Tian–Dokyeesun–Klavžar Conjecture 3.4 refutation

Conjecture 3.4 in section 3.2 of arXiv:2510.01294v2 states:

If x is not a cut vertex of a graph G, then gp_o(G-x) <= gp_o(G) + deg_G(x).

The formal claim ranges over all finite simple connected graphs and every non-cut vertex. Deletion is the actual induced graph on vertices other than x. Non-cut means that deletion does not increase the number of connected components. This retains K1, with empty deletion, zero empty outer number, and singleton outer number one.

A pair is Z-positionable when every shortest path between its endpoints has no internal vertex in Z. An outer general position set requires this for every pair with at least one endpoint in Z. The outer number is the maximum cardinality among all such subsets.

The theorem D5/S3/Combinatorics/Graph/TianDokyeesunKlavzarOuterGeneralPositionRefutation.result negates the complete finite assertion. The counterexample has 19 vertices, an actual non-cut vertex of degree two, deletion outer number at least eight, and original outer number at most five.

A replacement general upper bound is not determined. The local proof does not establish universal formulas for the outer numbers of a parameterized family, or the source's separate simplicial-vertex and conditional lower bounds.

Begin with a cycle of length twelve. Replace two opposite vertices by independent sets A and B of four vertices, preserving their cycle neighbors. This graph H has 18 vertices. Add x adjacent to one neighbor of A and the corresponding neighbor of B in the same cycle direction. The new graph G has 19 vertices, and H is isomorphic to the actual induced deletion G-x.

Exact finite certificates construct walks descending the proposed distance by one at each adjacent step and bound every walk below by that distance. Consequently the two distance tables are the native graph distances. The shortest-path condition is equivalent to excluding every geodesic triple with distinct selected endpoint and selected internal vertex.

The eight vertices in A union B satisfy this condition in H. Mapping walks in both directions through the deletion isomorphism transfers native distances and the eight-vertex set to G-x. For G, five colors suffice for an obstruction assignment: every distinct same-color pair admits a shortest path with one selected endpoint and the other vertex internally. Every outer set is therefore injectively colored, so it has size at most five. Both graphs are connected and x has degree two. The proposed bound would imply 8 <= 5+2.

An incorrect edge, failed distance certificate, missing same-color obstruction, failure of the induced-deletion isomorphism, or mismatch between the formal shortest-path predicate and the source definition would invalidate the argument. A restriction in a later source would limit attribution to that source, without changing the accessed v2 claim.

The frozen module defines induced deletion, the component-count non-cut predicate, all-shortest-path outer position, its finite maximum, and the full closed claim. Its sole public theorem is result : Not claim. All certificate and transfer facts are local to that proof. The axiom closure is propext, Classical.choice, and Quot.sound.

The finite universal bound is refuted by the formal result. This is a Tier 1 external named conjecture preregistered in issue 12543. The version-specific admission basis is open-problem-resolution; the proof shape is content, with the public refutation produced by the local graph certificate and walk-construction chain. Its computational use is certified-instance with the exact refutes relation from result to claim.

Qualification is confined to Conjecture 3.4 in section 3.2 of arXiv:2510.01294v2. The author-linked v2 PDF and the arXiv v2 PDF have the same SHA-256: 642fcda6fa53788475acad3dbc8232df3928be28707ec30093d370c46bf8db70. The author's publication page reports Last modified 2026-09-17. The later survey arXiv:2501.19385v5, revised 2026-08-16, cites the journal publication and summarizes removal results without settling this degree-dependent conjecture in the accessed summary. No exact settlement was identified in the accessible checked scope. Inaccessible bodies are outside that scope. This bounded qualification is not a worldwide absence or priority claim; a prior exact settlement would disqualify this route.

What the settlement shows

Proved in the frozen module: for this witness, adding x supplies a short route between the two replacement classes. Shortest paths then obstruct selected pairs in G, while deleting x restores the eight-vertex outer set. The two-neighbor degree budget therefore fails to bound the increase.

Source-attested neighboring statements, not settled by this module: Theorem 3.3 is a conditional lower bound and Proposition 3.5 concerns simplicial vertices. Here x has two nonadjacent neighbors and is not simplicial. This witness does not refute either separate statement. It supplies no verdict on other source results or questions; dependencies on the false bound require separate examination.

Open: a corrected general upper bound, sufficient structural restrictions, and any parameterized extension. The result does not prove general formulas for a family of graphs.

The body of the later journal publication, DOI 10.1016/j.dam.2026.02.044, has not been read here. Its wording, numbering and settlement status are unverified. Attribution and qualification are confined to the accessed arXiv v2. Novelty, exhaustive literature completeness and publication priority are unverified. The frozen refutation is D5/S3/Combinatorics/Graph/TianDokyeesunKlavzarOuterGeneralPositionRefutation.result, with result statement identity sha256:cbf825b91d7bb0631ec60152fd9cd48a3ac8267ec06d6014075741f7e5ffac71.