首页/论文/1002.4304

arXiv 1002.43042010-02math.CO

McKay-Radziszowski子图计数猜想的一个证明

A proof of the McKay-Radziszowski subgraph counting conjecture

Alexander Engstrom

备注：: 11 pages; typos fixed

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

摘要：

证明了一个关于如何计数图的邻域中的诱导子图的定理。然后利用它证明了McKay和Radziszowski在研究拉姆齐理论时提出的一个子图计数恒等式。

英文摘要：

We prove a theorem on how to count induced subgraphs in neighborhoods of graphs. Then we use it to prove a subgraph counting identity conjectured by McKay and Radziszowski in there work on Ramsey theory.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A bipartite version of the Erdős $-$ McKay conjecture

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, Ω(n^2)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, Ω(n^2)\}$.

2022-11-08math.CO

具有指定度的稠密随机图的子图

Subgraphs of dense random graphs with specified degrees

令d =（d1，d2，...，dn）是具有偶和的非负整数的向量。证明了度序列为d的随机图的结构的一些基本事实，包括给定子图或诱导子图的概率。虽然有很多这样的结果，他们被限制在稀疏的情况下，只有少数例外。相反，我们的重点是平均度近似为n的常数分数的情况。我们的方法是多维鞍点法。这扩展了McKay和Wormald（1990）的枚举工作，并且类似于Greenhill和McKay（arXiv：math/0701600，2009）为二部图开发的理论。

2010-11-27math.COmath.PR

Ramsey graphs induce subgraphs of quadratically many sizes

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. Motivated by an old problem of Erdős and McKay, recently Narayanan, Sahasrabudhe and Tomon conjectured that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of $Θ(n^2)$ different sizes. In this paper we prove this conjecture.

2021-09-07math.CO

Ramsey图导出许多不同大小的子图

Ramsey graphs induce subgraphs of many different sizes

一个$n$顶点上的图称为\emph{$C$-Ramsey}，如果图的每个团或独立集的大小至多为$C \log n$。唯一已知的Ramsey图的构造在本质上是概率性的，并且通常认为这样的图具有许多与稠密随机图相同的性质。在这里，我们证明了这样一个性质：对于任何固定的$C>0$，每个$C$ -拉姆齐图上的$n$顶点诱导至少$n^{2-o(1)}$不同大小的子图。这个接近最优的结果与两个未解决的问题密切相关，第一个是由Erdés和McKay提出的，第二个是由Erdés、Faudree和Sós提出的，都是1992年的。

2017-09-07math.CO

R(4,5)=25 的形式化证明

A Formal Proof of R(4,5)=25

本文在HOL4中形式化证明Ramsey数R(4,5)=25，利用MiniSat SAT接口、泛化算法及带图同构检查的定制求解器，将计算证明的可信基础缩小至HOL4内核。

2024-06-12cs.LOmath.CO

稠密随机正则图中的子图分布

Subgraph distributions in dense random regular graphs

本研究解决McKay提出的问题，证明非星形连通图在稠密随机正则图中的拷贝数渐近高斯分布，分析方差与子图结构的关联，并推广至顶点置换不变有界度统计量的分布控制。

2023-05-08math.COmath.PR

更多相关论文收起更多论文

Rainbow spanning trees in random subgraphs of dense regular graphs

We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p…

2023-01-08math.CO

Gallai-Ramsey number for $K_{5}$

Given a graph $H$, the $k$-colored Gallai Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $H$. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erd{\H o}s-Hajnal conjecture for rainbow triangles. J. Combin. Theory Ser. B, 111:75-125, 2015.] conjectured the value of the Gallai Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when $H = K_{4}$. In this paper we attack the n…

2019-01-11math.CO

具有指定度数和禁止边的随机图中的子图

Subgraphs in random graphs with specified degrees and forbidden edges

该研究通过更精确的边切换论证，改进了带指定度数和禁止边的简单随机图及二部随机图中，子图包含与边禁止条件概率的界，扩展了适用的度序列范围和已有研究结果。

2025-10-28math.CO

Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs

We study subgraphs of Paley graphs of prime order $p$ induced on the sets of vertices extending a given independent set of size $a$ to a larger independent set. Using a sufficient condition proved in the author's recent companion work, we show that a family of character sum estimates would imply that, as $p \to \infty$, the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability $2^{-a}$, namely, a Kesten-McKay law wi…

2023-03-29math.COmath.NT

New strongly regular graphs derived from the G2(4) graph

We consider simple loopless finite undirected graphs. Such a graph is called strongly regular with parameter set (v,k,l,m), for short a srg(v,k,l,m), iff it has exactly v vertices, each of them has exactly k neighbours, and the number of common neighbours of any two different vertices is l if they are neighbours and m otherwise. The G2(4) graph is a well-known srg(416,100,36,20). In this article, we explicitly construct it and a certain subgraph E induced by 320 vertices in the same way as in an older article by this author. We discover some interesting properties of E and derive five strongly…

2018-05-09math.CO

基于类型的随机图的正则诱导子图的Sharp渐近行为

The lower-bound problem for regular induced subgraphs of type-based random graphs

该研究确定了基于类型的随机图模型族中正则诱导子图最大阶数相关函数f(n)的最优渐近常数√(2e)，指出相关猜想的反例需来自非基于类型的构造。

2026-08-17math.COmath.PR