The minimal density of triangles in tripartite graphs

TL;DR

This paper determines the minimal triangle density in tripartite graphs with fixed edge densities, extending previous characterizations of conditions that guarantee triangle existence.

Contribution

It precisely identifies the weighted tripartite graph configuration with the lowest triangle density for given edge densities, generalizing earlier results.

Findings

Identifies the minimal triangle density configuration in tripartite graphs.

Extends previous characterizations of edge densities guaranteeing triangles.

Provides a weighted graph construction with minimal triangle density.

Abstract

We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomass\'e and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from $K_{3,3,3}$ has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.