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.

Figures (3)

• Figure 1: The constant $c$ in the minimum degree fraction $1-\frac{1}{c\cdot q}$, for $b=\frac{q}{k+\alpha}$ with $k\in\mathbb{N}$ and $\alpha\in[0,1]$.
• Figure 2: Internal, cross and small edges in the join of $H$, $H$, $H$, and $I$.
• Figure 3: $H$-internal, cross, small and $I$-internal edges in the join of $H$, $H$, and $I$.

Theorems & Definitions (23)

• Conjecture 1.1: Nash-Williams nash1970unsolved
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5: Partite Version of Nash-Williams' Conjecture
• Conjecture 1.6: Partite Version of the Folklore Conjecture
• Theorem 1.7
• Proposition 2.1
• proof
• Theorem 2.2