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Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphis"},"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":"2606.18462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T20:12:05Z","cross_cats_sorted":[],"title_canon_sha256":"7191273ece5e85f6cd33158997794caed19e72fe79aafccc539b366ccca75a1c","abstract_canon_sha256":"c11c9374a83fda49988995e29d22e4899c1b9b30c1503f9fe4e2b170bc3774ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:01.784361Z","signature_b64":"Q4KK5KZavGKG90slyOhBXCUL0ndFvPKAQ8SSXPnD5SXQUYoubMrnibNVeg8thZ0FT6BgtYEL5h0QeoLEbfF+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff5bb88c5f0c537dcb9b97b402a9961cb1847340b52dd5a3aaca6043339883f9","last_reissued_at":"2026-06-19T16:11:01.783999Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:01.783999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact 6-cut rigidity and small-order superconnectivity for the 6-regular case of Dirac's k=4 problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alper Ferudun","submitted_at":"2026-06-16T20:12:05Z","abstract_excerpt":"Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. 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