European Journal of Combinatorics最新文献

英文中文

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

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

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

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

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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引用次数: 0

On non-degenerate Turán problems for expansions 关于膨胀的非退化图兰问题

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-23 DOI: 10.1016/j.ejc.2024.104071

The

r

-uniform expansion

F^{(r) +}

of a graph

F

is obtained by enlarging each edge with

r - 2

new vertices such that altogether we use

(r - 2) | E (F) |

new vertices. Two simple lower bounds on the largest number

{ex}_{r} (n, F^{(r) +})

r

-edges in

F^{(r) +}

-free

r

-graphs are

Ω (n^{r - 1})

(in the case

F

is not a star) and

ex (n, K_{r}, F)

, which is the largest number of

r

-cliques in

n

-vertex

F

-free graphs. We prove that

{ex}_{r} (n, F^{(r) +}) = ex (n, K_{r}, F) + O (n^{r - 1})

. The proof comes with a structure theorem that we use to determine

{ex}_{r} (n, F^{(r) +})

exactly for some graphs

F

, every

r < χ (F)

and sufficiently large

n

图 F 的 r-uniform 扩展 F(r)+ 是通过用 r-2 个新顶点扩大每条边而得到的，这样我们总共使用了 (r-2)|E(F)| 个新顶点。关于无 F(r)+ r 图中 r 边的最大数量 exr(n,F(r)+) 的两个简单下限是 Ω(nr-1)（在 F 不是星形的情况下）和 ex(n,Kr,F)，后者是无 n 个顶点的 F 图中 r 簇的最大数量。我们证明，exr(n,F(r)+)=ex(n,Kr,F)+O(nr-1)。该证明包含一个结构定理，我们用它来精确确定某些图 F、每个 r<χ(F)和足够大的 n 的 exr(n,F(r)+)。

引用次数: 0

Induced subdivisions with pinned branch vertices 带有针状分支顶点的诱导细分区

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-22 DOI: 10.1016/j.ejc.2024.104072

We prove that for all

r \in N \cup {0}

and

s, t \in N

, there exists

Ω = Ω (r, s, t) \in N

with the following property. Let

G

be a graph and let

H

be a subgraph of

G

isomorphic to a

(\leq r)

-subdivision of

K_{Ω}

. Then either

G

contains

K_{t}

K_{t, t}

as an induced subgraph, or there is an induced subgraph

J

G

isomorphic to a proper

(\leq r)

-subdivision of

K_{s}

such that every branch vertex of

J

is a branch vertex of

H

. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.

我们证明，对于所有 r∈N∪{0} 和 s,t∈N，存在具有以下性质的 Ω=Ω(r,s,t)∈N。设 G 是图，设 H 是 G 的子图，与 KΩ 的（≤r）细分同构。那么要么 G 包含作为诱导子图的 Kt 或 Kt,t，要么 G 的诱导子图 J 与 Ks 的适当 (≤r)- 细分同构，使得 J 的每个分支顶点都是 H 的分支顶点。事实上，我们证明了分支顶点和它们之间对应于细分边的路径都可以保留。

引用次数: 0

Set partitions, tableaux, and subspace profiles under regular diagonal matrices 正则对角矩阵下的集合分区、表格和子空间剖面

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-14 DOI: 10.1016/j.ejc.2024.104060

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At $- 1$ , they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$ -Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions.

For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the $q$ -Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings.

我们引入了一系列以整数分区为索引的单变量多项式。在质数幂时，它们计算有限向量空间中以特定方式在规则对角矩阵下变换的子空间的数量。这个枚举公式是对本德、科利、罗宾斯和拉姆齐提出的一个问题的组合式解答。在 1 时，他们计算具有指定块大小的集合分区。在 0 时，他们计算指定形状的标准表格。在-1 时，它们计算指定形状的标准移位表格。这些多项式是由集合分区的新统计量（称为交错数）以及标准台格的多项式统计量产生的。对于部分最多为两个的分区，这些多项式是与 q-Hermite 正交多项式序列相关联的加泰罗尼亚三角形的非零项。特别是，当所有部分都等于二时，这些多项式与根据交叉数枚举弦图的 Touchard 定义的多项式重合。

