Pith Number

pith:3EQCJ3G2

pith:2026:3EQCJ3G2JJ7YEREGIUIFMNY2PM

not attested not anchored not stored refs pending

Reconstructing a graph from its Bell colouring graph

Brian Hearn

Every n-vertex graph with no vertex of degree n-1 is uniquely determined by its Bell colouring graph.

arxiv:2604.13005 v2 · 2026-04-14 · math.CO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{3EQCJ3G2JJ7YEREGIUIFMNY2PM}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

every n-vertex graph G with no vertices of degree n-1 is uniquely determined by its Bell colouring graph B(G), and by its upper-Bell colouring graph B_{≥k}(G) if k≤n-2

C2weakest assumption

The graphs are finite and the uniqueness holds precisely when there are no vertices of degree n-1 (or the degree bound for the k-coloring case); if this degree condition fails the reconstruction may not be unique.

C3one line summary

Graphs without vertices of degree n-1 are uniquely determined by their Bell colouring graphs, which encode partitions into independent sets.

Receipt and verification

First computed	2026-06-25T01:18:38.087401Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d92024ecda4a7f824486451056371a7b0efbe0947133e1a7a88c999ab063162e

Aliases

arxiv: 2604.13005 · arxiv_version: 2604.13005v2 · doi: 10.48550/arxiv.2604.13005 · pith_short_12: 3EQCJ3G2JJ7Y · pith_short_16: 3EQCJ3G2JJ7YEREG · pith_short_8: 3EQCJ3G2

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3EQCJ3G2JJ7YEREGIUIFMNY2PM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d92024ecda4a7f824486451056371a7b0efbe0947133e1a7a88c999ab063162e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "189fe367fb6183e179bf23c0302197ffaf4c1c69657062f81718d8d65e857083",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-04-14T17:40:00Z",
    "title_canon_sha256": "3152921f4ece93c008c1230b8a3b44255be131fb99d50dbb5568dfd47046efe9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.13005",
    "kind": "arxiv",
    "version": 2
  }
}