{"paper":{"title":"Coloring the square of a sparse graph $G$ with almost $\\Delta(G)$ colors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthew Yancey","submitted_at":"2015-02-10T21:44:25Z","abstract_excerpt":"For a graph $G$, let $G^2$ be the graph with the same vertex set as $G$ and $xy \\in E(G^2)$ when $x \\neq y$ and $d_G(x,y) \\leq 2$. Bonamy, L\\'ev\\^{e}que, and Pinlou conjectured that if $mad (G) < 4 - \\frac{2}{c+1}$ and $\\Delta(G)$ is large, then $\\chi_\\ell(G^2) \\leq \\Delta(G) + c$. We prove that if $c \\geq 3$, $mad (G) < 4 - \\frac{4}{c+1}$, and $\\Delta(G)$ is large, then $\\chi_\\ell(G^2) \\leq \\Delta(G) + c$. Dvo\\v{r}\\'ak, Kr\\'{a}\\soft{l}, Nejedl\\'{y}, and \\v{S}krekovski conjectured that $\\chi(G^2) \\leq \\Delta(G) +2$ when $\\Delta(G)$ is large and $G$ is planar with girth at least $5$; our result"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03132","kind":"arxiv","version":2},"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"}