Counting triangles in regular graphs

TL;DR

This paper determines the exact minimum number of triangles in large $n$-vertex, $k$-regular graphs for the entire range where $k$ is between $rac{2n}{5}$ and $rac{n}{2}$, confirming a recent conjecture.

Contribution

It provides the precise value of $t(n,k)$ for all $k$ in the specified range, extending previous results and confirming a conjecture for large $n$.

Findings

Exact value of $t(n,k)$ for $rac{2n}{5}<k<rac{n}{2}$

Bridges the gap between previous bounds and results

Confirms a conjecture for sufficiently large $n$

Abstract

In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andr\'asfai-Erd\H{o}s-S\'os Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}<k<\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\frac{2n}{5}<k<\frac{n}{2}$. This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large $n$.