{"paper":{"title":"Coloring graphs with two odd cycle lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Jie Ma","submitted_at":"2015-12-20T16:10:44Z","abstract_excerpt":"In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let $G$ be a graph and $L(G)$ be the set of all odd cycle lengths of $G$. We prove that: (1) If $L(G)=\\{3,3+2l\\}$, where $l\\geq 2$, then $\\chi(G)=\\max\\{3,\\omega(G)\\}$; (2) If $L(G)=\\{k,k+2l\\}$, where $k\\geq 5$ and $l\\geq 1$, then $\\chi(G)=3$. These, together with the case $L(G)=\\{3,5\\}$ solved in \\cite{W}, give a complete solution to the general problem addressed in \\cite{W,CS,KRS}. Our results also improve a classical theorem of Gy\\'{a}rf\\'{a}s which asserts that $\\chi(G)\\le 2|L(G)|+2$ for any graph $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06393","kind":"arxiv","version":2},"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"}