Pith Number

pith:ZS6QABNH

pith:2026:ZS6QABNHXN7P2RG2IKCRBJHBR3

not attested not anchored not stored refs resolved

A Ridge-Saturation Characterization of $\alpha$-Critical $\mathbf {W}_p$ Graphs

Do Trong Hoang, Eugen Mandrescu, Kevin Pereyra, Vadim E. Levit

Graphs that are α-critical and in W_p have three equivalent characterizations in graph, complex, and complement terms.

arxiv:2605.16838 v1 · 2026-05-16 · math.CO

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Record completeness

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4 Citations open

5 Replications open

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Claims

C1strongest claim

We characterize the graphs which are simultaneously α-critical and members of the class W_p. The characterization is stated in three equivalent languages. ... This gives an exact formula for the largest p for which a well-covered graph belongs to W_p.

C2weakest assumption

The three descriptions (graph-theoretic fibers, ridge degrees in the independence complex, and (r-1)-clique codegrees in the complement) are equivalent for α-critical members of W_p, relying on the prior definitions of α-criticality and the class W_p without additional verification steps shown in the abstract.

C3one line summary

Multi-language characterization of α-critical W_p graphs with saturation consequences, p-bounds, and sharp examples refuting a recent local sufficient condition outside the triangle-free case.

References

27 extracted · 27 resolved · 0 Pith anchors

[1] B. Andrásfai, P. Erdős, and V. T. Sós, On the connection be tween chro- matic number, maximal clique and minimal degree of a graph, Discrete Mathematics 8 (1974), no. 3, 205–218. 12 1974

[2] L. W. Beineke, F. Harary, and M. D. Plummer, On the critica l lines of a graph, Paciﬁc Journal of Mathematics 22 (1967), no. 2, 205–212. 2, 5 1967

[3] Berge, Some common properties for regularizable grap hs, edge- critical graphs and B-graphs, Annals of Discrete Mathematics 12 (1982), 31–44 1982

[4] S. Bermudo and H. Fernau, Lower bounds on the diﬀerential of a graph, Discrete Mathematics 312 (2012), 3236–3250. 2 2012

[5] I. D. Castrillón, R. Cruz, and E. Reyes, On well-covered, vertex decom- posable and Cohen–Macaulay graphs, Electronic Journal of Combina- torics 23 (2016), no. 2, 17 pp. 2 2016

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:25.443892Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ccbd0005a7bb7efd44da428510a4e18eddd887983671373ef56357bae2f15aef

Aliases

arxiv: 2605.16838 · arxiv_version: 2605.16838v1 · doi: 10.48550/arxiv.2605.16838 · pith_short_12: ZS6QABNHXN7P · pith_short_16: ZS6QABNHXN7P2RG2 · pith_short_8: ZS6QABNH

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZS6QABNHXN7P2RG2IKCRBJHBR3 \
  | 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: ccbd0005a7bb7efd44da428510a4e18eddd887983671373ef56357bae2f15aef

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3aa8f44668e110176887b93fc9b18b08695b9510e7cf54ba13bee961f38ea9f9",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-16T06:43:26Z",
    "title_canon_sha256": "e0a0e21c87ed69b775f8ab0282c88e388b86b81a3f02275e7e881c5d44ab8353"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16838",
    "kind": "arxiv",
    "version": 1
  }
}