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arXiv · 1712.00633

The exact minimum number of triangles in graphs of given order and size

Hong Liu·Oleg Pikhurko·Katherine Staden
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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 non-trivial 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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Hong Liu, Oleg Pikhurko, Katherine Staden. 2020-04-24. The exact minimum number of triangles in graphs of given order and size. https://doi.org/10.1017/fmp.2020.7

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