{"paper":{"title":"Deterministic Simple $(\\Delta+\\varepsilon\\alpha)$-Edge-Coloring in Near-Linear Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ariel Khuzman, Michael Elkin","submitted_at":"2024-01-19T07:33:48Z","abstract_excerpt":"We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(\\Delta+1)$-edge-coloring in $O(m\\cdot n)$ deterministic time. This running time was improved to $O\\left(m\\cdot\\min\\left\\{\\Delta\\cdot\\log n,\\sqrt{n}\\right\\}\\right)$, and very recently to randomized $\\tilde{O}\\left(m\\cdot n^{1/3}\\right)$. A randomized $(1+\\varepsilon)\\Delta$-edge-coloring algorithm can be computed in $O\\left(m\\cdot\\frac{\\log^6 n}{\\varepsilon^2}\\right)$ time, and f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.10538","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.10538/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}