The Maximum Number of Triangles in a Graph of Given Maximum Degree

TL;DR

This paper proves a precise upper bound on the maximum number of triangles in a graph with a given maximum degree, resolving a conjecture and advancing understanding of graph triangle counts.

Contribution

It establishes an exact upper bound on triangle counts in graphs with bounded degree, confirming a conjecture by Gan-Loh-Sudakov.

Findings

Derived the maximum number of triangles as a function of degree and vertices

Confirmed the conjecture of Gan-Loh-Sudakov

Provided a tight bound applicable to all graphs with given maximum degree

Abstract

We prove that any graph on $n$ vertices with max degree $d$ has at most $q{d+1 \choose 3}+{r \choose 3}$ triangles, where $n = q(d+1)+r$, $0 \le r \le d$. This resolves a conjecture of Gan-Loh-Sudakov.