{"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). 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5019","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":""},"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"}