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In a very recent paper, we gave a lower bound, $f_k(n) \\geq F(k,n)$, that is sharp for every $n=1\\,({\\rm mod}\\, k-1)$. It is also sharp for $k=4$ and every $n\\geq 6$. In this note, we present a simple proof of the bound for $k=4$. It implies the case $k=4$ of the conjecture by Ore from 1967 that for every $k\\geq 4$ and $n\\geq k+2$, $f_k(n+k-1)=f(n)+\\frac{k-1}{2}(k - \\frac{2}{k-1})$. We also"},"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":"1209.1173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-06T04:14:38Z","cross_cats_sorted":[],"title_canon_sha256":"9b205df937c17f0b3ee2e9d167247d32201245056df731e89526e14b04cd3f8c","abstract_canon_sha256":"3644a184353d80e33c7f77cd1f88464514ad85255ef5fb01c16f8248e220a8d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:06.351553Z","signature_b64":"5hiJJUaQKz9fj0nKYaTL1AJhKWQ0nKWnujUF+KqKH6ixLD4VzJJKzE5tx/twFp41nPBxc2575FxUl+TRDaIRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4223f374a51b6f65abb519c23b3106fdd0c97bd086db15f4749da322b30ad73","last_reissued_at":"2026-05-18T03:46:06.350857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:06.350857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ore's Conjecture for $k=4$ and Gr\\\" otzsch Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Matthew Yancey","submitted_at":"2012-09-06T04:14:38Z","abstract_excerpt":"A graph $G$ is $k$-{\\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. In a very recent paper, we gave a lower bound, $f_k(n) \\geq F(k,n)$, that is sharp for every $n=1\\,({\\rm mod}\\, k-1)$. It is also sharp for $k=4$ and every $n\\geq 6$. In this note, we present a simple proof of the bound for $k=4$. It implies the case $k=4$ of the conjecture by Ore from 1967 that for every $k\\geq 4$ and $n\\geq k+2$, $f_k(n+k-1)=f(n)+\\frac{k-1}{2}(k - \\frac{2}{k-1})$. 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