{"paper":{"title":"List-coloring the Squares of Planar Graphs without 4-Cycles and 5-Cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bobby Jaeger, Daniel W. Cranston","submitted_at":"2015-05-13T00:14:51Z","abstract_excerpt":"Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $\\Delta\\ge 32$. We prove that $\\chi_{\\ell}(G^2)\\le \\Delta+3$. For arbitrarily large maximum degree $\\Delta$, there exist planar graphs $G_{\\Delta}$ of girth 6 with $\\chi(G_{\\Delta}^2)=\\Delta+2$. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for $L(p,q)$-labeling. Specifically, $\\lambda_{2,1}(G)\\le \\Delta+8$ and, more generally, $\\lambda_{p,q}(G)\\le (2q-1)\\Delta+6p-2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03197","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"}