{"paper":{"title":"Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Necessary conditions for (a,b)-perfect sets in normal Cayley graphs are expressed using irreducible characters of the group.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peter J. Cameron, R. A. Bailey, Sanming Zhou","submitted_at":"2026-05-17T10:37:36Z","abstract_excerpt":"An equitable partition of a graph $\\Ga$ is a partition $\\{V_1, \\ldots, V_m\\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a, b)$-perfect set in $\\Ga$, where $a$ is the number of neighbours in $V_1$ of each vertex in $V_1$, and $b$ is the number of neighbours in $V_1$ of each vertex in $V_2$. In this paper we first derive general necessary conditions for a regular graph to admit two equitable partitions. As a corollary we obtain necessary conditions for the existence of an $(a,b)$-perfect se"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives necessary conditions for equitable partitions of regular graphs and for (a,b)-perfect sets in normal Cayley graphs using irreducible characters.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Necessary conditions for (a,b)-perfect sets in normal Cayley graphs are expressed using irreducible characters of the group.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fb0e3d1eb2ef919e036719926a2f507859dd9599b791c15ea80cb57e0ba57944"},"source":{"id":"2605.17376","kind":"arxiv","version":1},"verdict":{"id":"3d489c30-98a2-4045-98bb-7c02186d736e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:02:47.816668Z","strongest_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.","one_line_summary":"Derives necessary conditions for equitable partitions of regular graphs and for (a,b)-perfect sets in normal Cayley graphs using irreducible characters.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"Necessary conditions for (a,b)-perfect sets in normal Cayley graphs are expressed using irreducible characters of the group."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17376/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.060194Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:13:05.437691Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.773832Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.711128Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"93aa9256c99f353308bca43269634e556a9340b6c7417be7ae332bd7908a1c81"},"references":{"count":35,"sample":[{"doi":"","year":2019,"title":"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","work_id":"f8f821f2-5cca-4413-ac87-baf5c6117f56","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"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","work_id":"53ad484e-4c1f-4a66-b804-703af408fc59","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"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","work_id":"646c31a4-f378-42b9-94ca-90966688eeb3","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"N. L. Biggs, Perfect codes in graphs,J. Combin. Theory Ser. 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