¿Are critical k-forests tight?

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $k\geq2$ and a finite set $V$ with $|V|\geq k$. A $k$-uniform hypergraph is $H=(V,E)$ with $E\subseteq\binom Vk$. A $k$-colouring means a surjection $c:V\to[k]=\{1,\ldots,k\}$. An edge is rainbow when $c(e)=[k]$. The hypergraph is a $k$-forest if each $e\in E$ is the unique rainbow edge for some $k$-colouring. It is critical (saturated) if adding any missing $k$-element edge on the same vertex set destroys this property. It is tight if every surjective $k$-colouring has a rainbow edge; a tight $k$-forest is a $k$-tree.

Statement recorded in the supplied source.

Must every critical $k$-forest be tight, and hence be a $k$-tree? This is the historical conjecture; the cited primary paper gives a saturated $3$-forest on six vertices which is not tight. [S2] [S3]

Short English statement

Does being inclusion-maximal among these hypergraph forests force the corresponding connectivity property? The historical assertion has a six-vertex counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3293 (OPG-547). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, ¿Are critical k-forests tight?, node 547, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/are_critical_k_forests_tight.
• S3 Heidi Gebauer, Anna Gundert, Robin A. Moser and Yoshio Okamoto, Not All Saturated 3-Forests Are Tight (2011), Section 2 and Theorem 2.2. https://arxiv.org/pdf/1109.3390.