Discrete Mathematics最新文献

英文中文

Adversarial graph burning densities 逆向图燃烧密度

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114253

Graph burning is a discrete-time process that models the spread of influence in a network. Vertices are either burning or unburned, and in each round, a burning vertex causes all of its neighbours to become burning before a new fire source is chosen to become burning. We introduce a variation of this process that incorporates an adversarial game played on a nested, growing sequence of graphs. Two players, Arsonist and Builder, play in turns: Builder adds a certain number of new unburned vertices and edges incident to these to create a larger graph, then every vertex neighbouring a burning vertex becomes burning, and finally Arsonist ‘burns’ a new fire source. This process repeats forever. Arsonist is said to win if the limiting fraction of burning vertices tends to 1, while Builder is said to win if this fraction is bounded away from 1.

The central question of this paper is determining if, given that Builder adds $f (n)$ vertices at turn n, either Arsonist or Builder has a winning strategy. In the case that $f (n)$ is asymptotically polynomial, we give threshold results for which player has a winning strategy.

图燃烧是一个离散时间过程，是网络中影响力传播的模型。顶点要么燃烧，要么未燃烧，在每一轮中，一个燃烧的顶点会导致其所有相邻顶点燃烧，然后再选择一个新的火源燃烧。我们引入了这一过程的变体，在嵌套的、不断增长的图序列中加入了对抗游戏。两名玩家（纵火者和建造者）轮流进行游戏：建造者添加一定数量的新的未燃烧顶点和与之相连的边，以创建一个更大的图，然后每个与燃烧顶点相邻的顶点都变成燃烧顶点，最后纵火犯 "燃烧 "一个新的火源。这个过程永远重复。如果燃烧顶点的极限分数趋向于 1，那么 "纵火者 "就获胜了；如果这个分数远离 1，那么 "建造者 "就获胜了。本文的核心问题是确定，如果 "建造者 "在第 n 轮增加了 f(n) 个顶点，那么 "纵火者 "或 "建造者 "是否都有获胜策略。在 f(n) 是渐近多项式的情况下，我们给出了哪一方有获胜策略的阈值结果。

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引用次数: 0

On the spum and sum-diameter of paths 关于路径的空间和总直径

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.disc.2024.114257

In a sum graph, the vertices are labeled with distinct positive integers, and two vertices are adjacent if the sum of their labels is equal to the label of another vertex. In 1990, Harary showed that not all graphs G can be labeled as a sum graph but the union of G and at least some $σ (G)$ isolated vertices can be. The spum of a graph G is defined as the minimum difference between the largest and smallest labels of a sum graph that consists of the union of G and exactly $σ (G)$ isolated vertices. More recently, Li introduced the sum-diameter of a graph G, which modifies the definition of spum by removing the requirement that the number of isolated vertices must be $σ (G)$ . In this paper, we settle conjectures by Singla, Tiwari, and Tripathi and a conjecture by Li by evaluating the spum and the sum-diameter of paths.

在和图中，顶点用不同的正整数标注，如果两个顶点的标签之和等于另一个顶点的标签，那么这两个顶点就是相邻的。1990 年，哈拉里证明了并非所有图 G 都可以标记为和图，但 G 和至少某些 σ(G) 孤立顶点的结合可以标记为和图。图 G 的 spum 定义为由 G 和正好 σ(G) 个孤立顶点的结合组成的和图的最大标签和最小标签之间的最小差值。最近，Li 引入了图 G 的总和直径，它修改了 spum 的定义，取消了孤立顶点数必须为 σ(G)的要求。在本文中，我们通过评估 spum 和路径的总直径，解决了 Singla、Tiwari 和 Tripathi 的猜想和 Li 的猜想。

引用次数: 0

Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials 广义劳伦双正交多项式的组合解释所产生的广义施罗德路径

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.disc.2024.114230

Lattice paths called ℓ-Schröder paths are introduced. They are paths on the upper half-plane consisting of $ℓ + 2$ types of steps: $(i, ℓ - i)$ for $i = 0, \dots, ℓ$ , and $(1, - 1)$ . Those paths generalize Schröder paths and some variants, such as m-Schröder paths by Yang and Jiang and Motzkin-Schröder paths by Kim and Stanton. We show that ℓ-Schröder paths arise naturally from a combinatorial interpretation of the moments of generalized Laurent bi-orthogonal polynomials introduced by Wang, Chang, and Yue. We also show that some generating functions of non-intersecting ℓ-Schröder paths can be factorized in closed forms.

