European Journal of Combinatorics最新文献

英文中文

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

Dániel Gerbner

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

Sepehr Hajebi

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

Amritanshu Prasad , Samrith Ram

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。

{"title":"Set partitions, tableaux, and subspace profiles under regular diagonal matrices","authors":"Amritanshu Prasad , Samrith Ram","doi":"10.1016/j.ejc.2024.104060","DOIUrl":"10.1016/j.ejc.2024.104060","url":null,"abstract":"<div>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 <math><mrow><mo>−</mo><mn>1</mn></mrow></math>, 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 <math><mi>q</mi></math>-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 <math><mi>q</mi></math>-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.</div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104060"},"PeriodicalIF":1.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142229590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

Kazumasa Nomura , Paul Terwilliger

<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

Venkata Raghu Tej Pantangi

<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

Hyunwoo Lee

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

Binzhou Xia

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

Quasi-transitive K∞-minor free graphs 准传递 K∞-minor 自由图

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-09-05 DOI: 10.1016/j.ejc.2024.104056

Matthias Hamann

We prove that every locally finite quasi-transitive graph that does not contain $K_{\infty}$ as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their recent result that such graphs are quasi-isometric to some planar graph of bounded degree.

我们证明了每一个不包含 K∞ 作为次要部分的局部有限准传递图都与某个平面准传递局部有限图准等距。这解决了 Esperet 和 Giocanti 的一个问题，并改进了他们最近的结果，即这类图与某些有界平面图准等距。

引用次数: 0

Chromatic quasisymmetric class functions for combinatorial Hopf monoids 组合霍普夫单体的色度准对称类函数

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-08-27 DOI: 10.1016/j.ejc.2024.104055

Jacob A. White

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$ , and an $H$ -structure $h$ on a set $N$ , there are proper colorings of $h$ , generalizing graph colorings and poset partitions. We show that the automorphism group of $h$ acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.

我们研究线性化组合霍普夫单元的色度准对称类函数。给定一个线性化组合霍普夫单元 H 和一个集合 N 上的 H 结构 h，就有 h 的适当着色，即图形着色和正集分割的一般化。我们证明了 h 的自变群作用于适当着色的集合。色度准对称类函数枚举了这一作用的定点，用一个单项式对每个着色进行加权。对于图的 Hopf monoid，这个不变量概括了斯坦利的色度对称函数，并特化为卡梅隆和卡伊比的轨道色度多项式。我们推导出了相关轨道多项式不变量的各种不等式。我们将这些结果应用于与枚举图着色、poset 分区、矩阵上的泛函或广义 permutohedra 等相关的几个例子中。

{"title":"Chromatic quasisymmetric class functions for combinatorial Hopf monoids","authors":"Jacob A. White","doi":"10.1016/j.ejc.2024.104055","DOIUrl":"10.1016/j.ejc.2024.104055","url":null,"abstract":"<div>We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid <math><mi>H</mi></math>, and an <math><mi>H</mi></math>-structure <math><mi>h</mi></math> on a set <math><mi>N</mi></math>, there are proper colorings of <math><mi>h</mi></math>, generalizing graph colorings and poset partitions. We show that the automorphism group of <math><mi>h</mi></math> acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.</div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104055"},"PeriodicalIF":1.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001409/pdfft?md5=63ba3288644e8ff2f9de5b9b878244e1&pid=1-s2.0-S0195669824001409-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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

On balanceable and simply balanceable regular graphs 关于可平衡和简单可平衡正则图

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-08-24 DOI: 10.1016/j.ejc.2024.104045

Milad Ahanjideh , Martin Milanič , Mary Servatius

We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs $G$ such that any 2-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of $G$ (that is, a copy with half the edges of each color). While the problem of recognizing balanceable graphs was conjectured to be $NP$ -complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph’s edges and another one such that the corresponding cut contains half of all the graph’s edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks’ theorem, we show that every 4-regular graph with order divisible by 4 is balanceable. Finally, we show that it is $NP$ -complete to determine if a 9-regular graph is simply balanceable.

我们将继续研究可平衡图，卡洛、汉斯伯格和蒙特哈诺在 2021 年将可平衡图定义为这样的图 G：在一个包含足够多每种颜色的边的足够大的完整图中，边的任何 2 次着色都包含 G 的一个平衡副本（即每个颜色的边各占一半的副本）。尽管戴利、汉斯伯格和文图拉在 2021 年猜想识别可平衡图的问题是 NP-complete，但可平衡图有一个优雅的组合特征：如果且仅如果存在两个顶点子集，其中一个子集包含该图所有边的一半，另一个子集的相应切割包含该图所有边的一半，则该图是可平衡的。我们考虑了这一属性的一种特殊情况，即当两个集合中的一个是顶点盖时，我们称相应的图为简单可平衡图。我们证明了一系列关于可平衡和简单可平衡正则图的结果。首先，我们通过一个涉及图的独立性数的条件来描述简单可平衡正则图。其次，我们解决了 Dailly、Hansberg 和 Ventura 在 2021 年提出的一个问题，并证明了每个立方图都是可平衡的。第三，利用布鲁克斯定理，我们证明了每个阶数能被 4 整除的 4 规则图都是可平衡的。最后，我们证明了确定一个 9 规则图是否简单可平衡是 NP-complete。

{"title":"On balanceable and simply balanceable regular graphs","authors":"Milad Ahanjideh , Martin Milanič , Mary Servatius","doi":"10.1016/j.ejc.2024.104045","DOIUrl":"10.1016/j.ejc.2024.104045","url":null,"abstract":"<div>We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs <math><mi>G</mi></math> such that any 2-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of <math><mi>G</mi></math> (that is, a copy with half the edges of each color). While the problem of recognizing balanceable graphs was conjectured to be <math><mi>NP</mi></math>-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph’s edges and another one such that the corresponding cut contains half of all the graph’s edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks’ theorem, we show that every 4-regular graph with order divisible by 4 is balanceable. Finally, we show that it is <math><mi>NP</mi></math>-complete to determine if a 9-regular graph is simply balanceable.</div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104045"},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001306/pdfft?md5=129ec5810b9eb2e86760507f0c3a96ba&pid=1-s2.0-S0195669824001306-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142048283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

European Journal of Combinatorics

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