{"paper":{"title":"k-Coloring is Faster than Computing the Chromatic Number","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Or Zamir","submitted_at":"2026-07-28T16:53:58Z","abstract_excerpt":"We prove that $k$-coloring on $n$-vertex graphs has a randomized algorithm running in time $(2-\\varepsilon_k)^n$, where $\\varepsilon_k>0$ for every fixed $k$. Previously, only the cases $k\\leq 6$ were known to have faster solutions than the general $O^\\star\\bigl(2^n\\bigr)$ time algorithm of [Bj\\\"{o}rklund, Husfeldt, Koivisto, SICOMP 2009] that computes the chromatic number.\n  We resolve this long-standing open problem by generalizing and combining tools from the $(k+2)$-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25973","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.25973/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"}