{"paper":{"title":"Null and non--rainbow colorings of projective plane and sphere triangulations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Amanda Montejano, Jorge L. Arocha","submitted_at":"2012-10-25T13:38:38Z","abstract_excerpt":"For maximal planar graphs of order $n\\geq 4$, we prove that a vertex--coloring containing no rainbow faces uses at most $\\lfloor\\frac{2n-1}{3}\\rfloor$ colors, and this is best possible. For maximal graph embedded on the projective plane, we obtain the analogous best bound $\\lfloor\\frac{2n+1}{3}\\rfloor$. The main ingredients in the proofs are classical homological tools. By considering graphs as topological spaces, we introduce the notion of a null coloring, and prove that for any graph $G$ a maximal null coloring $f$ is such that the quotient graph $G/f$ is a forest."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6831","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"}