Metadata
bibkey: wangtian2026strongsaturation
authors: Qinglin Wang; Yingzhi Tian
year: 2026
title: Extremal Problems for the Family of k-Strongly Connected Digraphs
doi: 10.48550/arXiv.2605.01269
url: https://arxiv.org/abs/2605.01269v1
claim: "Section 4, Conjecture 1 states the unrestricted exact eventual extremal arc formula for the family of k-strongly connected strict digraphs."
strata_touched:
- D5/S3/Combinatorics/Graph/Saturation/WangTianStrongSaturationRefutation
license: citation-only
triage: anchor
Exact strong-digraph saturation formula
The source is arXiv:2605.01269v1. Section 1 defines finite strict directed graphs, directed dipaths, strong connectivity, separators and vertex-strong connectivity. Reciprocal arcs are permitted. A separator leaves a graph that is not strongly connected or is trivial. The finite model uses singleton for trivial and excludes the empty graph from positive strong connectivity; the source does not separately specify null connectivity. This convention preserves the consequence that a k-strong graph has at least k+1 vertices used in the proof of Lemma 2.2. The cited textbook’s null-graph convention is not asserted.
Saturation excludes every k-strong subdigraph and requires that adding any missing ordered arc creates one. Subdigraphs permit arbitrary vertex and arc subsets, with isolated vertices retained. The extremal number is the maximum arc cardinality among saturated n-vertex hosts, rather than among induced subgraphs or only spanning forbidden copies.
Section 4, Conjecture 1 (printed PDF p. 13) states:
For sufficiently large n,
The displayed formula has no rounding, divisibility condition or error term. The full formal claim quantifies every positive k, a threshold depending on k, and every n beyond that threshold. The local Section 4 context uses k at least two; the k=3 obstruction applies to that reading as well. Theorem 4.2’s construction has a divisibility-sensitive equality condition, which is not present in the conjecture.
The repository’s argument uses k=3 and n=2N+5 for every proposed threshold N. Twice the putative extremal value would be 2(choose(2N+5,2)+5N+7)+1, contradicting its evenness. The formal extremum is an actual finite maximum with a proved nonempty saturated family, an attaining host and domination over every saturated host. The proof is a repository derivation from integrality; the paper is the source of the target, not an attribution of this refutation to Wang and Tian. The labelled Fin n presentation and its definition-level bijective transport are explained in the corresponding Blueprint.
This argument only addresses the literal unrestricted exact equality. It does not settle corrected rounded, divisible-order, upper-bound or asymptotic variants. The related Bernshteyn–Kostochka work, arXiv:1504.03758v2, concerns an undirected upper-bound problem and supplies no graph theorem used here. The original Mader chapter is not invoked as proof. The bounded qualification of this target found no previous exact correction or refutation in the inspected sources; unavailable citation-graph routes give no negative evidence. No exhaustive literature search or global novelty assertion is made.
Verified locator
- DOI: https://doi.org/10.48550/arXiv.2605.01269
- URL: https://arxiv.org/abs/2605.01269v1
- HTML: https://arxiv.org/html/2605.01269v1#Thmconj1
- PDF: https://arxiv.org/pdf/2605.01269v1
- Section 1 definitions, Lemma 2.2, Section 4 Theorem 4.2 and Conjecture 1; conjecture on printed PDF page 13.