertex tournament should contain a copy of every n-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an

(n + o (n))

-vertex tournament may be sufficient.

In this paper, we give new results on these problems. Specifically, we show

i)
for every $α > 0$ , there exists $n_{0} \in N$ such that, whenever $n ⩾ n_{0}$ , every $((1 + α) n + k)$ -vertex tournament contains a copy of every n-vertex oriented tree with k leaves, and
ii)
for every $α > 0$ , there exists $c > 0$ and $n_{0} \in N$ such that, whenever $n ⩾ n_{0}$ , every $(1 + α) n$ -vertex tournament contains a copy of every n-vertex oriented tree with maximum degree $Δ (T) ⩽ c n$ .

Our first result gives an asymptotic form of a conjecture by Havet and Thomassé, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree.

萨姆纳的通用锦标赛猜想指出，每一个 (2n-2)- 顶点锦标赛都应该包含每一棵 n 个顶点的定向树的副本。如果我们知道一棵定向树的叶子数或它的最大度数，我们能否保证锦标赛中会有顶点数较少的定向树的副本呢？由于海格奎斯特（Häggkvist）和托马森（Thomason）（针对树叶数）以及库恩（Kühn）、迈克罗夫特（Mycroft）和奥斯特胡斯（Osthus）（针对最大度）所做的工作，我们知道在某些情况下可以改进萨姆纳猜想，实际上有时一个（n+o(n)）顶点锦标赛可能就足够了。具体地说，我们证明i)对于每一个 α>0, 都存在 n0∈N 这样的情况：当 n⩾n0 时，每一个 ((1+α)n+k)-vertex tournament 都包含每一个有 k 个叶子的 n-vertex 定向树的副本；ii)对于每一个 α>；0，存在 c>0 和 n0∈N 这样的情况：当 n⩾n0 时，每一个 (1+α)n 顶点锦标赛都包含每一棵具有最大度 Δ(T)⩽cn 的 n 顶点定向树的副本。我们的第一个结果给出了 Havet 和 Thomassé 猜想的渐近形式，第二个结果改进了 Mycroft 和 Naia 的一个结果，该结果适用于最大度为多对数的树。

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引用次数: 0

Erdős-Szekeres type theorems for ordered uniform matchings 有序均匀匹配的 Erdős-Szekeres 型定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-10-14 DOI: 10.1016/j.jctb.2024.09.004

Andrzej Dudek , Jarosław Grytczuk , Andrzej Ruciński

For

r, n ⩾ 2

, an ordered r-uniform matching of size n is an r-uniform hypergraph on a linearly ordered vertex set V, with

| V | = r n

, consisting of n pairwise disjoint edges. There are

\frac{1}{2} (\begin{matrix} 2 r \\ r \end{matrix})

different ways two edges may intertwine, called here patterns. Among them we identify

3^{r - 1}

collectable patterns P, which have the potential of appearing in arbitrarily large quantities called P-cliques.

We prove an Erdős-Szekeres type result guaranteeing in every ordered r-uniform matching the presence of a P-clique of a prescribed size, for some collectable pattern P. In particular, in the diagonal case, one of the P-cliques must be of size

Ω (n^{3^{1 - r}})

. In addition, for each collectable pattern P we show that the largest size of a P-clique in a random ordered r-uniform matching of size n is, with high probability,

Θ (n^{1 / r})

对于 r,n⩾2，大小为 n 的有序 r-Uniform 匹配是线性有序顶点集 V 上的 r-Uniform 超图，|V|=rn，由 n 条成对不相交的边组成。两条边有 12(2rr) 种不同的交织方式，在此称为模式。我们证明了 Erdős-Szekeres 类型的结果，即对于某个可收集模式 P，保证在每个有序 r-uniform 匹配中存在规定大小的 P-clique。此外，对于每个可收集模式 P，我们证明在大小为 n 的随机有序 r-uniform 匹配中，P-clique 的最大大小很有可能是 Θ(n1/r)。

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引用次数: 0

EPPA numbers of graphs EPPA 图表数量

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-10-03 DOI: 10.1016/j.jctb.2024.09.003

David Bradley-Williams , Peter J. Cameron , Jan Hubička , Matěj Konečný

If G is a graph, A and B its induced subgraphs, and

f : A \to B

an isomorphism, we say that f is a partial automorphism of G. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph G there is a finite graph H, an EPPA-witness for G, such that G is an induced subgraph of H and every partial automorphism of G extends to an automorphism of H.

