{"paper":{"title":"Injective colorings of graphs with low average degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Daniel W. Cranston, Gexin Yu, Seog-Jin Kim","submitted_at":"2010-06-18T19:06:34Z","abstract_excerpt":"Let $\\mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $\\Delta\\geq 4$ and $\\mad(G)<\\frac{14}5$, then $\\chi_i(G)\\leq\\Delta+2$. When $\\Delta=3$, we show that $\\mad(G)<\\frac{36}{13}$ implies $\\chi_i(G)\\le 5$. In contrast, we give a graph $G$ with $\\Delta=3$, $\\mad(G)=\\frac{36}{13}$, and $\\chi_i(G)=6$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3776","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"}