{"paper":{"title":"Injective colorings of sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel W. Cranston, Gexin Yu, Seog-Jin Kim","submitted_at":"2010-07-05T23:58:47Z","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 $mad(G) \\leq 5/2$, then $\\chi_i(G)\\leq\\Delta(G) + 1$; and if $mad(G) < 42/19$, then $\\chi_i(G)=\\Delta(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $\\Delta(G)\\geq 4$. We prove that if $g(G)\\geq 9$, then $\\chi_i(G)\\leq\\Delta(G)+1$; similarly, if $g(G)\\geq 13$, then $\\chi_i(G)=\\Delta(G)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0786","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"}