{"paper":{"title":"On Coloring Properties of Graph Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Taherkhani, Hossein Hajiabolhassan","submitted_at":"2011-04-22T08:42:31Z","abstract_excerpt":"This paper studies some coloring properties of graph powers. We show that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$ provided that $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \\over 2s+1} \\leq {\\chi_c(G) \\over 3(\\chi_c(G)-2)}$, then $\\chi_c(G^{^{\\frac{2r+1}{2s+1}}})=\\frac{(2s+1)\\chi_c(G)}{(s-r)\\chi_c(G)+2r+1}$. In particular, $\\chi_c(K_{3n+1}^{^{1\\over3}})={9n+3\\over 3n+2}$ and $K_{3n+1}^{^{1\\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\\over 2n+1}$. This provides a negative answer to a questio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4411","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"}