A Proof of Nash-Williams' Conjecture

Abstract: A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $0.75 n$ has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree $cn$ for some constant $c\ge 0.75$, then Nash-Williams' Conjecture holds for any constant $c' > c$. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with $c= \frac{7+\sqrt{21}}{14} \approx 0.82733$. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if $G$ is a graph on $n$ vertices with minimum degree at least $\frac{3n}{4}$, then $G$ has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no fractional $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. We use this to prove that if a triangle-divisible graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.