Discrete Mathematics最新文献

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On graphs with maximum difference between game chromatic number and chromatic number 关于游戏色度数与色度数最大差值的图形

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-26 DOI: 10.1016/j.disc.2024.114271

In the vertex colouring game on a graph G, Maker and Breaker alternately colour vertices of G from a palette of k colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker seeks to make some vertex impossible to colour. The game chromatic number of G,

χ_{g} (G)

, is the minimum number k of colours for which Maker has a winning strategy for the vertex colouring game.

Matsumoto proved in 2019 that

χ_{g} (G) - χ (G) \leq ⌊ n / 2 ⌋ - 1

, and conjectured that the only equality cases are some graphs of small order and the Turán graph

T (2 r, r)

. We resolve this conjecture in the affirmative by considering a modification of the vertex colouring game wherein Breaker may remove a vertex instead of colouring it.

Matsumoto further asked whether a similar result could be proved for the vertex marking game, and we provide an example to show that no such nontrivial result can exist.

在图 G 的顶点着色游戏中，制造者和破坏者交替从 k 种颜色中为 G 的顶点着色，相邻两个顶点不能着相同的颜色。制造者希望给整个图着色，而破坏者则希望让某个顶点无法着色。松本在 2019 年证明了χg(G)-χ(G)≤⌊n/2⌋-1，并猜想唯一相等的情况是一些小阶图和图兰图 T(2r,r)。我们考虑了顶点着色博弈的一种修改，即破坏者可以移除一个顶点而不是给它着色，从而肯定地解决了这一猜想。松本进一步询问是否可以为顶点标记博弈证明类似的结果，我们举例说明不可能存在这种非难结果。

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引用次数: 0

Stabbing boxes with finitely many axis-parallel lines and flats 具有有限多条轴平行线和平面的刺盒

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-26 DOI: 10.1016/j.disc.2024.114269

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in

R^{d}

to be pierceable by finitely many axis-parallel k-flats, where

0 \leq k < d

. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner

(p, q)

-problem.

在这篇短文中，我们提供了一个必要条件和充分条件，即 Rd 中轴对称的无穷盒集合可被有限多个轴对称的 k 平面（其中 0≤k<d 时）穿透。我们还考虑了上述结果的丰富多彩的一般化，并确定了它们的可行性。本文考虑的问题是哈德维格-德布鲁纳（p,q）问题的无限变体。

引用次数: 0

Transversal coalitions in hypergraphs 超图中的横向联盟

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.disc.2024.114267

A transversal in a hypergraph H is set of vertices that intersect every edge of H. A transversal coalition in H consists of two disjoint sets of vertices X and Y of H, neither of which is a transversal but whose union

X \cup Y

is a transversal in H. Such sets X and Y are said to form a transversal coalition. A transversal coalition partition in H is a vertex partition

Ψ = {V_{1}, V_{2}, \dots, V_{p}}

such that for all

i \in [p]

, either the set

V_{i}

is a singleton set that is a transversal in H or the set

V_{i}

forms a transversal coalition with another set

V_{j}

for some j, where

j \in [p] ∖ {i}

. The transversal coalition number

C_{τ} (H)

in H equals the maximum order of a transversal coalition partition in H. For

k \geq 2

a hypergraph H is k-uniform if every edge of H has cardinality k. Among other results, we prove that if

k \geq 2

and H is a k-uniform hypergraph, then

C_{τ} (H) \leq ⌊ \frac{1}{4} k^{2} ⌋ + k + 1

. Further we show that for every

k \geq 2

, there exists a k-uniform hypergraph that achieves equality in this upper bound.

超图 H 中的横向是指与 H 的每条边相交的顶点集合。H 中的横向联盟由 H 的两个不相交的顶点集合 X 和 Y 组成，这两个集合都不是横向，但它们的结合 X∪Y 是 H 中的横向。H 中的横向联盟分区是一个顶点分区Ψ={V1,V2,...,Vp}，对于所有 i∈[p]，要么集合 Vi 是 H 中横向的单子集，要么集合 Vi 与某个 j 的另一个集合 Vj 形成横向联盟，其中 j∈[p]∖{i}。H 中的横向联盟数 Cτ(H) 等于 H 中横向联盟分区的最大阶数。对于 k≥2 的超图 H，如果 H 中的每条边都有 cardinality k，那么 H 就是 k-uniform 的。我们进一步证明，对于每一个 k≥2，都存在一个达到这个上界相等的 k-uniform 超图。

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引用次数: 0

Fibonacci and Catalan paths in a wall 墙上的斐波纳契和加泰罗尼亚路径

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.disc.2024.114268

We study the distribution of some statistics (width, number of steps, length, area) defined for paths contained in walls. We present the results by giving generating functions, asymptotic approximations, as well as some closed formulas. We prove algebraically that paths in walls of a given width and ending on the x-axis are enumerated by the Catalan numbers, and we provide a bijection between these paths and Dyck paths. We also find that paths in walls with a given number of steps are enumerated by the Fibonacci numbers. Finally, we give a constructive bijection between the paths in walls of a given length and peakless Motzkin paths of the same length.

