Pith Number

pith:2EDXS5UM

pith:2026:2EDXS5UMW3XA4VWW6SAGF3VO4N

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Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs

Alexander Milner, Dimitra Kosta, Oliver Clarke

For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.

arxiv:2605.13662 v1 · 2026-05-13 · math.AC · math.CO

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Claims

C1strongest claim

For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved.

C2weakest assumption

The toric ideals are homogeneous and the bouquet matrix is a monomial curve in A^3; without homogeneity the preservation across bouquets may fail.

C3one line summary

For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.

References

22 extracted · 22 resolved · 0 Pith anchors

[1] Springer Science & Business Media, 2012 2012

[2] Distance-reducing markov bases for sampling from a discrete sample space.Bernoulli, 11(5):793–813, 2005 2005

[3] Minimal systems of binomial generators and the indispensable complex of a toric ideal.Proceedings of the American Mathematical Society, pages 3443–3451, 2007 2007

[4] Markov complexity of monomial curves.Journal of Algebra, 417:391–411, 2014 2014

[5] Distance reducing markov bases.Journal of Pure and Applied Algebra, 229(10):108057, 2025 2025

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:17.301153Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42

Aliases

arxiv: 2605.13662 · arxiv_version: 2605.13662v1 · doi: 10.48550/arxiv.2605.13662 · pith_short_12: 2EDXS5UMW3XA · pith_short_16: 2EDXS5UMW3XA4VWW · pith_short_8: 2EDXS5UM

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N \
  | 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: d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d6f5cb6463c67c6488c1129caac6637315610715bbdd53ab7d03db1ac524619d",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AC",
    "submitted_at": "2026-05-13T15:19:19Z",
    "title_canon_sha256": "b7575570dd8e3cad34953ad59beb6f5d23cd17671fbe915a7520510b2c3640b8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13662",
    "kind": "arxiv",
    "version": 1
  }
}