Journal of Combinatorial Theory Series B最新文献

英文中文

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 : 2025-03-01 Epub 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

Orientably-regular embeddings of complete multigraphs 完全多图的可定向正则嵌入

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-03-01 Epub Date: 2024-12-11 DOI: 10.1016/j.jctb.2024.11.004

Štefan Gyürki, Soňa Pavlíková, Jozef Širáň

An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge pairs of the graph. A classification of orientably-regular embeddings of complete graphs was obtained by L.D. James and G.A. Jones (1985) [10], pointing out interesting connections to finite fields and Frobenius groups. By a combination of graph-theoretic methods and tools from combinatorial group theory we extend results of James and Jones to classification of orientably-regular embeddings of complete multigraphs with arbitrary edge-multiplicity.

一个图在可定向曲面上的嵌入是可定向正则的（或旋转的，在一个等价的术语中），如果该嵌入的保持方向的自同构群在图的相关顶点边对上是可传递的（因此是正则的）。L.D. James和G.A. Jones(1985)[10]给出了完全图的可定向正则嵌入的分类，指出了与有限域和Frobenius群的有趣联系。结合图论方法和组合群论工具，将James和Jones的结果推广到具有任意边多重性的完全多图的可定向正则嵌入的分类。

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 : 2025-03-01 Epub 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

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 : 2025-03-01 Epub 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

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 : 2025-01-01 Epub 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

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 : 2025-01-01 Epub 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

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 : 2025-01-01 Epub 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

Lift theorems for representations of matroids over pastures 牧场矩阵表征的提升定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-01 Epub Date: 2024-09-05 DOI: 10.1016/j.jctb.2024.08.004

Matthew Baker , Oliver Lorscheid

Pastures are a class of field-like algebraic objects which include both partial fields and hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh–van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand–Rybnikov–Stone inspired by Tutte's homotopy theorem.

牧场是一类类似于场的代数对象，包括部分场和超场，具有很好的分类性质。我们证明了牧场上矩阵表示的几个提升定理，包括 Pendavingh 和 van Zwam 的部分域提升定理的一般化。通过将先前的理论嵌入到一个更一般的框架中，我们甚至能够在部分域的提升情况下建立新的结果，例如 Pendavingh-van Zwam 的猜想，即他们的提升构造是幂等的。我们给出了许多关于矩阵表示的应用，例如，我们证明了在投影等价性范围内，由六边形表示和方向组成的每一对都能唯一地提升到近规则表示。证明与彭达文和范兹瓦姆使用的论证不同，而是依赖于格尔方-里布尼科夫-斯通受图特同调定理启发而得出的结果。

引用次数: 0

Crux, space constraints and subdivisions 核心、空间限制和分区

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-01 Epub Date: 2024-09-19 DOI: 10.1016/j.jctb.2024.08.005

Seonghyuk Im , Jaehoon Kim , Younjin Kim , Hong Liu

For a given graph H, its subdivisions carry the same topological structure. The existence of H-subdivisions within a graph G has deep connections with topological, structural and extremal properties of G. One prominent example of such a connection, due to Bollobás and Thomason and independently Komlós and Szemerédi, asserts that the average degree of G being d ensures a $K_{Ω (\sqrt{d})}$ -subdivision in G. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph for many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux.

Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph G, which is determined by both its average degree and its crux size. As corollaries, we obtain

•
a characterization of extremal graphs for which the square-root bound above is tight: they are essentially disjoint unions of graphs having crux size linear in d;
•
a unifying approach to find a clique subdivision of almost optimal size in graphs which do not contain a fixed bipartite graph as a subgraph;
•
and that the clique subdivision size in random graphs $G (n, p)$ witnesses a dichotomy: when $p = ω (n^{- 1 / 2})$ , the barrier is the space, while when $p = o (n^{- 1 / 2})$ , the bottleneck is the density.

对于给定的图 H，其细分图具有相同的拓扑结构。图 G 中 H 细分的存在与 G 的拓扑、结构和极值特性有着深刻的联系。这种联系的一个突出例子是由 Bollobás 和 Thomason 以及 Komlós 和 Szemerédi 提出的，他们断言 G 的平均度数为 d 可以确保 G 中存在 KΩ(d)-细分。我们研究了crux（一种捕捉图的基本顺序的概念）与大簇细分的存在之间的联系。这揭示了所研究的各类图中所有这些改进的统一原因。我们的主要结果给出了通用图 G 中最大簇细分大小的渐近最优约束，该约束由其平均度和簇大小共同决定。作为推论，我们得到了极值图的特征，对于这些极值图，上述平方根约束是紧密的：它们本质上是轴心大小与 d 成线性关系的图的不相交联合体；- 在不包含固定二方图作为子图的图中找到几乎最优大小的簇细分的统一方法；- 随机图 G(n,p) 中的簇细分大小呈现二分法：当 p=ω(n-1/2) 时，障碍是空间，而当 p=o(n-1/2) 时，瓶颈是密度。

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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 : 2025-01-01 Epub 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":"2025-01-01","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}

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