{"paper":{"title":"Cycles with two blocks in $k$-chromatic digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boram Park, Jie Ma, Ringi Kim, Seog-Jin Kim","submitted_at":"2016-10-19T01:36:06Z","abstract_excerpt":"Let $k$ and $\\ell$ be positive integers. A cycle with two blocks $c(k,\\ell)$ is an oriented cycle which consists of two internally (vertex) disjoint directed paths of lengths at least $k$ and $\\ell$, respectively, from a vertex to another one. A problem of Addario-Berry, Havet and Thomass\\'e (2007) asked if, given positive integers $k$ and $\\ell$ such that $k+\\ell\\ge 4$, any strongly connected digraph $D$ containing no $c(k,\\ell)$ has chromatic number at most $k+\\ell-1$. In this paper, we show that such digraph $D$ has chromatic number at most $O((k+\\ell)^2)$, improving the previous upper boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05839","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"}