Not All Saturated 3-Forests Are Tight

A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This resolves an open problem posed by Strausz.

💡 Research Summary

The paper investigates whether the classical graph‑theoretic fact “every inclusion‑maximal forest is connected (i.e., a tree)” extends to higher‑dimensional hypergraphs. Using the notion of a k‑forest introduced by Graham and Lovász, a k‑uniform hypergraph H is a k‑forest if for each edge e there exists a k‑coloring of the vertex set in which e is the unique rainbow edge (all its vertices receive distinct colors). This generalises the idea that every edge of a graph forest is a cut edge.

For connectivity the authors adopt the concept of tightness from Arocha, Bracho and Neumann‑Lara. The heterochromatic number hc(H) is the smallest integer t such that every t‑coloring of H contains a rainbow edge. A k‑graph is called tight when hc(H)=k. In ordinary graphs (k=2) tightness coincides with connectivity, so an inclusion‑maximal 2‑forest (a maximal forest) is automatically a tree.

Strausz asked whether the same implication holds for k>2: does every saturated (inclusion‑maximal) k‑forest have to be tight? The authors answer this question negatively for k=3 by constructing an explicit counterexample.

Construction. Let V={1,2,3,4,5,6} and define the 3‑uniform hypergraph H with eight edges: {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {2,3,6}, {2,5,6}, {3,4,6}, {3,5,6}.

(i) H is a 3‑forest. For each edge e the authors exhibit a concrete 3‑coloring γ(e) in which e is the only rainbow triple. A table of these colorings shows that any other triple of vertices receives at most two colors, hence is not an edge of H. Therefore the defining property of a 3‑forest holds.

(ii) H is saturated. Let (\bar{E}) be the set of all 3‑subsets of V that are not edges of H. For each existing edge e they compute the set Φ(e) of all 3‑colorings that make e the unique rainbow edge. They then determine Δ(e), the collection of non‑edges that become rainbow under every coloring in Φ(e). By exhaustive analysis they find that every non‑edge belongs to Δ(e) for some e; consequently, adding any non‑edge to H would destroy the 3‑forest property because the new hypergraph would lack a coloring that isolates a single rainbow edge. Hence H is inclusion‑maximal (saturated).

(iii) H is not tight. A single 3‑coloring c with assignments (1→3, 2→2, 3→2, 4→2, 5→2, 6→1) makes all edges bichromatic or monochromatic, so no edge is rainbow. This shows that hc(H)≥4, i.e., the heterochromatic number exceeds 3, and therefore H is not tight.

Thus the hypergraph H is a saturated 3‑forest that fails to be tight, disproving the conjectured equivalence for k=3. The paper also notes that for 4 or 5 vertices every saturated 3‑forest is tight, so the phenomenon first appears at six vertices. Moreover, the example demonstrates that saturated k‑forests can have fewer than (\left\lfloor\frac{n-1}{k-1}\right\rfloor) edges, contradicting a naïve extension of the graph case.

In conclusion, the authors resolve Strausz’s open problem by providing a concrete counterexample, thereby showing that the relationship between maximal acyclicity (forest) and connectivity (tightness) diverges in higher dimensions. The result invites further investigation into stronger notions of connectivity for hypergraphs and suggests that the classical tree‑forest duality does not generalise straightforwardly beyond graphs.