The number of triangles is more when they have no common vertex

TL;DR

This paper improves lower bounds on the number of triangles in graphs under certain conditions, extending classical theorems and proposing conjectures for further generalizations.

Contribution

It provides a new lower bound on the number of triangles in graphs with no universal vertex, and introduces conjectures for broader cases involving complete subgraphs and multiple extra edges.

Findings

Graphs without a universal vertex contain at least n-2 triangles.

Improves Erdős's bound from at least n/2 to n-2 under specific conditions.

Proposes conjectures for generalizations involving complete graphs and multiple constraints.

Abstract

By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erd\H{o}s gives a strengthening: there are not only one, but at least $\lfloor\frac{n}{2}\rfloor$ triangles. We give a further improvement: if there is no vertex contained by all triangles then there are at least $n-2$ of them. There are some natural generalizations when $(a)$ complete graphs are considered (rather than triangles), $(b)$ the graph has $t$ extra edges (not only one) or $(c)$ it is supposed that there are no $s$ vertices such that every triangle contains one of them. We were not able to prove these generalizations, they are posed as conjectures.