Counting triangles in regular graphs

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 Andrásfai-Erdős-Sós 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 $n$.

Figures (2)

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Theorems & Definitions (12)

• Theorem 1.1: Lo, Lo09
• Conjecture 1.2: Cambie, de Verclos and Kang, CdVK
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Theorem 3.1
• Lemma 3.2: Lo Lo09
• Proposition 4.1
• proof
• Proposition 4.2