Metadata

bibkey: dovgosheyrovenska2026centraledges
authors: Oleksiy Dovgoshey and Olga Rovenska
year: 2026
title: On the Graph Corresponding to Center of Distances of Metric Space
doi: 10.48550/arXiv.2610.11596
url: https://arxiv.org/abs/2610.11596v1
claim: "Conjecture 4.2 characterizes three-central ultrametric edge minima against all metric spaces of the same order."
strata_touched:
  - D5/S3/Combinatorics/Geometry/DovgosheyRovenskaCentralEdges
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2610.11596

Source: https://arxiv.org/abs/2610.11596v1

Distance centers and Conjecture 4.2

Definition 1.1, p. 1

Let (Y,ρ) be a metric space. The center of distances of (Y,ρ) is the subset C(Y)⊆ℝ⁺, such that t∈ C(Y) if and only if the equation ρ(y,p)=t has a solution for every p∈ Y.

Definition 1.2, p. 1

Let (Y,ρ) be a metric space and let C(Y) be the center of distances of (Y,ρ). The central graph CG_Y of (Y,ρ) is a simple graph such that V(CG_Y)=Y and {u,v}∈ E(CG_Y) ⇔ ρ(u,v)∈ C(Y) for all u,v∈ Y. We will write that the central graph CG_Y is k-central if k=|C(Y)|.

Definition 2.14, p. 4

Let (Y,ρ) be a finite metric space with |Y|≥ 2. The diametrical graph G_Y of (Y,ρ) is a graph such that V(G)=Y and, for all u,v∈ Y, {u,v}∈ E(G_{Y}) ⇔ ρ(u,v)=diamY.

Conjecture 4.2, p. 13

Let (X,d) be a finite ultrametric space with |X|≥ 4, let the central graph CG_X be 3-central, and let G_X be the diametrical graph of (X,d). Then the following statements are equivalent. (i) The diametrical graph G_X is complete bipartite with parts X_1, X_2 satisfying the equality min{|X_1|,|X_2|}=2, and the central graph CG_X is the union G_X with star graphs S_1 and S_2, CG_X=G_X∪ S_1 ∪ S_2, such that V(S_1)=X_1 and V(S_2)=X_2. (ii) The inequality |E(CG_X)| ≤ |E(CG_Y)| holds for all metric spaces (Y,ρ) which have 3-central CG_Y and satisfy |Y|=|X|.

The source uses $R^{+} = [0, \infty)$ , including zero. Edges are unordered pairs of distinct vertices. The Lean predicates use MetricSpace, SimpleGraph.fromRel, Metric.diam, completeBipartiteGraph, subtype stars mapped into the original carrier, and graph joins. Nat.card counts center values and unordered edges. The comparator in clause (ii) has no ultrametric hypothesis.

The four-point refutation uses the induced 2-adic metric on the values $0, 1, 2, 3$ and the induced real metric on $0, 4, 5, 9$ . Their centers are ${0, 1/2, 1}$ and ${0, 4, 5}$ , and their central graphs have six and four edges, respectively. The first metric satisfies clause (i).

The literature screen recorded in issue #22906 found no settlement in its specified sources. Completeness of that screen, including OpenAlex citation coverage, is ASSUMED-UNVERIFIED. The restricted comparison with ultrametrics, the sharp unrestricted lower bound and the all-order realization are not settled by the Lean module.

The finite-family experiment is in trureturing-experiments, commit b43e249214c62d9db0aa173ebce49fee6c792470, under docs/reports/central-distance-ultrametric-edges/. The recorded command is python3 docs/reports/central-distance-ultrametric-edges/check.py; its exit code is 0 and its final line is ALL_CHECKS_OK. The checker SHA256 is 632f89f2254e25666c83fff36e78c82d263199b8f0f2091484c1bc4378b56005. It checks the separate all-order construction for orders 4 through 32 using exact rational arithmetic. These finite computations do not establish the unbounded statements in Lean.

Keyboard shortcuts