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Let $\\chi'_k(G)$ denote the minimum number of colors required, and let $\\chi'_k$ be the supremum of $\\chi'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\\chi'_k(G)\\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\\geq0$ and $k\\geq3c+2$, we"},"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":"2608.02239","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T13:51:55Z","cross_cats_sorted":[],"title_canon_sha256":"ee4ff4bd1945789b23d465c4e5588bcd35ce76ad57f5faabd86b861d5dd96579","abstract_canon_sha256":"a0fe5ca5308a73d6fc878050037011bd4004a0596d30c2bd29f2603cfecf7836"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:11:28.352096Z","signature_b64":"XsRKIPBt0GefHaRudzySgBs0b2oldkR2j1J9rCFV1JqHkcB5lmJ51PL9R09FAN54zBOdIknQ/bi+rY+IgpNEAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87b910e0da41474cd73a5a1cddf2708eaf7f64b5b5d47503431d2792707f9264","last_reissued_at":"2026-08-04T02:11:28.344057Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:11:28.344057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linear Lower Bounds for the Modular Chromatic Index","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boyan Xu, Xiao-Chuan Liu, Xu Yang","submitted_at":"2026-08-03T13:51:55Z","abstract_excerpt":"Let $k\\geq2$ be an integer. A $1\\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\\chi'_k(G)$ denote the minimum number of colors required, and let $\\chi'_k$ be the supremum of $\\chi'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\\chi'_k(G)\\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. 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