The exact minimum number of triangles in graphs of given order and size
Combinatorics
2020-04-27 v2
Abstract
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s in 1955; it is now known as the Erd\H{o}s-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~, which in this range confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.
Keywords
Cite
@article{arxiv.1712.00633,
title = {The exact minimum number of triangles in graphs of given order and size},
author = {Hong Liu and Oleg Pikhurko and Katherine Staden},
journal= {arXiv preprint arXiv:1712.00633},
year = {2020}
}
Comments
Published in Forum of Mathematics, Pi, Volume 8, e8 (2020)