{"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 $U$. It is easy to see that $\\xi(G) \\le \\min\\{\\delta(G), \\lfloor \\Phi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.06458","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}