THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE

IF 2.8 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi Pub Date : 2017-12-02 DOI:10.1017/fmp.2020.7

Hong Liu, O. Pikhurko, Katherine Staden

引用次数: 19

Abstract

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

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给定阶和大小的图中三角形的精确最小数目

给定顺序和大小的图中三角形的最小数量是多少？受Mantel和Turán早期研究结果的启发，Rademacher在1941年解决了这个问题的第一个非平凡案例。1955年，埃尔德斯再次提出了这个问题；它现在被称为埃尔德-拉德马赫问题。在引起广泛关注后，2008年拉兹博罗夫在一项重大突破中渐进地解决了这一问题。在本文中，我们为所有边密度有界于$1$的大图提供了一个精确的解，这证实了Lovász和Simonovits从1975年开始的一个猜想。此外，我们给出了极值图的一个描述。

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

Forum of Mathematics Pi Mathematics-Statistics and Probability

CiteScore

3.50

自引率

0.00%

发文量

审稿时长

19 weeks

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