Pandey–Patra subgraph core conjecture

Dinesh Pandey and Kamal Lochan Patra, Some new central parts of connected graphs, arXiv:2110.00738v1, Conjecture 3.25, page 9:

The subgraph core of a graph G is contained in a block of G.

The native domain is every finite simple connected undirected graph. The subgraph number counts all actual connected subgraphs containing a vertex, including singleton subgraphs and distinct edge choices. The conclusion is one native maximal no-cut block containing the entire set of maximizing vertices. Singleton, bridge-edge, empty-deletion, and unbounded-order cases remain in scope.

Conjecture 3.25 asks whether the vertices maximizing the number of connected subgraphs containing a vertex share one native block. The formal result settles that whole question while retaining the source's actual subgraph carrier and its maximal-block convention. The cited Library note is D5/L/Geometry/pandeypatra2021subgraphcore; the note's strata_touched entry remains the owning graph-invariant module.

Before this resolution, the cited paper stated the containment assertion as Conjecture 3.25. The remaining publication boundary is documentary: the formal source and frozen records already identify the exact theorem, while the two public theorem Reg audits remain unfinished under the linked-issue exception recorded below.

SubgraphCoreBlock.subgraphNumber uses the existing countContaining carrier, so it preserves both vertex and edge choices. subgraphCore is the complete maximizer set, and NativeBlock expresses native no-cut connectivity together with maximality under actual subgraph inclusion. The proof first bounds root-omitting subgraphs by an attachment injection, then uses exact one-root gluing to exclude maximizers in different deletion components. The union of all simple paths between maximizers has no cut vertex; finiteness extends it to a maximal native block. The path-hull lemma is a consumed helper, not another problem.

This resolution would fail to answer the source question if the formal carrier counted only induced or spanning subgraphs, if it selected only one maximizer, if NativeBlock weakened native maximality or deletion, or if the proof excluded singleton, bridge-edge, empty-deletion, or some finite order. It would also fail if the path-hull helper were asserted as a separate external problem. The delivered statement makes none of those substitutions.

The exact statement is proved by D5/S3/Combinatorics/Graph/Eccentric/SubgraphCoreBlock.result, with direct reuse of the existing De–Patra counting modules. No induced-only, spanning-only, single-maximizer, subclass, or order-cutoff statement is substituted. The formal axioms are the standard propext, Classical.choice, and Quot.sound closure reported for the source modules.

The consumed helper D5/S3/Combinatorics/Graph/Eccentric/SubgraphCoreHull.exists_no_cut_cover constructs the path hull used by the block proof; it is not a second external problem. The proof's live path uses the attachment injection, exact one-root gluing, deletion-component separation, and the union of all simple paths between maximizers.

theorem; this is one named external problem and one typed OpenProblemResolutionClaim(Proved). The settlement proves the full native quantifier and supplies one shared maximal no-cut block for the complete maximizing set. It does not claim priority, originality, worldwide openness, or solved credit. No experimental program or numerical certificate is used; the utility classification is none because the delivered mathematics is a general symbolic graph construction.

The two new public theorem Reg audits are unfinished and remain covered only by the precise linked-issue exception at https://github.com/the-omega-institute/trureturing/issues/23700#issuecomment-6499492075; the exception permits delivery and is not registration evidence.

The dossier does not establish that the source's historical wording was still open worldwide beyond the pre-registered literature reading, and it makes no priority or originality claim. The linked Reg exception is unfinished audit evidence, not a declared validation. The formal theorem, source locator, and native boundary statements above are the delivered scope.