Two conjectures predicted that graphs with enough edges must contain 2-connected subgraphs of every intermediate size — that the spectrum of 2-connected subgraphs would be continuous, with no gaps between the smallest and largest.
Liu and Ning (arXiv:2603.11662) disprove both. The counterexamples come not from graph theory but from combinatorial design theory. Symmetric balanced incomplete block designs — objects defined by parameters (v, k, λ) satisfying a counting identity — provide the exact structure needed to create graphs with large 2-connected subgraphs and small 2-connected subgraphs but nothing in between.
The obstruction is combinatorial, not topological. The graphs are well-connected globally. They have high minimum degree. They contain many cycles. But the 2-connected subgraphs cluster at two size scales with a gap in the middle. The designs enforce this clustering because their regularity creates substructures of uniform size — blocks of exactly k vertices — that combine into larger 2-connected structures only at specific multiples.
The replacement conjecture that Liu and Ning propose has a cleaner threshold, informed by the structure of the counterexamples. The original conjectures assumed that edge density would force size interpolation. The block designs show that regularity — every pair of elements in exactly λ blocks, every element in exactly r blocks — creates a combinatorial rigidity that prevents interpolation despite having enough edges.
The tool that provided the counterexample came from a different branch of mathematics entirely. Graph theorists were looking for obstructions inside graph theory. The obstructions live in design theory — in the constraints that counting conditions impose on combinatorial structure.