On the minimum degree required for a triangle decomposition

TL;DR

This paper proves that for large graphs with minimum degree at least 0.852 times the number of vertices, a fractional triangle decomposition exists, refining previous bounds and linking to exact decompositions under certain conditions.

Contribution

It improves the minimum degree threshold for fractional triangle decompositions from 0.9n to 0.852n using refined methods.

Findings

Graphs with minimum degree ≥ 0.852n have fractional triangle decompositions.

The result approaches the exact decomposition threshold under parity and divisibility conditions.

Refinement of existing methods enables tighter bounds on graph decompositions.

Abstract

We prove that, for sufficiently large $n$, every graph of order $n$ with minimum degree at least $0.852n$ has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross to establish a bound of $0.9n$. By a result of Barber, K\"{u}hn, Lo and Osthus, our result implies that, for each $\epsilon >0$, every graph of sufficiently large order $n$ with minimum degree at least $(0.852+\epsilon)n$ has a triangle decomposition if and only if it has all even degrees and number of edges a multiple of three.