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We say that a graph $G$ has no $\\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\\in \\mathcal{K}_t^{-s}$. Jakobsen in 1971 proved that every graph with no $\\mathcal{K}_7^{-2}$ minor is $6$-colorable. In this paper we consider the next step and prove that every graph with no $\\mathcal{K}_8^{-4}$ min"},"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":"2208.07338","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-08-15T17:08:01Z","cross_cats_sorted":[],"title_canon_sha256":"8efb0de3b215dab185491feb259e25b25a5118771d0c02664ff37a5cf18146fa","abstract_canon_sha256":"9837c9cdd0844bee346b51f91373b0951bdeaef39ff5186e9d98932a80d17b31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:50:19.889501Z","signature_b64":"IzWoUVYWVEV4at3QJlOEvfJWm24lMGcJM1/Xg6K++vPn904lHkYPYJC3EFScDBWHEaswdBNSXm4pqbsRdOtuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6924ae818ca835784091eae972db2cc240cbe66aa9e46b32e1c4f446bf30fbf0","last_reissued_at":"2026-07-05T04:50:19.889080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:50:19.889080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Every graph with no $\\mathcal{K}_8^{-4}$ minor is $7$-colorable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Lafferty, Zi-Xia Song","submitted_at":"2022-08-15T17:08:01Z","abstract_excerpt":"Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\\ge 7$. 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