Counting triangles in regular graphs

Abstract

In this paper, we investigate the minimum number of triangles, denoted by $t\left(n,k\right)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andrásfai–Erdős–Sós Theorem has established that $t\left(n,k\right)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t\left(n,k\right)$ 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 $t\left(n,k\right)$ 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 $n$.