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For a non-3-colorable graph $G$, $W(G)$ is equal to the minimum $k$ such that $G$ can be distinguished from $K_3$ in the $k$-variable existential-positive first-order logic. We define the dynamic width of the 3-colorability problem as the function $W(n)=\\max_G W(G)$, where the maximum is taken "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.5937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-20T13:42:06Z","cross_cats_sorted":[],"title_canon_sha256":"f29accd2f5508da740ef40693a62fc4d4920b201799441d6b7ccd88296662edd","abstract_canon_sha256":"3b3b4a1405a684ee97250979c9876c7e5b661772385b27a7b4a09b8ff2261600"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:56.436273Z","signature_b64":"FBAQ1XGnz04/bKqUHW81X8DjVbL1IlMm/D9/8+mE/osFrIk4soOFX6k3Iv53mlbwwgEZog+AnIF5efQx8g7HDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b909b4cf35a55103bc41fa693834c85539000090b75c056ffe1dd65915d7afcc","last_reissued_at":"2026-05-18T02:58:56.435585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:56.435585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dynamic width of the 3-colorability problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Albert Atserias, Anuj Dawar, Oleg Verbitsky","submitted_at":"2013-12-20T13:42:06Z","abstract_excerpt":"A graph $G$ is 3-colorable if and only if it maps homomorphically to the complete 3-vertex graph $K_3$. 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