European Journal of Combinatorics最新文献

英文中文

On the faces of unigraphic 3-polytopes 关于单图式 3 多面体的面

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-16 DOI: 10.1016/j.ejc.2024.104081

Riccardo W. Maffucci

A 3-polytope is a 3-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other 3-polytope, up to graph isomorphism. The classification of unigraphic 3-polytopes appears to be a difficult problem.

In this paper we prove that, apart from pyramids, all unigraphic 3-polytopes have no

n

-gonal faces for

n \geq 10

. Our method involves defining several planar graph transformations on a given 3-polytope containing an

n

-gonal face with

n \geq 10

. The delicate part is to prove that, for every such 3-polytope, at least one of these transformations both preserves 3-connectivity, and is not an isomorphism.

3 多面体是一个 3 连接的平面图形。如果它的顶点度序列不与任何其他 3 多面体共享，直到图同构，那么它就被称为单图形。在本文中，我们证明了除金字塔外，所有单图形三多面体在 n≥10 时都没有 n 个球面。我们的方法是在一个给定的 3 多面体上定义几个平面图形变换，其中包含一个 n≥10 的 n 角面。最复杂的部分是证明，对于每一个这样的 3 多面体，这些变换中至少有一个既保留了 3 连通性，又不是同构。

引用次数: 0

Bounded unique representation bases for the integers 整数的有界唯一表示基

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-16 DOI: 10.1016/j.ejc.2024.104080

Yong-Gao Chen, Jin-Hui Fang

<div><div>For a nonempty set <math><mi>A</mi></math> of integers and an integer <math><mi>n</mi></math>, let <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> be the number of representations of <math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> with <math><mrow><mi>a</mi><mo>≤</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> and <math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math>, and let <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> be the number of representations of <math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> with <math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math>. Erdős and Turán (1941) posed the profound conjecture: if <math><mi>A</mi></math> is a set of positive integers such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math> for all sufficiently large <math><mi>n</mi></math>, then <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set <math><mi>A</mi></math> with logarithmic growth such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all integers <math><mi>n</mi></math>. In this paper, we prove that, for any positive function <math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> with <math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn></mrow></math> as <math><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math>, there is a bounded set <math><mi>A</mi></math> of integers such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all integers <math><mi>n</mi></math> and <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all positi

对于非空整数集合 A 和整数 n，设 rA(n) 是 n=a+a′ 的表示数，其中 a≤a′ 和 a,a′∈A ；设 dA(n) 是 n=a-a′ 的表示数，其中 a,a′∈A 。厄尔多斯和图兰（1941）提出了一个深刻的猜想：如果 A 是一个正整数集合，对于所有足够大的 n，rA(n)≥1，那么 rA(n) 是无界的。Nešetřil 和 Serra (2004) 引入了有界集的概念，并证实了 Erdős-Turán 对所有有界基的猜想。Nathanson (2003) 考虑了具有对数增长的集合 A 的存在性，即对于所有整数 n，rA(n)=1。在本文中，我们证明了对于任何正函数 l(x)，当 x→∞ 时，l(x)→0，存在一个有界的整数集合 A，使得对于所有整数 n，rA(n)=1；对于所有正整数 n，dA(n)=1；对于所有足够大的 x，A(-x,x)≥l(x)logx，其中 A(-x,x) 是具有 -x≤a≤x 的元素 a∈A 的个数。

引用次数: 0

Induced subgraph density. II. Sparse and dense sets in cographs 诱导子图密度II.cographs 中的稀疏集和密集集

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-09 DOI: 10.1016/j.ejc.2024.104075

Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour

<div><div>A well-known theorem of Rödl says that for every graph <math><mi>H</mi></math>, and every <math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math>, there exists <math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math> such that if <math><mi>G</mi></math> does not contain an induced copy of <math><mi>H</mi></math>, then there exists <math><mrow><mi>X</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> with <math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>δ</mi><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math> such that one of <math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math> has edge-density at most <math><mi>ɛ</mi></math>. But how does <math><mi>δ</mi></math> depend on <math><mi>ϵ</mi></math>? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all <math><mi>H</mi></math> there exists <math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math> such that for all <math><mi>ɛ</mi></math> with <math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>, Rödl’s theorem holds with <math><mrow><mi>δ</mi><mo>=</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow></math>. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs <math><mi>H</mi></math>. Our first result shows that it is true when <math><mrow><mi>H</mi><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math>. Indeed, in that case we can take <math><mrow><mi>δ</mi><mo>=</mo><mi>ɛ</mi></mrow></math>, and insist that one of <math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math> has maximum degree at most <math><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math>).</div><div>Second, we will show that every graph <math><mi>H</mi></math> that can be obtained by substitution from copies of <math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math> satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say <math><mi>H</mi></math> is viral if there exists <math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math> such that for all <math><mi>ɛ</mi></math> with <math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/

罗德尔（Rödl）的一个著名定理指出，对于每个图 H 和每个ɛ>0，都存在 δ>0，这样，如果 G 不包含 H 的诱导副本，则存在 X⊆V(G)，其中 |X|≥δ|G| 这样，G[X],G¯[X]中的一个边密度最多为ɛ。但是，δ 如何取决于ϵ？福克斯和苏达科夫猜想，这种依赖性最多是多项式的：对于所有 H，存在 c>0 这样的条件：对于所有 ɛ 且 0<ɛ≤1/2 时，罗德尔定理成立，δ=ɛc。我们的第一个结果表明，当 H=P4 时，这个猜想成立。事实上，在这种情况下，我们可以取 δ=ɛ，并坚持认为 G[X],G¯[X] 中的一个图的最大度最多为ɛ2|G|)。其次，我们将证明每个可以从 P4 的副本中通过替换得到的图 H 都满足福克斯-苏达科夫猜想。为了证明这一点，我们需要使用一个更强的性质。如果存在 c>0 这样的情况，即对于所有 0<ɛ≤1/2 的ɛ，如果 G 最多包含 H 的ɛc|G|||H|副本作为诱导子图，那么存在 X⊆V(G)，其中 |X|≥ɛc|G| 这样的情况，即 G[X],G¯[X] 中的一个边密度最多为ɛ。我们将利用 Alon 和 Fox 的 "多项式 P4-removal Lemma "来证明 P4 是病毒式的。最后，我们将给出罗德尔定理的另一个强化：我们将证明，如果 G 不包含 P4 的诱导副本，那么它的顶点最多可以划分为 480ɛ-4 个子集 X，使得 G[X],G¯[X] 中的一个子集的最大度最多为ɛ|X|。

引用次数: 0

The diameter of randomly twisted hypercubes 随机扭曲超立方体的直径

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-09 DOI: 10.1016/j.ejc.2024.104078

Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano

The

n

-dimensional random twisted hypercube

G_{n}

is constructed recursively by taking two instances of

G_{n - 1}

, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is

O (n log log log n / log log n)

with high probability and at least

(n - 1) / {log}_{2} n

. We improve their upper bound by showing that

diam (G_{n}) = (1 + o (1)) \frac{n}{{log}_{2} n}

with high probability.

n 维随机扭曲超立方体 Gn 是由两个具有任意联合分布的 Gn-1 实例，并在它们的顶点集之间添加一个随机完美匹配来递归构造的。Benjamini、Dikstein、Gross 和 Zhukovskii 证明了其直径为 O(nloglogn/loglogn)，且概率很高，至少为 (n-1)/log2n。我们通过证明 diam(Gn)=(1+o(1))nlog2n 的高概率，改进了他们的上限。

引用次数: 0

Intersection density of transitive groups with small cyclic point stabilizers 具有小循环点稳定子的传递群的交集密度

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-07 DOI: 10.1016/j.ejc.2024.104079

Ademir Hujdurović , István Kovács , Klavdija Kutnar , Dragan Marušič

For a permutation group

G

acting on a set

V

, a subset

F

G

is said to be an intersecting set if for every pair of elements

g, h \in F

there exists

v \in V

such that

g (v) = h (v)

. The intersection density

ρ (G)

of a transitive permutation group

G

is the maximum value of the quotient

| F | / | G_{v} |

where

G_{v}

is a stabilizer of a point

v \in V

and

F

runs over all intersecting sets in

G

. If

G_{v}

is a largest intersecting set in

G

then

G

is said to have the Erdős-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among others, constructions of an infinite family of transitive permutation groups having point stabilizer of order 3 with intersection density

4 / 3

and of infinite families of transitive permutation groups having point stabilizer of order 3 with arbitrarily large intersection density are given.

