{"paper":{"title":"Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Yuxue Yin","submitted_at":"2018-09-04T12:53:32Z","abstract_excerpt":"Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\\{4,5\\}$-cycles and $d^{\\Delta}\\ge d_0$ are $3$-choosable and planar graphs without $\\{4,5,6\\}$-cycles and $d^{\\Delta}\\ge d_1$ are $3$-choosable, where $d^{\\Delta}$ is the smallest distance between triangles. They showed that $2\\le d_0\\le 4$ and $d_1\\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\\{4,5\\}$-cycles and $d^{\\Delta}\\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\\{4,5,6\\}$-cycles"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00925","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"}