Nash-Williams' Triangle Decomposition Conjecture

• Nash-Williams' Conjecture is a pivotal result asserting that any triangle-divisible graph with a minimum degree of 3n/4 can be decomposed exactly into triangles.
• It introduces a fractional perspective by showing that a 3/4-minimum degree is necessary for achieving a fractional K3-decomposition, bridging the gap between theory and application.
• The study leverages absorption methods and discharging techniques, providing robust tools that inform related problems in extremal combinatorics and design theory.

Nash-Williams' Conjecture is a central result in extremal design theory concerning triangle decompositions of graphs. It asserts a precise minimum degree threshold for when a triangle-divisible graph can be edge-partitioned into triangles, and its full resolution (Delcourt et al., 9 Jun 2026) closes a decades-old question by identifying the exact barrier.

1. Statement and Motivation of Nash-Williams' Conjecture

A graph $G$ on $n$ vertices is triangle-divisible if:

• Every vertex $v\in V(G)$ has even degree: $d_G(v)\equiv 0\pmod2$,
• The total number of edges $e(G)$ is divisible by 3: $3 \mid e(G)$,

These constraints are plainly necessary for $G$ to have a $K_3$-decomposition, that is, an edge-partition into triangles.

Nash-Williams' Conjecture (1970).

Let $G$ be a triangle-divisible graph on $n$ vertices. If the minimum degree satisfies

$n$0

and $n$1 is sufficiently large, then $n$2 admits an exact $n$3-decomposition (an edge-partition into triangles).

This bound is tight: Graham's extremal construction (the join of two $n$4-vertex graphs, each $n$5-regular, joined by all cross-edges) demonstrates that $n$6 cannot be reduced in general.

The conjecture sits at the intersection of extremal combinatorics, design theory, and the theory of Steiner triple systems; it is the graph-theoretic analogue of the existence of $n$7-Steiner systems and generalizes the divisible complete graph case solved by Kirkman.

2. Fractional Relaxation: The Fractional Nash-Williams' Conjecture

A fractional $n$8-decomposition is a function $n$9 from triangles of $v\in V(G)$0 to $v\in V(G)$1 so that

$v\in V(G)$2

Define the fractional threshold

$v\in V(G)$3

Extremal examples show $v\in V(G)$4 is necessary.

The Fractional Nash-Williams' Conjecture posits that this lower bound is tight: If $v\in V(G)$5 is an $v\in V(G)$6-vertex graph with $v\in V(G)$7, then $v\in V(G)$8 admits a fractional $v\in V(G)$9-decomposition.

(Delcourt et al., 9 Jun 2026) provides the first proof of this conjecture, establishing $d_G(v)\equiv 0\pmod2$0.

Proof Outline for the Fractional Result

• By Farkas’ Lemma, failure of a fractional triangle decomposition leads to a certificate in the form of an assignment of real “charges” $d_G(v)\equiv 0\pmod2$1 to the edges such that every triangle has nonnegative total charge but $d_G(v)\equiv 0\pmod2$2.
• The key technique is discharging in the dual: positive and negative edges discharge according to whether triangles are “acute” or “obtuse”.
• The argument proceeds with a series of technical reductions: removing contributions outside common neighborhoods, introducing edge “values”, proving the induced filtered subgraph is $d_G(v)\equiv 0\pmod2$3-free, and bounding its total weight via a graph Lagrangian.
• These rules force every negative edge’s final charge to be nonnegative, contradicting the assumed certificate.

This result implies, through the absorption and reduction machinery of Barber–Kühn–Lo–Osthus, that the asymptotic Nash-Williams theorem holds for any $d_G(v)\equiv 0\pmod2$4.

3. Fractional Stability and Extremality

Proving mere existence of fractional decompositions is insufficient for a robust conversion to an exact decomposition. Fractional stability concerns the structural characterization of near-extremal graphs which lack a fractional decomposition.

Fractional Stability Theorem:

For any small $d_G(v)\equiv 0\pmod2$5, there exists $d_G(v)\equiv 0\pmod2$6 such that if $d_G(v)\equiv 0\pmod2$7 has

$d_G(v)\equiv 0\pmod2$8

but fails to have a fractional $d_G(v)\equiv 0\pmod2$9-decomposition, then $e(G)$0 is $e(G)$1-extremal in the following sense:

• At least $e(G)$2 vertices have degree $e(G)$3,
• $e(G)$4.

The proof extends the discharging method, tracks charge propagation, and iteratively shows that any negative edge must live in a section of the graph nearly isomorphic to the extremal join construction.

4. Conversion: From Fractional to Exact Decomposition via Absorption

To move from fractional to exact decompositions, the absorption method is used:

1. Absorber construction: In non-extremal graphs with $e(G)$5, one constructs a bounded absorber subgraph $e(G)$6 such that for every small balanced leftover $e(G)$7 compatible with divisibility, $e(G)$8 has a perfect triangle decomposition.
2. Random-greedy packing: Most edges are packed greedily into triangles following the fractional decomposition, leaving a sparse leftover.
3. Final absorption: Pre-selected absorbers are then used to cover all remaining edges exactly.

Postle–Delcourt’s absorber-based “black-box” theorem allows this to be implemented even for graphs with partitioned vertex sets and is robust to block-structured extremal configurations.

Applied to each of the finitely many structural cases, this method establishes the final step: For all sufficiently large $e(G)$9, every triangle-divisible $3 \mid e(G)$0 with $3 \mid e(G)$1 has a $3 \mid e(G)$2-decomposition (Delcourt et al., 9 Jun 2026).

5. Interplay, Tightness, and Broader Significance

The resolution of Nash-Williams’ conjecture is structured in three central layers:

• The fractional threshold is exactly $3 \mid e(G)$3.
• Any graph near this threshold which fails to admit a decomposition must closely mimic the construction of two $3 \mid e(G)$4-regular halves, joined.
• The absorption method, paired with the robust fractional structure, effects the transition from fractional to exact decomposition.

This result settles the minimum-degree barrier for decomposing large triangle-divisible graphs into triangles, providing the tight bound for all $3 \mid e(G)$5.

Table: Central Results and Techniques

Result/Theorem	Essential Content	Reference
Nash-Williams' Conjecture (1970)	$3 \mid e(G)$6 forces triangle decomposition for large $3 \mid e(G)$7	(Delcourt et al., 9 Jun 2026)
Fractional Nash-Williams Conjecture	$3 \mid e(G)$8 (sharp bound for fractional)	(Delcourt et al., 9 Jun 2026)
Fractional Stability Theorem	Structural characterization of near-extremal non-fractional	(Delcourt et al., 9 Jun 2026)
Absorption method for exact case	Random-greedy + absorber yields full $3 \mid e(G)$9-decomposition	(Delcourt et al., 9 Jun 2026)

6. Connections and Impact

Triangle decomposition thresholds are deeply connected to design theory, regularity/absorption paradigms, and extremal combinatorics. Matching the fractional threshold to the integral threshold demonstrates the efficacy of robust absorption techniques in bridging fractional relaxations and integral packings.

The Nash-Williams conjecture now serves as a template for higher-order decompositional problems—see, for example, current work on $G$0-decompositions, hypergraph generalizations, and analogues in design theory (Zhang et al., 9 Oct 2025, Henderson et al., 3 Dec 2025, Zhang et al., 21 May 2026). The new technical tools for fractional stability and absorber construction have broad applicability across these domains.

The proof’s layered approach—fractional result, extremal characterization, and absorber-fueled conversion—is expected to inform parallel work in hypergraph and multipartite settings, where divisibility and stability phenomena are likewise paramount.