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引用次数: 0

Spin models and distance-regular graphs of q-Racah type 自旋模型和 q-Racah 型距离不规则图

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-12 DOI: 10.1016/j.ejc.2024.104069

<div>Let <math><mi>Γ</mi></math> denote a distance-regular graph, with vertex set <math><mi>X</mi></math> and diameter <math><mrow><mi>D</mi><mo>≥</mo><mn>3</mn></mrow></math>. We assume that <math><mi>Γ</mi></math> is formally self-dual and <math><mi>q</mi></math>-Racah type. Let <math><mi>A</mi></math> denote the adjacency matrix of <math><mi>Γ</mi></math>. Pick <math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math>, and let <math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> denote the dual adjacency matrix of <math><mi>Γ</mi></math> with respect to <math><mi>x</mi></math>. The matrices <math><mrow><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math> generate the subconstituent algebra <math><mrow><mi>T</mi><mo>=</mo><mi>T</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math>. We assume that for every choice of <math><mi>x</mi></math> the algebra <math><mi>T</mi></math> contains a certain central element <math><mrow><mi>Z</mi><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> whose significance is illuminated by the following relations: <math><mrow><mi>A</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>B</mi><mi>C</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>C</mi><mi>B</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>,</mo><mspace></mspace><mi>B</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>C</mi><mi>A</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>C</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>,</mo><mspace></mspace><mi>C</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>A</mi><mi>B</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>B</mi><mi>A</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>.</mo></mrow></math> The matrices <math><mi>A</mi></math>, <math><mi>B</mi></math> satisfy <math><mrow><mi>A</mi><mo>=</mo><mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>ɛ</mi><mi>I</mi><mo>)</mo></mrow><mo>/</mo><mi>α</mi></mrow></math> and <math><mrow><mi>B</mi><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow

让 Γ 表示一个距离规则图，其顶点集为 X，直径为 D≥3。我们假设 Γ 是形式上自偶的 q-Racah 型图。让 A 表示 Γ 的邻接矩阵。选取 x∈X，让 A∗=A∗(x) 表示Γ 关于 x 的对偶邻接矩阵。矩阵 A、A∗ 生成子构成代数 T=T(x)。我们假设，对于 x 的每一种选择，代数 T 都包含某个中心元素 Z=Z(x)，其意义由以下关系揭示：A+qBC-q-1CBq2-q-2=Z,B+qCA-q-1ACq2-q-2=Z,C+qAB-q-1BAq2-q-2=Z。矩阵 A、B 满足 A=（A-ɛI）/α 和 B=（A∗-ɛI）/α，其中 α、ɛ 是复标量，用于描述 A 和 A∗ 的特征值。矩阵 C 是用第三个显示方程定义的。我们用 Z 来构建一个由 Γ 提供的自旋模型 W。我们研究了 Z 的组合意义，并逆转逻辑方向，从 W 中恢复了 Z。

引用次数: 0

All 3-transitive groups satisfy the strict-Erdős–Ko–Rado property 所有 3 传递群都满足严格的厄尔多斯-柯-拉多性质

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-11 DOI: 10.1016/j.ejc.2024.104057

