Tripartite Version of the Corr\'adi-Hajnal Theorem

TL;DR

This paper proves a tripartite version of the Corradi-Hajnal Theorem, establishing conditions under which a tripartite graph contains a perfect triangle packing or has a specific extremal structure.

Contribution

It extends the Corradi-Hajnal Theorem to tripartite graphs, identifying minimum degree conditions for perfect triangle packings in this setting.

Findings

If each vertex connects to at least 2/3 of vertices in other classes, a perfect triangle packing exists.

Alternatively, the graph has a precise extremal structure with exactly 2/3 adjacency.

The result characterizes the threshold for triangle packings in tripartite graphs.

Abstract

Let $G$ be a tripartite graph with $N$ vertices in each vertex class. If each vertex is adjacent to at least $(2/3)N$ vertices in each of the other classes, then either $G$ contains a subgraph that consists of $N$ vertex-disjoint triangles or $G$ is a specific graph in which each vertex is adjacent to exactly $(2/3)N$ vertices in each of the other classes.