引入了称为 ℓ-Schröder 路径的网格路径。它们是由 ℓ+2 种步长组成的上半平面路径：i=0,...,ℓ时的(i,ℓ-i)和(1,-1)。这些路径概括了施罗德路径和一些变体，例如杨和江的 m-Schröder 路径以及金和斯坦顿的 Motzkin-Schröder 路径。我们证明，ℓ-Schröder 路径自然产生于对王、张和岳提出的广义劳伦特双正交多项式矩的组合解释。我们还证明了一些非相交ℓ-Schröder 路径的生成函数可以用封闭形式因式分解。

引用次数: 0

Lagrangian densities of 4-uniform matchings and degree stability of extremal hypergraphs 4-Uniform matchings 的拉格朗日密度和极值超图的度稳定性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.disc.2024.114235

<div>The Lagrangian density of an r-uniform graph F is <math><msub><mrow><mi>π</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>sup</mi><mo>⁡</mo><mo>{</mo><mi>r</mi><mo>!</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mi>G</mi><mspace></mspace><mi>i</mi><mi>s</mi><mspace></mspace><mi>F</mi><mtext>-</mtext><mi>f</mi><mi>r</mi><mi>e</mi><mi>e</mi><mo>}</mo></math>, where <math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> is the Lagrangian of an r-uniform graph G. Hypergraph Lagrangian has been a helpful tool in extremal combinatorics. Let <math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> denote the r-uniform matching with size t. The well-known Erdős Matching conjecture proposed that the Turán number of <math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> is <math><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>e</mi><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>,</mo><mi>e</mi><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>}</mo></math>, where <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> is the complete r-graph on <math><mi>r</mi><mi>t</mi><mo>−</mo><mn>1</mn></math> vertices and <math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo></math> is the r-graph with vertex set <math><mo>[</mo><mi>n</mi><mo>]</mo></math> and with edge set <math><mi>E</mi><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mo>{</mo><mi>e</mi><mo>∈</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>:</mo><mo>|</mo><mi>e</mi><mo>∩</mo><mo>[</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>]</mo><mo>|</mo><mo>≥</mo><mn>1</mn><mo>}</mo></math>. Regarding Lagrangian density of hypergraph matchings, Jiang, Peng and Wu [22] (Wu [34] as well) conjectured that the property similar to Erdős Matching Conjecture holds, precisely, they conjectured that <math><msub><mrow><mi>π</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>=</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo>{</mo><mi>r</mi><mo>!</mo><mi>λ</mi><mo>(

r-均匀图 F 的拉格朗日密度为 πλ(F)=sup{r!λ(G):GisF-free}, 其中 λ(G) 是 r-均匀图 G 的拉格朗日。让 Mtr 表示大小为 t 的 r-uniform 匹配。著名的厄多斯匹配猜想提出，Mtr 的图兰数为 max{e(Ktr-1r),e(St-1r(n))} ，其中 Ktr-1r 是 rt-1 个顶点上的完整 r 图，St-1r(n) 是具有顶点集 [n] 和边集 E(St-1r(n))={e∈([n]r):|e∩[t-1]|≥1} 的 r 图。关于超图匹配的拉格朗日密度，蒋、彭和吴[22]（以及吴[34]）猜想与厄尔多斯匹配猜想类似的性质成立，确切地说，他们猜想πλ(Mtr)=max{r!λ(Ktr-1r),r!limn→∞λ(St-1r)(n)}。Hefetz 和 Keevash [18] 证实了 M23，Jiang、Peng 和 Wu [22] 证实了 Mt3，Wu、Peng 和 Chen [35] 证实了 M24，Bene Watts、Norin 和 Yepremyan [2] 证实了 r≥4 时的 M2r，Wu [34] 证实了 M34。在本文中，我们证明了当 t≥4 时，πλ(Mt4)=4!λ(K4t-14)，这就解决了对 Mt4 的猜想。我们的结果还有有趣的应用。将我们的结果与 Liu、Mubayi 和 Reiher 在 [26] 中的定理 1.12 结合起来，我们可以得到 Mt4 扩展的图兰密度（t≥4）以及极值超图的度稳定性（比边稳定性更强的稳定性）。结合我们的结果和 Keevash、Lenz 和 Mubayi [24] 在 [24] 中的定理 1.4，我们还可以得到，如果 G 是一个有 n 个顶点的无 Mt4 的 4-Uniform 超图，那么如果 α>1，n 足够大且 t≥4 时，G 的 α 谱半径不超过 K4t-14 的 α 谱半径。事实上，对于满足 Liu、Mubayi 和 Reiher [26] 的定理 1.12 或 Keevash、Lenz 和 Mubayi [24] 的定理 1.4 条件的超图，要获得相应极值超图的度稳定性或相应的 α 谱结果，只需确定相应超图的拉格朗日密度即可。这些联系为确定超图的拉格朗日密度增添了更多的动力。