对于作用于集合 V 的置换群 G，如果每一对元素 g、h∈F 都存在 v∈V，使得 g(v)=h(v) ，则称 G 的子集 F 为交集。跨正交置换群 G 的交集密度 ρ(G) 是商 |F|/|Gv| 的最大值，其中 Gv 是点 v∈V 的稳定子，而 F 遍历 G 中的所有交集。如果 Gv 是 G 中的最大交集，则称 G 具有厄尔多斯-科-拉多（EKR）属性。本文致力于研究具有素阶点稳定器的传递置换群，特别强调不具有 EKR 属性的 2 阶和 3 阶。除其他外，本文还给出了具有交集密度为 4/3 的 3 阶点稳定器的传递置换群无穷族的构造，以及具有任意大交集密度的 3 阶点稳定器的传递置换群无穷族的构造。

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引用次数: 0

Turán numbers of ordered tight hyperpaths 有序紧密超路径的图兰数

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-03 DOI: 10.1016/j.ejc.2024.104070

John P. Bright, Kevin G. Milans, Jackson Porter

An ordered hypergraph is a hypergraph

G

whose vertex set

V (G)

is linearly ordered. We find the Turán numbers for the

r

-uniform

s

-vertex tight path

{\vec{P}}_{s}^{(r)}

(with vertices in the natural order) exactly when

r \leq s < 2 r

and

n

is even; our results imply

\vec{ex} (n, {\vec{P}}_{s}^{(r)}) = (1 - \frac{1}{2^{s - r}} + o (1)) (\frac{n}{r})

when

r \leq s < 2 r

. When

s \geq 2 r

, the asymptotics of

\vec{ex} (n, {\vec{P}}_{s}^{(r)})

remain open. For

r = 3

, we give a construction of an

r

-uniform

n

-vertex hypergraph not containing

{\vec{P}}_{s}^{(r)}

which we conjecture to be asymptotically extremal.

有序超图是顶点集 V(G) 是线性有序的超图 G。当 r≤s<2r 且 n 为偶数时，我们精确地找到了 r-uniform s-vertex 紧路径 P→s(r)（顶点按自然顺序排列）的图兰数；当 r≤s<2r 时，我们的结果意味着 ex→(n,P→s(r))=(1-12s-r+o(1))nr。当 s≥2r 时，ex→(n,P→s(r)) 的渐近线仍未确定。对于 r=3，我们给出了一个不包含 P→s(r) 的 r-uniform n 顶点超图的构造，我们猜想它是渐近极值的。

引用次数: 0

Boundary rigidity of 3D CAT(0) cube complexes 三维 CAT(0) 立方体复合物的边界刚度

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-01 DOI: 10.1016/j.ejc.2024.104077

John Haslegrave , Alex Scott , Youri Tamitegama , Jane Tan

The boundary rigidity problem is a classical question from Riemannian geometry: if

(M, g)

is a Riemannian manifold with smooth boundary, is the geometry of

M

determined up to isometry by the metric

d_{g}

induced on the boundary

\partial M

? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in

R^{3}

can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

边界刚度问题是黎曼几何中的一个经典问题：如果 (M,g) 是一个具有光滑边界的黎曼流形，那么 M 的几何形状是否由边界 ∂M 上的度量 dg 决定？在本文中，我们将考虑这一问题的离散版本：我们能否根据有限立方体复数的边界距离确定其组合类型？与连续的情况一样，重构在一般情况下是不可能的，但我们期望在适当的收缩性和非正曲率条件下得到肯定的答案。事实上，在二维中，当复数是一个内部顶点度数不小于 4 的有限圆盘四曲面时，哈斯勒格拉夫给出了肯定答案。我们证明了这一结果的三维概括：一个嵌入 R3 的有限 CAT(0) 立方复数的组合类型可以从其边界距离中重建。此外，我们还证明了哈斯勒格拉夫结果的直接强化：任何有限二维 CAT(0) 立方复数的组合类型都可以从其边界距离中重建。

