Pith Number

pith:DVISGBZI

pith:2026:DVISGBZIMWXAYX2GPIATFSKKF6

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Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs

Peter J. Cameron, R. A. Bailey, Sanming Zhou

Necessary conditions for (a,b)-perfect sets in normal Cayley graphs are expressed using irreducible characters of the group.

arxiv:2605.17376 v1 · 2026-05-17 · math.CO

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Claims

C1strongest claim

With the help of these results we then obtain necessary conditions for the existence of an (a,b)-perfect set in a normal Cayley graph in terms of the irreducible characters of the underlying group.

C2weakest assumption

The derivations assume the graph is regular and that the equitable partitions exist in a form that allows direct application of character theory to the adjacency matrix or orbital structure of the normal Cayley graph.

C3one line summary

Derives necessary conditions for equitable partitions of regular graphs and for (a,b)-perfect sets in normal Cayley graphs using irreducible characters.

References

35 extracted · 35 resolved · 0 Pith anchors

[1] R. A. Bailey, P. J. Cameron, A. L. Gavrilyuk and S. V. Goryainov, Equitable partitions of Latin- square graphs,J. Combin. Des.27 (2019), no. 3, 142–160 2019

[2] R. A. Bailey, P. J. Cameron, D. Ferreira, S. S. Ferreira and C. Nunes, Designs for half-diallel experiments with commutative orthogonal block structure,J. Statist. Plann. Inference231 (2024), 106139 2024

[3] E. A. Bespalov, D. S. Krotov, A. A. Matiushev and K. V. Vorob’ev, Perfect 2-colorings of Hamming graphs,J Combin Des.29 (2021), no. 6, 367–396 2021

[4] N. L. Biggs, Perfect codes in graphs,J. Combin. Theory Ser. B15 (1973) 289–296 1973

[5] D. M. Cardoso, An overview of (κ, τ)-regular sets and their applications,Discrete Appl. Math.269 (2019) 2–10 2019

Formal links

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Receipt and verification

First computed	2026-05-20T00:03:55.422653Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1d5123072865ae0c5f467a0132c94a2f9c81b20654f0388ffe48753b0db8831f

Aliases

arxiv: 2605.17376 · arxiv_version: 2605.17376v1 · doi: 10.48550/arxiv.2605.17376 · pith_short_12: DVISGBZIMWXA · pith_short_16: DVISGBZIMWXAYX2G · pith_short_8: DVISGBZI

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DVISGBZIMWXAYX2GPIATFSKKF6 \
  | 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: 1d5123072865ae0c5f467a0132c94a2f9c81b20654f0388ffe48753b0db8831f

Canonical record JSON

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    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-17T10:37:36Z",
    "title_canon_sha256": "9b228948592c71e5f5b22444c73ffcd69a6b74a193a88c7c5bdec0d2e223b90b"
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}