Beyond Nash-Williams: Counterexamples to Clique Decomposition Thresholds for All Cliques Larger than Triangles

Abstract: A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every $K_3$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $3n/4$ has a $K_3$-decomposition. A folklore generalization of Nash-Williams' Conjecture extends this to all $q\ge 4$ by positing that every $K_q$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $\left(1-\frac{1}{q+1}\right)n$ has a $K_q$-decomposition. We disprove this conjecture for all $q\ge 4$; namely, we show that for each $q\ge 4$, there exists $c > 1$ such that there exist infinitely many $K_q$-divisible graphs $G$ with minimum degree at least $\left(1-\frac{1}{c\cdot(q+1)}\right)v(G)$ and no $K_q$-decomposition; indeed we construct them admitting no fractional $K_q$-decomposition thus disproving the fractional relaxation of this conjecture. Our result also disproves the more general partite version. Indeed, we even show the folklore conjecture is off by a multiplicative factor by showing that for every $\varepsilon > 0$ and every large enough integer $q$, there exist infinitely many $K_q$-divisible graphs $G$ with minimum degree at least $\bigg(1-\frac{1}{\left(\frac{1+\sqrt{2}}{2}-\varepsilon\right)\cdot (q+1)}\bigg)v(G)$ with no (fractional) $K_q$-decomposition.

• The paper presents explicit counterexamples that refute the generalized Nash-Williams threshold for Kq-decompositions for all q ≥ 4.
• It employs join, lexicographic product, and graph blow-up techniques to optimize minimum degree conditions and rule out both integral and fractional decompositions.
• The results reveal a significant gap between absorber methods and fractional relaxations, prompting a reevaluation of clique decomposition thresholds in design theory.

Counterexamples to Clique Decomposition Thresholds Beyond Nash-Williams

Introduction

This paper addresses a central problem in extremal design theory: the minimum degree threshold required for the decomposition of graphs into cliques of size $q$ ($K_q$-decomposition), generalizing the Nash-Williams Conjecture for triangle decompositions ($K_3$). The authors construct explicit counterexamples disproving the folklore generalization of Nash-Williams' Conjecture for all $q \geq 4$, including its fractional and partite relaxations. The results demonstrate that the conjectured threshold is not only incorrect, but also off by a multiplicative factor, fundamentally altering the landscape of clique decomposition theory.

Background and Conjectures

The Nash-Williams Conjecture posits that every $K_3$-divisible graph on $n$ vertices with minimum degree at least $3n/4$ admits a $K_3$-decomposition. Its folklore generalization for $K_q$-decompositions ($q \geq 4$) suggests a threshold of $(1 - 2/(q+1))n$ for the minimum degree. Previous constructions (Graham, Gustavsson, Garaschuk) supported the tightness of these thresholds for $K_3$ and $K_q$ respectively, and fractional relaxations were believed to be similarly tight.

The partite version, motivated by Latin squares and transversal designs, extends these conjectures to balanced $q$-partite graphs, with analogous divisibility and minimum degree conditions.

Main Results

The authors prove that for every $q \geq 4$, there exist infinitely many $K_q$-divisible graphs with minimum degree at least $(1 - c/(q+1))v(G)$ (for some $c > 1$) that do not admit a $K_q$-decomposition, nor even a fractional $K_q$-decomposition. Furthermore, for every $\epsilon > 0$ and sufficiently large $q$, counterexamples exist with minimum degree at least $1 - (1 + \sqrt{2} - \epsilon)/(q+1)$. The partite versions of these conjectures are also disproved via tensor product constructions.

Construction Methodology

The core construction is based on the join of multiple copies of a regular graph $H$ (with parameters $d$, $v$) and, for small $q$, additional "fractional parts" (either independent sets or regular graphs). The lexicographic product and graph blow-up techniques are used to generate infinite families of counterexamples from a single base graph.

The key technical insight is a pigeonhole principle argument: in the join of $b$ copies of $H$, every $K_q$ must contain at least $q-b$ internal edges. If the ratio of internal to cross edges is insufficient, no fractional $K_q$-decomposition exists. The authors optimize parameters to maximize the minimum degree while maintaining the non-existence of such decompositions.

For $q \geq 6$, the construction is straightforward; for $q = 4,5$, specialized constructions are required, involving joins with fractional parts and careful degree calculations.

Implications and Contradictory Claims

The paper establishes that the folklore threshold $(1 - 2/(q+1))$ is not only non-optimal, but the true threshold is strictly higher by a multiplicative factor approaching $1 + \sqrt{2} \approx 1.207$ for large $q$. This contradicts decades of intuition and prior results in the field, including the tightness of fractional relaxations. Notably, the threshold for embedding absorbers (as in Glock et al.) is below the fractional threshold, a phenomenon not previously observed in graph decomposition problems.

The results also show that the behavior for $K_3$ (triangles) is exceptional; the Nash-Williams Conjecture may still hold for $q = 3$, but fails for all larger cliques.

Theoretical and Practical Implications

The findings necessitate a reevaluation of minimum degree thresholds in design theory, particularly for clique decompositions and related packing problems. The disproval of the fractional and partite relaxations impacts the study of Latin squares, transversal designs, and orthogonal arrays, with consequences for combinatorial design, coding theory, and graph algorithms.

From a theoretical perspective, the paper opens new questions regarding the optimal threshold for $K_q$-decompositions, the structure of extremal graphs, and the interplay between fractional and absorber-based methods. The techniques developed may be applicable to other decomposition and packing problems, including hypergraphs and regular subgraph packings.

Future Directions

Determining the exact minimum degree threshold for $K_q$-decompositions remains open. The constructions suggest that further improvements may be possible, and the gap between absorber and fractional thresholds warrants deeper investigation. Extensions to hypergraph decompositions, approximate packings, and algorithmic aspects of clique decompositions are promising avenues for future research.

Conclusion

This work fundamentally alters the understanding of clique decomposition thresholds in extremal design theory, disproving longstanding conjectures for all cliques larger than triangles. The explicit constructions and technical arguments provide a new framework for analyzing decomposition problems, with significant implications for combinatorics and related fields. The exceptional nature of triangle decompositions and the newly discovered gap between fractional and absorber thresholds highlight the complexity and richness of graph decomposition theory.