<div>A subset <math><mi>S</mi></math> of a transitive permutation group <math><mrow><mi>G</mi><mo>≤</mo><mi>Sym</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> is said to be an intersecting set if, for every <math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>S</mi></mrow></math>, there is an <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math> such that <math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow></math>. The stabilizer of a point in <math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math> and its cosets are intersecting sets of size <math><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>/</mo><mi>n</mi></mrow></math>. Such families are referred to as canonical intersecting sets. A result by Meagher, Spiga, and Tiep states that if <math><mi>G</mi></math> is a 2-transitive group, then <math><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>/</mo><mi>n</mi></mrow></math> is the size of an intersecting set of maximum size in <math><mi>G</mi></math>. In some 2-transitive groups (for instance <math><mrow><mi>Sym</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><mi>Alt</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>), every intersecting set of maximum possible size is canonical. A permutation group, in which every intersecting family of maximum possible size is canonical, is said to satisfy the strict-EKR property. In this article, we investigate the structure of intersecting sets in 3-transitive groups. A conjecture by Meagher and Spiga states that all 3-transitive groups satisfy the strict-EKR property. Meagher and Spiga showed that this is true for the 3-transitive group <math><mrow><mi>PGL</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math>. Using the classification of 3-transitive groups and some results in the literature, the conjecture reduces to showing that the 3-transitive group <math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math> satisfies the strict-EKR property. We show that <math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math> satisfies the strict-EKR property and as a consequence, we prove Meagher and Spiga’s conjecture. We also prove a stronger result for <math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math> by showing that “large” intersecting sets in <math><mrow><mi>A

如果对每一个 g1,g2∈S 都存在一个 i∈[n]，使得 g1(i)=g2(i) ，则称传递置换群 G≤Sym(n) 的子集 S 为交集。[n]中某点的稳定子及其余集是大小为 |G|/n 的交集。这样的族被称为典型相交集。Meagher、Spiga 和 Tiep 的一个结果指出，如果 G 是一个 2 传递群，那么 |G|/n 是 G 中最大相交集的大小。在某些 2 传递群（例如 Sym(n)、Alt(n)）中，每个最大可能大小的相交集都是典型的。如果一个置换群中，每个最大可能大小的交集族都是典型的，那么这个置换群就满足严格-EKR 属性。本文将研究 3 传递群中相交集的结构。Meagher 和 Spiga 的猜想指出，所有 3 传递群都满足严格-EKR 性质。Meagher 和 Spiga 证明了这一点在 3 传递群 PGL(2,q) 中是正确的。利用 3 传递群的分类和文献中的一些结果，这一猜想简化为证明 3 传递群 AGL(n,2) 满足严格-EKR 性质。我们证明了 AGL(n,2) 满足严格-EKR 属性，从而证明了 Meagher 和 Spiga 的猜想。通过证明 AGL(n,2) 中的 "大 "相交集必须是一个典型相交集的子集，我们还证明了 AGL(n,2) 的一个更强的结果。这种现象被称为稳定性。

引用次数: 0

Spanning subdivisions in dense digraphs 密集图中的跨细分

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-06 DOI: 10.1016/j.ejc.2024.104059

We prove that an $n$ -vertex digraph $D$ with minimum semi-degree at least $(\frac{1}{2} + ɛ) n$ and $n \geq C m$ contains a subdivision of all $m$ -arc digraphs without isolated vertices. Here, $C$ is a constant only depending on $ɛ .$ This is the best possible and settles a conjecture raised by Pavez-Signé (2023) in a stronger form.

我们证明，最小半度至少为 12+ɛn 且 n≥Cm 的 n 个顶点数图 D 包含所有无孤立顶点的 m 弧数图的细分。这里，C 是一个常数，只取决于 ɛ。这是可能的最佳结果，并以更强的形式解决了 Pavez-Signé (2023) 提出的猜想。

引用次数: 0

Graphical regular representations of (2,p)-generated groups (2,p)生成群的图形正则表达式

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-06 DOI: 10.1016/j.ejc.2024.104058

For groups $G$ that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of $G$ to be a graphical regular representation (GRR), that is, for the Cayley graph to have full automorphism group isomorphic to $G$ . This condition enables one to show the existence of GRRs of prescribed valency for a large class of groups, and in this paper, $k$ -valent GRRs of finite nonabelian simple groups with $k \geq 5$ are considered.

对于可以由一个内卷和一个奇素数元素生成的群 G，本文给出了一个充分条件，即 G 的某个 Cayley 图是一个图形正则表达式（GRR），也就是 Cayley 图具有与 G 同构的全自形群。

引用次数: 0

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