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This generalizes a Kempe-Heawood result for d=2 and is work related to the conjecture that all d-spheres have chromatic number d+1 or d+2 which is based on the geometric conjecture that every d-sphere can be embedded in an Eule"},"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":"1501.03116","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-12T13:29:29Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"1c1f6a138e2ecb93ff6f0d364e9ae6887ca42bda03769ae5f19d214c154dadf7","abstract_canon_sha256":"e5be04ec0f8a227b3db11084d887113d64cf31f4c2ab307575c38d2207640ce8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:30.475239Z","signature_b64":"qcpRfWAAyjpvYQxkGjZzuV2E7GZRyBsjLDMnHPfOy2LEQGz/aYG5bQJmQvKprfjMcwivE6nD4L528wFx6rQQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e72e94802d19a61d44a7bf8186445ab7264a36e75b8cc9e0a677380535759b01","last_reissued_at":"2026-05-18T02:29:30.474796Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:30.474796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graphs with Eulerian unit spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2015-01-12T13:29:29Z","abstract_excerpt":"d-spheres in graph theory are inductively defined as graphs for which all unit spheres S(x) are (d-1)-spheres and that the removal of one vertex renders the graph contractible. 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