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Let $\\delta(G)$ be the minimum degree of $G$ and let $\\Phi(G)$ be the {\\em co-density} of $G$, defined by \\[\\Phi(G)=\\min \\Big\\{\\frac{2|E^+(U)|}{|U|+1}:\\,\\, U \\subseteq V, \\,\\, |U|\\ge 3 \\hskip 2mm {\\rm and \\hskip 2mm odd} \\Big\\},\\] where $E^+(U)$ is the set of all edges of $G$ with at least one end in $U$. It is easy to see that $\\xi(G) \\le \\min\\{\\delta(G), \\lfloor \\Phi"},"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":"1906.06458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-15T03:07:57Z","cross_cats_sorted":[],"title_canon_sha256":"6f51a4d7f1d288c47d6cf5dc79d6f9e5da68620e77ce48216a257b1c2cbb1500","abstract_canon_sha256":"fa7ae11751abc0ed612da5556e7e122f0d0ae99359bcd19dcb7f6f362efe56e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:13.612914Z","signature_b64":"/K8v926GUmz5oPUwC73KRPs8akjFtcnfPRS5+0PbXZz8wPfnSFBBbFIuXdtBZJka26JSnyhkGt829bqY60LrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9e2dc53998c636e80c753f2b28797a701ceb9e5a96f70d6d8105a13a8a7c446","last_reissued_at":"2026-05-17T23:43:13.612453Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:13.612453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Gupta's Co-density Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guangming Jing, Guantao Chen, Guoli Ding, Wenan Zang, Yan Cao","submitted_at":"2019-06-15T03:07:57Z","abstract_excerpt":"Let $G=(V,E)$ be a multigraph. The {\\em cover index} $\\xi(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. Let $\\delta(G)$ be the minimum degree of $G$ and let $\\Phi(G)$ be the {\\em co-density} of $G$, defined by \\[\\Phi(G)=\\min \\Big\\{\\frac{2|E^+(U)|}{|U|+1}:\\,\\, U \\subseteq V, \\,\\, |U|\\ge 3 \\hskip 2mm {\\rm and \\hskip 2mm odd} \\Big\\},\\] where $E^+(U)$ is the set of all edges of $G$ with at least one end in $U$. 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