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We prove that {\\sc #ET} is #P-complete even for planar 4-regular graphs.\n  A closely related problem is that of counting A-trails ({\\sc #A-trails}) in graphs with rotational embedding schemes (so called maps). Kotzig \\cite{MR0248043} showed that {\\sc #A-trails} can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivale"},"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":"1009.5019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-09-25T15:40:39Z","cross_cats_sorted":[],"title_canon_sha256":"6b6f1d31c26374cae939903d8b3055a550811684b08011025c606add513014cf","abstract_canon_sha256":"4093b5261cdd0cd01ea8bfcb032a57dc46b120ef2a8916c3ba8ad75ad6f25a67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:19.377246Z","signature_b64":"jfGOHFIYWWn1U5faZaJ/hnRe/5lZEVQ/WfayfcUiB8w2oRw8sitW7Yrxtunhg7IIRPMM/rUqxfI/ApuO6WsaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eaafe94df8d2bc100c55764f8859328c2be4784e02a6ecb98db59e9803f5094a","last_reissued_at":"2026-05-18T04:40:19.376515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:19.376515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Complexity of Counting Eulerian Tours in 4-Regular Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Daniel Stefankovic, Qi Ge","submitted_at":"2010-09-25T15:40:39Z","abstract_excerpt":"We investigate the complexity of counting Eulerian tours ({\\sc #ET}) and its variations from two perspectives---the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions \\cite{MR2044886}). We prove that {\\sc #ET} is #P-complete even for planar 4-regular graphs.\n  A closely related problem is that of counting A-trails ({\\sc #A-trails}) in graphs with rotational embedding schemes (so called maps). 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