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引用次数: 0

Decks of rooted binary trees 有根二叉树甲板

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-25 DOI: 10.1016/j.ejc.2024.104076

Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke

We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree

T

refers to the set (resp. multiset) of leaf-induced binary subtrees of

T

. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on

n

leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for

k

-universal trees, i.e., rooted binary trees that contain all

k

-leaf rooted binary trees as leaf-induced subtrees.

我们考虑与有根二叉树（又称有根系统树形）的甲板和多甲板相关的极值问题。在这里，树 T 的甲板（或多甲板）指的是 T 的叶诱导二叉子树的集合（或多集合）。一方面，我们考虑从树的（多）甲板重建树。我们给出了唯一编码 n 个树叶上有根二叉树所需的最小（多）甲板大小的下限和上限。另一方面，我们还考虑了与牌面明度相关的问题。特别是，我们描述了具有最小尺寸和最大尺寸牌面的树的特征。最后，我们介绍了一些 k 通用树的详尽计算，即包含所有 k 叶有根二叉树作为叶诱导子树的有根二叉树。

引用次数: 0

Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole 诱导子图和树分解 XIV.洞中的非相邻邻图

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-24 DOI: 10.1016/j.ejc.2024.104074

Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl

A clock is a graph consisting of an induced cycle

C

and a vertex not in

C

with at least two non-adjacent neighbours in

C

. We show that every clock-free graph of large treewidth contains a “basic obstruction” of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.

我们证明，每一个大树宽的无时钟图都包含一个大树宽的 "基本障碍 "诱导子图：一个完整的图、一堵墙的细分图或一堵墙的细分图的线图。

引用次数: 0

Zig-zag Eulerian polynomials 之字形欧拉多项式

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-24 DOI: 10.1016/j.ejc.2024.104073

T. Kyle Petersen , Yan Zhuang

For any finite partially ordered set

P

, the

P

-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of

P

, and is closely related to the order polynomial of

P

arising in the theory of

P

-partitions. Here we study the

P

-Eulerian polynomial where

P

is a naturally labeled zig-zag poset; we call these zig-zag Eulerian polynomials. A result of Brändén implies that these polynomials are gamma-nonnegative, and hence their coefficients are symmetric and unimodal. The zig-zag Eulerian polynomials and the associated order polynomials have appeared fleetingly in the literature in a wide variety of contexts—e.g., in the study of polytopes, magic labelings of graphs, and Kekulé structures—but they do not appear to have been studied systematically.

In this paper, we use a “relaxed” version of

P

-partitions to both survey and unify results. Our technique shows that the zig-zag Eulerian polynomials also capture the distribution of “big returns” over the set of (up-down) alternating permutations, as first observed by Coons and Sullivant. We develop recurrences for refined versions of the relevant generating functions, which evoke similarities to recurrences for the classical Eulerian polynomials. We conclude with a literature survey and open questions.

对于任何有限部分有序集合 P，P-Eulerian 多项式是 P 的线性扩展集合上下降数的生成函数，它与 P 分区理论中出现的 P 的阶多项式密切相关。在这里，我们研究 P-Eulerian 多项式，其中 P 是一个自然标注的之字形正集；我们称这些之字形欧拉多项式为欧拉多项式。Brändén 的一个结果意味着这些多项式是伽马负的，因此它们的系数是对称和单模态的。之字形欧拉多项式和相关的阶多项式曾在各种文献中昙花一现--例如，在研究多面体、图的魔法标注和凯库雷结构时，但似乎还没有对它们进行过系统的研究。我们的技术表明，"之 "字形欧拉多项式也能捕捉到（上下）交替排列集合上 "大回报 "的分布，正如库恩斯和苏利文首次观察到的那样。我们为相关生成函数的改进版本建立了递推关系，这与经典欧拉多项式的递推关系相似。最后，我们将对文献进行梳理，并提出一些开放性问题。

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