Discrete Mathematics最新文献

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On the size and structure of t-representable sumsets 关于可表示 t 的和集的大小和结构

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-21 DOI: 10.1016/j.disc.2024.114295

Christian Táfula

Let

A \subseteq Z_{\geq 0}

be a finite set with minimum element 0, maximum element m, and ℓ elements strictly in between. Write

{(h A)}^{(t)}

for the set of integers that can be written in at least t ways as a sum of h elements of A. We prove that

{(h A)}^{(t)}

is “structured” for

h \geq (1 + o (1)) \frac{1}{e} m ℓ t^{1 / ℓ}

(as

ℓ \to \infty

t^{1 / ℓ} \to \infty

), and prove a similar theorem on the size and structure of

A \subseteq Z^{d}

for h sufficiently large. Moreover, we construct a family of sets

A = A (m, ℓ, t) \subseteq Z_{\geq 0}

for which

{(h A)}^{(t)}

is not structured for

h ≪ m ℓ t^{1 / ℓ}

设 A⊆Z≥0 是一个有限集合，最小元素为 0，最大元素为 m，ℓ 元素严格介于两者之间。我们证明 (hA)(t) 在 h≥(1+o(1))1emℓt1/ℓ （如 ℓ→∞, t1/ℓ→∞）时是 "结构化 "的，并证明了关于 h 足够大时 A⊆Zd 的大小和结构的类似定理。此外，我们还构造了一个集合族 A=A(m,ℓ,t)⊆Z≥0，其中 (hA)(t) 在 h≪mℓt1/ℓ 时没有结构。

引用次数: 0

On the number of small Steiner triple systems with Veblen points 关于具有维布伦点的小型斯坦纳三重系统的数量

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-21 DOI: 10.1016/j.disc.2024.114294

Giuseppe Filippone , Mario Galici

The concept of Schreier extensions of loops was introduced in the general case in [11] and, more recently, it has been explored in the context of Steiner loops in [6]. In the latter case, it gives a powerful method for constructing Steiner triple systems containing Veblen points. Counting all Steiner triple systems of order v is an open problem for

v > 21

. In this paper, we investigate the number of Steiner triple systems of order 19, 27 and 31 containing Veblen points and we present some examples.

在一般情况下，[11] 引入了循环的施赖尔扩展概念，最近，[6] 又在斯坦纳循环的背景下对其进行了探索。在后一种情况下，它为构建包含维布伦点的斯坦纳三重系统提供了一种强有力的方法。对于 v>21 而言，计算所有 v 阶的斯坦纳三重系统是一个未决问题。在本文中，我们研究了含有 Veblen 点的 19、27 和 31 阶 Steiner 三重系统的数量，并给出了一些示例。

引用次数: 0

The average connectivity matrix of a graph 图形的平均连接矩阵

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.disc.2024.114290

Linh Nguyen , Suil O

<div><div>For a graph G and for two distinct vertices u and v, let <math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math> be the maximum number of vertex-disjoint paths joining u and v in G. The average connectivity matrix of an n-vertex connected graph G, written <math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>, is an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> matrix whose <math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math>-entry is <math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math> and let <math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math> be the spectral radius of <math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>. In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any n-vertex connected graph G, we have <math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>4</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mfrac></math>, which implies a result of Kim and O [8] stating that for any connected graph G, we have <math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math>, where <math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mfrac><mrow><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></mfrac></math> and <math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math> is the maximum size of a matching in G; equality holds only when G is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely <math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mro

对于图 G 和两个不同的顶点 u 和 v，设 κ(u,v) 是连接 G 中 u 和 v 的顶点不相交路径的最大数目。n 个顶点连通图 G 的平均连通性矩阵，记为 Aκ‾(G)，是一个 n×n 矩阵，其 (u,v) 项为 κ(u,v)/(n2)，设 ρ(Aκ‾(G)) 为 Aκ‾(G) 的谱半径。本文将研究矩阵的一些谱性质。特别是，我们证明了对于任意 n 个顶点的连通图 G，ρ(Aκ‾(G))≤4α′(G)n，这意味着 Kim 和 O [8] 的一个结果，即对于任意连通图 G、κ‾(G)≤2α′(G)，其中κ‾(G)=∑u,v∈V(G)κ(u,v)(n2)，α′(G)是 G 中匹配的最大大小；只有当 G 是具有奇数个顶点的完整图时，相等关系才成立。此外，对于二叉图，我们改进了边界，即 ρ(Aκ‾(G))≤(n-α′(G))(4α′(G)-2)n(n-1) ，只有当 G 是完整的平衡二叉图时，边界相等才成立。

引用次数: 0

Equitable coloring in 1-planar graphs 1 平面图形中的等价着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.disc.2024.114286

Daniel W. Cranston, Reem Mahmoud

For every

r \geq 13

, we show every 1-planar graph G with

Δ (G) \leq r

has an equitable r-coloring.