引用次数: 0

q-ary (1,k)-overlap-free codes with given restrictions 具有给定限制条件的 q-ary (1,k) - 无重叠编码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.disc.2024.114236

Two words u and v have a t-overlap if the length t prefix of u is equal to the length t suffix of v, or vice versa. A code $C$ is t-overlap-free if no two words u and v in $C$ (including $u = v$ ) have a t-overlap. A code of length n is said to be $(t_{1}, t_{2})$ -overlap-free if it is t-overlap-free for all t such that $1 ⩽ t_{1} ⩽ t ⩽ t_{2} ⩽ n - 1$ . A $(1, n - 1)$ -overlap-free code of length n is called non-overlapping, which has applications in DNA-based data storage systems and frame synchronization. In this paper, we initialize the study for codes of length n which are simultaneously $(1, k)$ -overlap-free and $(n - k, n - 1)$ -overlap-free, and establish lower and upper bounds for the size of balanced and error-correcting $(1, k)$ -overlap-free codes.

如果 u 的前缀长度 t 等于 v 的后缀长度 t，或反之亦然，则两个词 u 和 v 有 t 重叠。如果 C 中没有两个词 u 和 v（包括 u=v）有 t 重叠，则代码 C 无 t 重叠。如果长度为 n 的代码对所有 t 都是无 t 重叠的，且 1⩽t1⩽t⩽t2⩽n-1。长度为 n 的（1,n-1）无重叠编码称为无重叠编码，它可应用于基于 DNA 的数据存储系统和帧同步。本文初步研究了长度为 n、同时具有 (1,k)-overlap-free 和 (n-k,n-1)-overlap-free 的编码，并建立了平衡编码和纠错 (1,k)-overlap-free 编码大小的下限和上限。

引用次数: 0

Noncommutative symmetric functions and skewing operators 非交换对称函数和偏斜算子

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.disc.2024.114255

Skewing operators play a central role in the symmetric function theory because of the importance of the product structure of the symmetric function space. The theory of noncommutative symmetric functions is a useful tool for studying expansions of a given symmetric function in terms of various bases. In this paper, we establish a further development of the theory for studying skewing operators. Using this machinery, we are able to easily reproduce the Littlewood–Richardson rule and provide recurrence relations for chromatic quasisymmetric functions, which generalize Harada–Precup's recurrence.

由于对称函数空间乘积结构的重要性，偏斜算子在对称函数理论中发挥着核心作用。非交换对称函数理论是研究给定对称函数在各种基上展开的有用工具。在本文中，我们进一步发展了研究偏斜算子的理论。利用这一机制，我们能够轻松地重现 Littlewood-Richardson 规则，并提供色度准对称函数的递推关系，这是对 Harada-Precup 递推关系的概括。

引用次数: 0

Edge-apexing in hereditary classes of graphs 遗传图类中的边缘apexing

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-04 DOI: 10.1016/j.disc.2024.114234

A class $G$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $G^{epex}$ the class of graphs that are at most one edge away from being in $G$ . We note that $G^{epex}$ is hereditary and prove that if a hereditary class $G$ has finitely many forbidden induced subgraphs, then so does $G^{epex}$ .

The hereditary class of cographs consists of all graphs G that can be generated from $K_{1}$ using complementation and disjoint union. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. For the class of edge-apex cographs our main result bounds the order of such forbidden induced subgraphs by 8 and finds all of them by computer search.

