{"paper":{"title":"Complexity Classification of Colouring Problems with Parity Constraints","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Haiko M\\\"uller, Juan Pablo Bravo, Noleen K\\\"ohler, R\\'emy Belmonte","submitted_at":"2026-07-18T16:34:49Z","abstract_excerpt":"We study variants of graph colouring with parity constraints. More specifically, we consider $q$-colourings $c\\colon V(G)\\rightarrow \\{1,\\dots,q\\}$ of a graph $G$ where, for every vertex $v\\in V(G)$, the number of neighbours $w$ of $v$ with $c(w)=c(v)$ is restricted to be odd, even, positive, zero or a combination thereof. For every colour $i\\neq c(v)$ the number of neighbours $w$ of $v$ with $c(w)=i$ is restricted by a constraint of similar type. Many known colouring problems such as proper colouring, defective colouring, exact defective colouring, odd colouring, and strong odd colouring can "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16879","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16879/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}