Pith Number

pith:KWRKIDHJ

pith:2026:KWRKIDHJRG44FOPB56KPJRC63B

not attested not anchored not stored refs resolved

Graceful Labeling of Two Families of Spiders

Singling Shan, Yucheng Zhong

Spiders with sufficiently rapidly increasing leg lengths are graceful.

arxiv:2605.14295 v1 · 2026-05-14 · math.CO

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\usepackage{pith}
\pithnumber{KWRKIDHJRG44FOPB56KPJRC63B}

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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

Any spider with legs L1,...,Ls (s >= 1) satisfying |E(L2)| >= 2|E(L1)| + 4 and |E(Li+1)| >= 2|E(Li)| + 2 for i in {2,...,s-1} is graceful.

C2weakest assumption

The condition that n is not congruent to 1 modulo 4 in the relaxed path-attachment theorem, together with the leg-length inequalities that define the spider families.

C3one line summary

Spiders whose leg lengths satisfy |E(L2)| >= 2|E(L1)| + 4 and |E(Li+1)| >= 2|E(Li)| + 2 are graceful, with an explicit construction also given for spiders having one arbitrary-length leg and all others of length at most two.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] P. Bahls, S. Lake, and A. Wertheim. Gracefulness of families of spiders.Involve, 3(3):241–247, 2010 2010

[2] R. Cattell. Graceful labellings of paths.Discrete Math., 307(24):3161–3176, Nov. 2007 2007

[3] J. A. Gallian. A dynamic survey of graph labeling.Electronic Journal of Combinatorics, DS6, 2025. 28th edition, October 30, 2025 2025

[4] S. W. Golomb. How to number a graph.Graph theory and computing, pages 23–37, 1972 1972

[5] Hrnc´ ıar and A 2001

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-17T23:39:10.157685Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

55a2a40ce989b9c2b9e1ef94f4c45ed84eb12d9c0608236b8a7b573e1bf209e2

Aliases

arxiv: 2605.14295 · arxiv_version: 2605.14295v1 · doi: 10.48550/arxiv.2605.14295 · pith_short_12: KWRKIDHJRG44 · pith_short_16: KWRKIDHJRG44FOPB · pith_short_8: KWRKIDHJ

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/KWRKIDHJRG44FOPB56KPJRC63B \
  | 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: 55a2a40ce989b9c2b9e1ef94f4c45ed84eb12d9c0608236b8a7b573e1bf209e2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0c653e14031fdbb423077aa319a36fec73eea111e39de54da43d48ab0da10074",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-14T02:55:56Z",
    "title_canon_sha256": "289f966d73d945c4cef0be05bc9794162202fe180ff174fa4358237a0292cdd1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14295",
    "kind": "arxiv",
    "version": 1
  }
}