{"paper":{"title":"Exact Complexity of Exact-Four-Colorability","license":"","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"J\\\"org Rothe","submitted_at":"2001-09-14T15:07:08Z","abstract_excerpt":"Let $M_k \\seq \\nats$ be a given set that consists of $k$ noncontiguous integers. Define $\\exactcolor{M_k}$ to be the problem of determining whether $\\chi(G)$, the chromatic number of a given graph $G$, equals one of the $k$ elements of the set $M_k$ exactly. In 1987, Wagner \\cite{wag:j:min-max} proved that $\\exactcolor{M_k}$ is $\\bhlevel{2k}$-complete, where $M_k = \\{6k+1, 6k+3, >..., 8k-1 \\}$ and $\\bhlevel{2k}$ is the $2k$th level of the boolean hierarchy over $\\np$. In particular, for $k = 1$, it is DP-complete to determine whether $\\chi(G) = 7$, where $\\DP = \\bhlevel{2}$. Wagner raised the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0109018","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"}