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

引用次数: 0

摘要

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Star-critical Ramsey numbers involving large books

For graphs

F, G

and H, let

F \to (G, H)

signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number

r (G, H)

is defined to be the smallest integer r such that

K_{r} \to (G, H)

. Let

B_{n}^{(k)}

be the book graph which consists of n copies of

K_{k + 1}

all sharing a common

K_{k}

, and let

G : = K_{p + 1} (a_{1}, a_{2}, . . ., a_{p + 1})

be the complete

(p + 1)

-partite graph with

a_{1} = 1

a_{2} | (n - 1)

and

a_{i} \leq a_{i + 1}

In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed

p \geq 1

k \geq 2

and sufficiently large n,

K_{p (n + a_{2} k - 1) + 1} ∖ K_{1, n + a_{2} - 2} \to (G, B_{n}^{(k)})

. This implies that the star-critical Ramsey number

r_{⁎} (G, B_{n}^{(k)}) = (p - 1) (n + a_{2} k - 1) + a_{2} (k - 1) + 1

. As a corollary,

r_{⁎} (G, B_{n}^{(k)}) = (p - 1) (n + k - 1) + k

for

a_{1} = a_{2} = 1

and

a_{i} \leq a_{i + 1}

. This solves a problem proposed by Hao and Lin (2018) [11] in a stronger form.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.

期刊最新文献

Spectral upper bounds for the Grundy number of a graph Transitive (q − 1)-fold packings of PGn(q) Truncated theta series related to the Jacobi Triple Product identity Explicit enumeration formulas for m-regular simple stacks The e−positivity of some new classes of graphs