Fractional triangle decompositions in graphs with large minimum degree

TL;DR

This paper proves that large graphs with minimum degree slightly above 0.9n have fractional triangle decompositions, improving previous bounds and implying actual triangle decompositions for certain graphs.

Contribution

It improves the minimum degree threshold for fractional triangle decompositions from 0.956n to just above 0.9n, advancing understanding of graph decompositions.

Findings

Large graphs with minimum degree > (0.9 + ε)n have fractional triangle decompositions.

Improves previous minimum degree bounds from 0.956n to just above 0.9n.

Results imply triangle decompositions for certain graphs with minimum degree above 0.9n.

Abstract

A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the triangles containing any given edge is one. We prove that for all $\epsilon > 0$, every large enough graph graph on $n$ vertices with minimum degree at least $(0.9 + \epsilon)n$ has a fractional triangle decomposition. This improves a result of Garaschuk that the same result holds for graphs with minimum degree at least $0.956n$. Together with a recent result of Barber, K\"{u}hn, Lo and Osthus, this implies that for all $\epsilon > 0$, every large enough triangle divisible graph on $n$ vertices with minimum degree at least $(0.9 + \epsilon)n$ admits a triangle decomposition.