The EPPA number of a graph G, denoted by

eppa (G)

, is the smallest number of vertices of an EPPA-witness for G, and we put

eppa (n) = \max {eppa (G) : | G | = n}

. In this note we review the state of the area, prove several lower bounds (in particular, we show that

eppa (n) \geq \frac{2^{n}}{\sqrt{n}}

, thereby identifying the correct base of the exponential) and pose many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and

K_{k}

-free graphs.

如果 G 是一个图，A 和 B 是它的诱导子图，f:A→B 是同构，我们就说 f 是 G 的部分自动形。1992 年，赫鲁晓夫斯基证明了图具有部分自动态的扩展性质（EPPA，又称赫鲁晓夫斯基性质），即对于每个有限图 G，都有一个有限图 H（G 的 EPPA 见证），使得 G 是 H 的诱导子图，并且 G 的每个部分自动态都扩展为 H 的一个自动态。图 G 的 EPPA 数（用 eppa(G) 表示）是 G 的 EPPA 证图的最小顶点数，我们将 eppa(n)=max{eppa(G):|G|=n} 放为 eppa(n)=max{eppa(G):|G|=n}。在本说明中，我们回顾了这一领域的现状，证明了几个下界（特别是，我们证明了 eppa(n)≥2nn ，从而确定了指数的正确基数），并提出了许多开放性问题。我们还简要讨论了超图、有向图和无 Kk 图的 EPPA 数。

{"title":"EPPA numbers of graphs","authors":"David Bradley-Williams , Peter J. Cameron , Jan Hubička , Matěj Konečný","doi":"10.1016/j.jctb.2024.09.003","DOIUrl":"10.1016/j.jctb.2024.09.003","url":null,"abstract":"<div><div>If G is a graph, A and B its induced subgraphs, and <math><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></math> an isomorphism, we say that f is a partial automorphism of G. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph G there is a finite graph H, an EPPA-witness for G, such that G is an induced subgraph of H and every partial automorphism of G extends to an automorphism of H.</div><div>The EPPA number of a graph G, denoted by <math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math>, is the smallest number of vertices of an EPPA-witness for G, and we put <math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>=</mo><mi>n</mi><mo>}</mo></math>. In this note we review the state of the area, prove several lower bounds (in particular, we show that <math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></math>, thereby identifying the correct base of the exponential) and pose many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and <math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math>-free graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"170 ","pages":"Pages 203-224"},"PeriodicalIF":1.2,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Volume rigidity and algebraic shifting 体积刚性和代数移动

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-09-27 DOI: 10.1016/j.jctb.2024.09.002

Denys Bulavka , Eran Nevo , Yuval Peled

We study the generic volume-rigidity of

(d - 1)

-dimensional simplicial complexes in

R^{d - 1}

, and show that the volume-rigidity of a complex can be identified in terms of its exterior shifting. In addition, we establish the volume-rigidity of triangulations of several 2-dimensional surfaces and prove that, in all dimensions >1, volume-rigidity is not characterized by a corresponding hypergraph sparsity property.

我们研究了 Rd-1 中 (d-1)-dimensional 简单复数的一般体积刚度，并证明复数的体积刚度可以通过其外部移动来确定。此外，我们还建立了几个二维曲面三角形的体积刚度，并证明在所有维数>1中，体积刚度并不以相应的超图稀疏性为特征。

引用次数: 0

Sufficient conditions for perfect mixed tilings 完美混合倾斜的充分条件

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2024-09-24 DOI: 10.1016/j.jctb.2024.08.007

Eoin Hurley , Felix Joos , Richard Lang

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs H with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect F-tilings (for an arbitrary fixed graph F) by replacing the F-tiling with the aforementioned graphs H. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.

我们开发了一种方法来研究完美混合倾斜的充分条件。我们的框架允许嵌入具有亚线性阶成分的有界阶图 H。作为推论，我们恢复并扩展了库恩（Kühn）和奥斯特胡斯（Osthus）的工作，即用上述图 H 替换 F-tiling，从而获得完美 F-tiling（对于任意固定图 F）的最小阶数充分条件。最后，我们在强意义上渐近地解决了孔洛斯的一个猜想。

引用次数: 0

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