如果一类图 G 在取诱导子图时是封闭的，则该类图 G 称为遗传类图。我们用 Gepex 表示离 G 最多只有一条边的一类图。我们注意到 Gepex 是遗传的，并证明如果一个遗传类 G 有有限多个禁止的诱导子图，那么 Gepex 也是遗传的。Cographs 正是没有 4 顶点路径作为诱导子图的图。对于无边 Cographs 类，我们的主要结果将此类禁止诱导子图的阶数限定为 8，并通过计算机搜索找到所有这些子图。

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引用次数: 0

A note on clique immersion of strong product graphs 关于强积图的簇嵌入的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-04 DOI: 10.1016/j.disc.2024.114237

Let $G, H$ be graphs, and $G ⁎ H$ represent a specific graph product of G and H. Define $i m (G)$ as the largest t for which G contains a $K_{t}$ -immersion. Collins, Heenehan, and McDonald posed the question: given $i m (G) = t$ and $i m (H) = r$ , how large can $i m (G ⁎ H)$ be? They conjectured $i m (G ⁎ H) \geq t r$ when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when H contains $K_{r}$ as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.

设 G、H 为图，G⁎H 表示 G 和 H 的特定图积。定义 im(G) 为 G 包含 Kt-imersion 的最大 t。柯林斯、希尼汉和麦克唐纳提出了这样一个问题：给定 im(G)=t 和 im(H)=r，im(G⁎H)可以有多大？他们猜想，当⁎表示强积时，im(G⁎H)≥tr。在本注释中，我们肯定了这一猜想在具有特定浸入的图中成立，尤其是当 H 包含 Kr 作为子图时。因此，我们还为 Guyer 和 McDonald 的一个结果提供了另一种论证，证明恒多重性多图的线图满足 Abu-Khzam 和 Langston 最初提出的猜想。

引用次数: 0

Covering the edges of a graph with triangles 用三角形覆盖图形边缘

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114226

In a graph G, let $ρ_{△} (G)$ denote the minimum size of a set of edges and triangles that cover all edges of G, and let $α_{1} (G)$ be the maximum size of an edge set that contains at most one edge from each triangle. Motivated by a question of Erdős, Gallai, and Tuza, we study the relationship between $ρ_{△} (G)$ and $α_{1} (G)$ and establish a sharp upper bound on $ρ_{△} (G)$ . We also prove Nordhaus-Gaddum-type inequalities for the considered invariants.

在图 G 中，让 ρ△(G) 表示覆盖 G 所有边的边集和三角形的最小大小，让 α1(G) 表示每个三角形中最多包含一条边的边集的最大大小。受 Erdős、Gallai 和 Tuza 问题的启发，我们研究了 ρ△(G) 和 α1(G) 之间的关系，并建立了 ρ△(G) 的尖锐上界。我们还证明了所考虑的不变式的 Nordhaus-Gaddum 型不等式。

引用次数: 0

Degree powers and number of stars in graphs with a forbidden broom 带禁忌扫帚的图形中的度数幂和星数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114232

Given a graph G with degree sequence $d_{1}, \dots, d_{n}$ and a positive integer r, let $e_{r} (G) = \sum_{i = 1}^{n} d_{i}^{r}$ . We denote by ${ex}_{r} (n, F)$ the largest value of $e_{r} (G)$ among n-vertex F-free graphs G, and by $ex (n, S_{r}, F)$ the largest number of stars $S_{r}$ in n-vertex F-free graphs. The broom $B (ℓ, s)$ is the graph obtained from an ℓ-vertex path by adding s new leaves connected to a penultimate vertex v of the path.

We determine ${ex}_{r} (n, B (ℓ, s))$ for $r \geq 2$ , any $ℓ, s$ and sufficiently large n, proving a conjecture of Lan, Liu, Qin and Shi. We also determine $ex (n, S_{r}, B (ℓ, s))$ for $r \geq 2$ , any $ℓ, s$ and sufficiently large n.

给定一个阶数为 d1，...，dn 的图 G 和一个正整数 r，设 er(G)=∑i=1ndir。我们用 exr(n,F) 表示无 n 个顶点的 F 图 G 中 er(G) 的最大值，用 ex(n,Sr,F) 表示无 n 个顶点的 F 图中星星 Sr 的最大数目。对于 r≥2、任意 ℓ,s 和足够大的 n，我们确定了 exr(n,B(ℓ,s))，证明了 Lan、Liu、Qin 和 Shi 的猜想。对于 r≥2、任意 ℓ,s 和足够大的 n，我们还确定了 ex(n,Sr,B(ℓ,s))。

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