对于每一个 r≥13，我们都能证明Δ(G)≤r 的每一个 1 平面图形 G 都有一个公平的 r 着色。

引用次数: 0

Weak submodularity implies localizability: Local search for constrained non-submodular function maximization 弱次模化意味着本地化：受约束非次模块化函数最大化的局部搜索

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.disc.2024.114287

Majun Shi , Qingyong Zhu , Bei Liu , Yuchao Li

Local search algorithms are commonly employed to address a variety of problems in the domain of operations research and combinatorial optimization. Most of the literature on the maximization of constrained monotone non-submodular functions is based on a greedy strategy, and few designs of local search approach are involved. In this paper, we explore the problem of maximizing a monotone non-submodular function under a p-matroid (

p \geq 1

) constraint with local search algorithms. And we indicate that weak submodularity implies localizability of set function optimization which can be used to offer provable approximation guarantees of local search algorithms.

局部搜索算法通常用于解决运筹学和组合优化领域的各种问题。大多数关于受约束单调非次模化函数最大化的文献都基于贪婪策略，很少涉及局部搜索方法的设计。在本文中，我们用局部搜索算法探讨了在 p-matroid (p≥1) 约束下单调非次模化函数最大化的问题。我们指出，弱次模性意味着集合函数优化的可局部性，可用于为局部搜索算法提供可证明的近似保证。

引用次数: 0

Localized version of hypergraph Erdős-Gallai Theorem 超图厄尔多斯-加莱定理的本地化版本

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.disc.2024.114293

Kai Zhao, Xiao-Dong Zhang

The weight function of an edge in an n-vertex uniform hypergraph

H

is defined with respect to the number of edges in the longest Berge path containing the edge. We prove that the summation of the weight function values for all edges in

H

is at most n, and characterize all extremal hypergraphs that attain this bound. This result strengthens and extends the hypergraph version of the classic Erdős-Gallai Theorem, providing a local version of this theorem.

在 n 个顶点的均匀超图 H 中，边的权函数是根据包含该边的最长 Berge 路径中的边数定义的。我们证明了 H 中所有边的权函数值之和最多为 n，并描述了达到这一界限的所有极值超图的特征。这一结果加强并扩展了经典厄尔多斯-加莱定理的超图版本，提供了该定理的局部版本。

引用次数: 0

A note proving the nullity of block graphs is unbounded 证明块图的无效性是无界的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-17 DOI: 10.1016/j.disc.2024.114289

Michael Cary

Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a

K_{2}

-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of

K_{2}

-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.

块图是大量群落检测和其他网络划分模型的重要基准结构。奇异图在物理科学中有许多重要用途。最近有人提出了一个猜想，即无 K2 的块图的无效性不可能大于 1。在本文中，我们利用实对称矩阵的考奇交错定理构建了一个反例族，从而证明该猜想是错误的。在此过程中，我们证明了更强的声明，即无 K2 块图的无效性是无界的。最后，我们讨论了这一结果对计算网络理论文献的影响。

引用次数: 0

Bipartite Ramsey number pairs that involve combinations of cycles and odd paths 涉及循环和奇数路径组合的二方拉姆齐数对

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-15 DOI: 10.1016/j.disc.2024.114283

Ernst J. Joubert

<div><div>For bipartite graphs <math><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><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math>, the bipartite Ramsey number <math><mi>b</mi><mo>(</mo><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></math>, <math><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math> is the least positive integer b, so that any coloring of the edges of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>b</mi></mrow></msub></math> with k colors, will result in a copy of <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math> in the ith color, for some i. For bipartite graphs <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, the bipartite Ramsey number pair <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math>, denoted by <math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><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><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math>, is an ordered pair of integers such that for any blue-red coloring of the edges of <math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math>, with <math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></math>, either a blue copy of <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> exists or a red copy of <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> exists if and only if <math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>a</mi></math> and <math><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>b</mi></math>. In [4], Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in [7] and [8], that <math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>

