{"paper":{"title":"Maximal-clique partitions and the Roller Coaster Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan Cutler, Luke Pebody","submitted_at":"2014-12-15T13:55:11Z","abstract_excerpt":"A graph $G$ is {\\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence sequence $(i_0(G), i_1(G), \\ldots, i_q(G))$ was unimodal for any well-ordered graph $G$ with independence number $q$. Michael and Traves disproved this conjecture. Instead they posited the so-called ``Roller Coaster\" Conjecture: that the terms \\[\n  i_{\\left\\lceil\\frac{q}2\\right\\rceil}(G), i_{\\left\\lceil\\frac{q}2\\right\\rceil+1}(G), \\ldots, i_q(G) \\] could be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4595","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"}