我们研究了为墙内路径定义的一些统计量（宽度、步数、长度、面积）的分布。我们通过给出生成函数、渐近近似值以及一些封闭公式来呈现结果。我们用代数方法证明，在给定宽度的墙壁中，以 x 轴为终点的路径可以用加泰罗尼亚数枚举，并提供了这些路径与戴克路径之间的双射关系。我们还发现，具有给定步数的墙内路径可以用斐波那契数枚举。最后，我们给出了给定长度的墙内路径与相同长度的无峰莫兹金路径之间的构造偏射。

引用次数: 0

On the inclusion chromatic index of a Halin graph 论哈林图的包含色度指数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.disc.2024.114266

An inclusion-free edge-coloring of a graph G with

δ (G) \geq 2

is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of G is called the

i n c l u s i o n

f r e e

c h r o m a t i c i n d e x

, denoted by

χ_{\subset}^{'} (G)

. In this paper, we show that for a Halin graph G with maximum degree

Δ \geq 4

, if G is isomorphic to a wheel

W_{Δ + 1}

where Δ is odd, then

χ_{\subset}^{'} (G) = Δ + 2

, otherwise

χ_{\subset}^{'} (G) = Δ + 1

. We also show a special cubic Halin graph with

χ_{\subset}^{'} (G) = 5

δ(G)≥2的图 G 的无包含边着色是一种适当的边着色，使得任何顶点的颜色集合都不包含在其任何相邻顶点的颜色集合中。G 的无包含边染色所需的最少颜色数称为无包含色度指数，用 χ⊂′(G)表示。在本文中，我们证明了对于最大度数为 Δ≥4 的 Halin 图 G，如果 G 与 Δ 为奇数的轮 WΔ+1 同构，则 χ⊂′(G)=Δ+2 ，否则 χ⊂′(G)=Δ+1。我们还展示了一个特殊的立方哈林图，其χ⊂′(G)=5。

引用次数: 0

Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations 非交叉 (1,2) 构型上的循环筛分和二面筛分

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.disc.2024.114262

<div>Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>-configurations (denoted by <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>), which is a class of set partitions of <math><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math>. More precisely, Thiel proved that, with a natural action of the cyclic group <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math> on <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, the triple <math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>)</mo></math> exhibits the CSP, where <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>≔</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>[</mo><mtable><mtr><mtd><mn>2</mn><mi>n</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math> is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring <math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, Jesse Kim found a combinatorial basis for <math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math> indexed by <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. In this paper, we continue to study <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and obtain the following results:<ul><li>(1)We define a statistic on <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> whose generating function is <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>, which answers a problem of Thiel.</li><li>(2)We show that <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math> is equivalent to<math><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>k</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>k</mi><mo>+</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></munder><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi><mo>−</mo><mn

为了验证普罗普和赖纳关于循环筛分现象（CSP）的猜想，蒂尔（M. Thiel）引入了一个称为非交叉（1,2）配置（用 Xn 表示）的加泰罗尼亚对象，它是 [n-1] 的一类集合分区。更确切地说，蒂尔证明了在循环群 Cn-1 对 Xn 的自然作用下，三重（Xn,Cn-1,Catn(q)）展示了 CSP，其中 Catn(q)≔1[n+1]q[2nn]q 是麦克马洪的 q 加泰罗尼亚数。最近，在对费米对角共变环 FDRn 的研究中，Jesse Kim 发现了以 Xn 为索引的 FDRn 组合基。(2)我们证明 Catn(q) 等价于∑k,x,y2k+x+y=n-1[n-12k,x,y]qCatk(q)qk+(x2)+(y2)+(n2) modulo qn-1-1，这回答了 Jesse Kim 的一个问题。(3)我们考虑二面筛分，它是 CSP 的广义化，最近由 Rao 和 Suk 提出。在二面群 I2(n-1)（偶数 n）的自然作用下，我们证明了 Xn 上的二面筛分结果。

引用次数: 0

Graphs with the minimum spectral radius for given independence number 给定独立数时具有最小谱半径的图形

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.disc.2024.114265

Let $G_{n, α}$ be the set of connected graphs with order n and independence number α. The graph with the minimum spectral radius among $G_{n, α}$ is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in $G_{n, α}$ appears to be a tough problem. Recently, Lou and Guo (2022) [14] proved that the minimizer graph in $G_{n, α}$ must be a tree if $α \geq ⌈ \frac{n}{2} ⌉$ . In this paper, we further give the structural features for the minimizer graph in detail, and then provide a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in $G_{n, α}$ along with their spectral radius for any given $α \geq ⌈ \frac{n}{2} ⌉$ . As an application, we determine all the minimizer graphs in $G_{n, α}$ for $α = n - 5, n - 6$ along with their spectral radius.