对于双胞图 G1,G2,...,Gk，双胞拉姆齐数 b(G1,G2,...,Gk)是最小的正整数 b，使得 Kb,b 的边的任何着色都有 k 种颜色，在第 i 种颜色下将得到 Gi 的副本。对于双分部图 G1 和 G2，双分部拉姆齐数对（a,b）（用 bp(G1,G2)=(a,b) 表示）是一对有序整数，对于 Ka′、b′，当且仅当 a′≥a，b′≥b 时，要么存在 G1 的蓝色副本，要么存在 G2 的红色副本。在 [4] 中，Faudree 和 Schelp 考虑了涉及路径的双方位拉姆齐数对。最近，Joubert、Hattingh 和 Henning 在 [7] 和 [8] 中证明，对于足够大的正整数 s，bp(C2s,C2s)=(2s,2s-1) 和 b(P2s,C2s)=2s-1。具体来说，假设 s 和 r 是足够大的正整数。我们将证明，如果 r≥s+1 时，bp(C2s,P2r+1)=(s+r,s+r-1)；如果 r=s+1 时，bp(P2s+1,C2r)=(s+r,s+r)；如果 r≥s+2 时，bp(P2s+1,C2r)=(s+r-1,s+r-1)。

引用次数: 0

A method for constructing graphs with the same resistance spectrum 构建具有相同电阻谱的图形的方法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-14 DOI: 10.1016/j.disc.2024.114284

Si-Ao Xu, Huan Zhou, Xiang-Feng Pan

Let G be a simple graph with vertex set

V (G)

and edge set

E (G)

. The resistance distance

R_{G} (x, y)

between two vertices

x, y

of G, is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance spectrum

RS (G)

of a graph G is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer k, there exist at least

2^{k}

graphs with the same resistance spectrum. Furthermore, it is shown that for

n \geq 10

, there are at least

2 ((n - 10) p (n - 9) + q (n - 9))

pairs of graphs of order n with the same resistance spectrum, where

p (n - 9)

and

q (n - 9)

are the numbers of partitions of the integer

n - 9

and simple graphs of order

n - 9

, respectively.

假设 G 是一个简单图，具有顶点集 V(G) 和边集 E(G)。G 的两个顶点 x,y 之间的电阻距离 RG(x,y) 定义为两个顶点之间在相应电网络中的有效电阻，其中 G 的每条边都由一个单位电阻代替。图 G 的电阻谱 RS(G) 是图中所有顶点对之间电阻距离的多集。本文提出了一种构建具有相同电阻谱的图的新方法。结果表明，对于任意正整数 k，至少存在 2k 个具有相同电阻谱的图。此外，对于 n≥10，至少存在 2((n-10)p(n-9)+q(n-9)) 对具有相同阻力谱的 n 阶图形，其中 p(n-9) 和 q(n-9) 分别是整数 n-9 和 n-9 阶简单图形的分区数。

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引用次数: 0

Star-critical Ramsey numbers involving large books 涉及大型图书的星形临界拉姆齐数字

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-11 DOI: 10.1016/j.disc.2024.114270

Xun Chen , Qizhong Lin , Lin Niu

<div><div>For graphs <math><mi>F</mi><mo>,</mo><mi>G</mi></math> and H, let <math><mi>F</mi><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math> signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number <math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math> is defined to be the smallest integer r such that <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math>. Let <math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math> be the book graph which consists of n copies of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math> all sharing a common <math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math>, and let <math><mi>G</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math> be the complete <math><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math>-partite graph with <math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math>, <math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> and <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math>.</div><div>In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed <math><mi>p</mi><mo>≥</mo><mn>1</mn></math>, <math><mi>k</mi><mo>≥</mo><mn>2</mn></math> and sufficiently large n, <math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>2</mn></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo></math>. This implies that the star-critical Ramsey number <math><msub><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><m

对于图 F、G 和 H，让 F→(G,H)表示 F 的任何红/蓝边着色都包含一个红色的 G 或一个蓝色的 H。拉姆齐数 r(G,H) 定义为 Kr→(G,H)的最小整数 r。设 Bn(k) 是由 n 份 Kk+1 组成的书本图，所有 Kk+1 都共享一个共同的 Kk；设 G:=Kp+1(a1,a2,...,ap+1) 是完整的 (p+1) 部分图，其中 a1=1, a2|(n-1) 且 ai≤ai+1 。本文避免使用 Szemerédi 的正则性 Lemma，证明对于任意固定的 p≥1，k≥2 和足够大的 n，Kp(n+a2k-1)+1ȖK1,n+a2-2→(G,Bn(k))。这意味着星形临界拉姆齐数 r⁎(G,Bn(k))=(p-1)(n+a2k-1)+a2(k-1)+1。由此推论，当 a1=a2=1 且 ai≤ai+1 时，r⁎(G,Bn(k))=(p-1)(n+k-1)+k。这就以更强的形式解决了郝和林（2018）[11]提出的问题。

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