¿Are critical k-forests tight?

Discussion

We say that a hypergraph $ H=(V,E) $ is a $ k $ -graph if it is $ k $ -uniform, and denote its order by $ n=|V| $ and its size by $ m=|E| $ . Laszlo Lovasz introduced the following concept: a $ k $ -graph $ H=(V,E) $ is said to be a $ k $ -forest if for every edge $ e\in E $ there exists a $ k $ -colouing $ \varsigma\colon V\to[k] $ such that $ \varsigma(e')=[k]\Leftrightarrow e'=e $ ; that is, such that only the edge $ e $ receives the $ k $ colours in its vertices. Clearly a $ 2 $ -forest is simply a forest in the usual sense (i.e., an acyclic graph). Lovasz proved that Theorem A $ k $ -forest has size at most $ m\leq{n-1\choose k-1} $ . On the other hand, Victor Neumann-Lara introduced the following invariant: the heterochromatic number of a $ k $ -graph $ H=(V,G) $ is the minimum number of colours $ c $ such that, in every colouring $ \varsigma\colon V\to[c] $ there is an edge wich receives different colours in each of its vertices; that is, there exists $ e\in E $ such that $ |\varsigma(e)|=k $ . If the heterochromatic number and the rank are equal, the hypergraph is said to be tight . Clearly a $ 2 $ -graph is tight if and only if it is connected . A tight $ k $ -forest is called a $ k $ -tree . I can prove the following Theorem If a $ k $ -forest has size $ m={n-1\choose k-1} $ then it is tight — and therefore a $ k $ -tree. Finally, we say that a $ k $ -forest is critical if no edge can be added to it without loosing the property of being a $ k $ -forest; it is maximal (in size) with such a property. Observe that there are critical $ k $ -forests of size $ m<{n-1\choose k-1} $ , whenever $ k>2 $ . So, the conjecture is to motivate the question: ¿are critical $ k $ -forests tight?