Erd\H{o}s meets Nash-Williams

TL;DR

This paper advances the understanding of triangle decompositions in graphs by reducing a combined Erdős-Nash-Williams conjecture to a fractional relaxation, achieving results for graphs with high minimum degree using refined absorption methods.

Contribution

It reduces the Erdős meets Nash-Williams conjecture to a fractional relaxation and proves it for graphs with minimum degree at least 0.82733n, introducing refined absorption techniques.

Findings

Proves the combined conjecture for graphs with minimum degree ≥ 0.82733n.

Generalizes previous results on Nash-Williams and Erdős conjectures.

Introduces refined absorption method as an alternative to iterative absorption.

Abstract

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In 1970, Nash-Williams conjectured that every graph $G$ on $n$ vertices with minimum degree at least $3n/4$ (for $n$ large enough and satisfying the necessary divisibility conditions) has a triangle decomposition. In 1973, Erd\H{o}s conjectured that for each integer $g$, there exists a Steiner triple system on $n$ vertices with girth at least $g$ (provided that $n\equiv 1,3 \mod 6$ is large enough compared to the fixed $g$). In 2021, Glock, K\"uhn, and Osthus conjectured the common generalization of these two conjectures, dubbing it the ``Erd\H{o}s meets Nash-Williams' Conjecture''. In this paper, we reduce the combined conjecture to the fractional relaxation of the Nash-Williams' Conjecture. Combined with the best known fractional bound of Delcourt and Postle, this proves the combined conjecture above when $G$ has minimum degree at least $0.82733n$. We note that our result generalizes the seminal work of Barber, K\"uhn, Lo, and Osthus on Nash-Williams' Conjecture and the resolution of Erd\H{o}s' Conjecture by Kwan, Sah, Sawhney, and Simkin. Both previous proofs of those results used the method of iterative absorption. Our proof instead proceeds via the newly developed method of refined absorption (and hence provides new independent proofs of both results).