A note on the random greedy triangle-packing algorithm

TL;DR

This paper analyzes a random greedy algorithm for constructing large partial Steiner-Triple-Systems, showing that it results in a triangle-free graph with a bounded number of edges, specifically at most on the order of n^{7/4} log^{5/4} n, with high probability.

Contribution

It provides probabilistic bounds on the size of the triangle-free graph produced by a random greedy process, a novel analysis for this type of stochastic algorithm.

Findings

Number of edges in the final graph is at most O(n^{7/4} log^{5/4} n) with high probability.

The process terminates with a triangle-free graph having significantly fewer edges than the complete graph.

The analysis offers new insights into the behavior of random greedy algorithms in combinatorial constructions.

Abstract

The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. We begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random from the collection of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph. In this note we show that with high probability the number of edges in the final graph is at most $ O\big( n^{7/4}\log^{5/4}n \big) $.