On fractional triangle decompositions of random graphs

TL;DR

This paper proves that random graphs with edge probability above a certain threshold almost surely admit a fractional triangle decomposition, improving previous bounds and providing an algorithmic construction method.

Contribution

It establishes a lower threshold for the existence of fractional triangle decompositions in random graphs and introduces an iterative algorithmic approach to find such decompositions.

Findings

Fractional triangle decompositions exist with high probability for p ≥ n^(-4/11+o(1))

The method improves previous bounds from p ≥ n^(-1/3+o(1))

An algorithmic process constructs the fractional decomposition iteratively

Abstract

We prove that with high probability $G(n,p)$ with $p \geq n^{-4/11 + o(1)}$ admits a fractional triangle decomposition (FTD), i.e., a nonnegative weighting of its triangles such that for each edge, the total weight of the triangles containing it equals one. This improves on the state of the art, due to Delcourt, Kelly, and Postle, that $p \geq n^{-1/3+o(1)}$ suffices. The proof is algorithmic: Given $G \sim G(n,p)$, we first construct an approximate FTD by taking a uniform weighting of the triangles. We then use specialized gadgets to iteratively shift weights and obtain successively better approximations of an FTD.