A critical probability for biclique partition of Gn,p

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-01-12 DOI:10.1016/j.jctb.2023.12.005

Tom Bohman , Jakob Hofstad

引用次数: 0

Abstract

The biclique partition number of a graph $G = (V, E)$ , denoted $b p (G)$ , is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that $b p (G) \leq n - α (G)$ , where $α (G)$ is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if $G = G_{n, 1 / 2}$ then $b p (G) = n - α (G)$ with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take $G = G_{n, p}$ , where p is constant and less than a certain threshold value $p_{0} \approx 0.312$ . This verifies a conjecture of Chung and Peng for these values of p. We also show that if $p_{0} < p < 1 / 2$ then $b p (G_{n, p}) = n - (1 + Θ (1)) α (G_{n, p})$ with high probability.

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Gn 的双斜分区临界概率 p

图 G=(V,E) 的双骰子分割数表示 bp(G)，它是 G 的成对边缘相交的完整双骰子图的最小数目，这样 G 的每条边都正好属于其中之一。很容易看出，bp(G)≤n-α(G)，其中α(G)是 G 的独立集的最大大小。埃尔德在上世纪 80 年代猜想，对于几乎所有的图 G 来说，等式都成立；也就是说，如果 G=Gn,1/2 那么 bp(G)=n-α(G) 的概率很高。阿隆证明了这是错误的。我们证明，如果我们取 G=Gn,p，其中 p 为常数且小于某个临界值 p0≈0.312，那么厄尔多斯的猜想就是真的。我们还证明，如果 p0<p<1/2，则 bp(Gn,p)=n-(1+Θ(1))α(Gn,p) 的概率很高。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.