Progress towards Nash-Williams' Conjecture on Triangle Decompositions

TL;DR

This paper advances the understanding of Nash-Williams' conjecture by proving that graphs with minimum degree at least approximately 0.8273n admit fractional triangle decompositions, bringing us closer to the conjecture's full resolution.

Contribution

The paper proves that graphs with minimum degree at least 0.827327n have fractional triangle decompositions, improving bounds towards Nash-Williams' conjecture.

Findings

Graphs with minimum degree ≥ 0.827327n admit fractional triangle decompositions.

This result, combined with existing work, implies large graphs with minimum degree ≥ 0.82733n have triangle decompositions.

Progress towards Nash-Williams' conjecture on triangle decompositions.

Abstract

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1. We show that for any graph on $n$ vertices with minimum degree at least $0.827327 n$ admits a fractional triangle decomposition. Combined with results of Barber, K\"{u}hn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on $n$ vertices with minimum degree at least $0.82733 n$ admits a triangle decomposition.