设 Gn,α 是阶数为 n 且独立数为 α 的连通图集合，Gn,α 中谱半径最小的图称为最小图。Stevanović 在其经典著作[Spectral Radius of Graphs, Academic Press, Amsterdam, 2015]中指出，确定 Gn,α 中的最小图似乎是一个难题。最近，Lou 和 Guo（2022）[14] 证明，如果α≥⌈n2⌉，则 Gn,α 中的最小图一定是树。本文进一步详细给出了最小图的结构特征，并给出了其构造定理。因此，我们从理论上确定了 Gn,α 中的最小化图，以及任意给定 α≥⌈n2⌉ 时它们的谱半径。作为应用，我们确定了 α=n-5,n-6 时 Gn,α 中的所有最小图及其谱半径。

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引用次数: 0

A note on rainbow stackings of random edge-colorings of hypergraphs 关于超图随机边着色的彩虹堆积的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114261

A rainbow stacking of r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete d-uniform hypergraph on n vertices is a way of superimposing $χ_{1}, \dots, χ_{m}$ so that no edges of the same color are superimposed on each other. The definition of rainbow stackings of graphs was proposed by Alon, Defant, and Kravitz, and they determined a sharp threshold for r (as a function of m and n) governing the existence and nonexistence of rainbow stackings of random r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete graph $K_{n}$ . In this paper, we extend their result to d-uniform hypergraph, obtain a sharp threshold for r controlling the existence and nonexistence of rainbow stackings of random r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete d-uniform hypergraph for $d \geq 3$ .

n 个顶点上的完整 d-Uniform 超图的 r 边颜色 χ1，...，χm 的彩虹叠加是一种叠加 χ1，...，χm 的方法，这样就不会有相同颜色的边相互叠加。图的彩虹叠加定义是由 Alon、Defant 和 Kravitz 提出的，他们确定了 r 的一个尖锐临界值（作为 m 和 n 的函数），该临界值决定了完整图 Kn 的随机 r 边颜色χ1,...,χm 的彩虹叠加存在与否。在本文中，我们将他们的结果推广到 d-uniform hypergraph，得到了一个控制完整 d-uniform hypergraph 的随机 r 边着色 χ1,...,χm 的彩虹堆叠存在与否的 r 的尖锐阈值，且 d≥3 时。

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引用次数: 0

Group divisible designs with block size 4 and group sizes 4 and 10 and some other 4-GDDs 组块大小为 4、组块大小为 4 和 10 的可分设计，以及其他一些 4-GDD 设计

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114254

In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 10. We show that a 4-GDD of type $4^{t} 10^{s}$ exists when the necessary conditions are satisfied, except possibly for a finite number of cases with $4 t + 10 s \leq 178$ . We also give some new examples of 4-GDDs for which the number of points is 51, 54 or some value less than or equal to 50.

在本文中，我们考虑了块大小为 4、组大小为 4 和 10 的组可分割设计（GDD）的存在性。我们证明，在满足必要条件的情况下，4t10s 类型的 4-GDD 是存在的，4t+10s≤178 的有限个例除外。我们还给出了一些点数为 51、54 或小于等于 50 的 4-GDD 的新例子。

引用次数: 0

Critically 3-frustrated signed graphs 关键的 3 个沮丧的签名图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114258

Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely indecomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given k there are only finitely many critically k-frustrated signed graphs of this kind.

Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building indecomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being indecomposable is necessary for our conjecture.

从 maxcut 的概念出发，研究有符号图的挫折指数是有符号图理论的基本问题之一。最近，两位作者发起了对临界挫折有符号图的研究。这是一种有符号图，其沮度指数随着任何边的移除而减小。本研究的重点是临界有符号图，这些图不是临界受挫有符号图（即不可分解有符号图）的边缘相交的联合体，也不是通过细分从其他临界受挫有符号图建立起来的。我们猜想，对于任何给定的 k，只有有限多个此类临界 k 受挫有符号图。为了支持这一猜想，我们证明了只有两个此类临界 3 受挫有符号图不存在一对边缘相接的负循环。同样，我们证明了正好有十个临界三挫折有符号平面图既不是可分解的，也不是其他临界三挫折有符号图的细分。我们提出了一种基于两个给定的此类有符号图形构建不可分解的临界受挫有符号图形的方法。我们还证明了不可分解的条件对于我们的猜想是必要的。

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