{"paper":{"title":"On the chromatic number of random geometric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Colin McDiarmid, Tobias M\\\"uller","submitted_at":"2011-01-31T20:16:00Z","abstract_excerpt":"Given independent random points $X_1,...,X_n\\in\\eR^d$ with common probability distribution $\\nu$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\\{1,...,n\\}$ where distinct $i$ and $j$ are adjacent when $\\norm{X_i-X_j}\\leq r$. Here $\\norm{.}$ may be any norm on $\\eR^d$, and $\\nu$ may be any probability distribution on $\\eR^d$ with a bounded density function. We consider the chromatic number $\\chi(G_n)$ of $G_n$ and its relation to the clique number $\\omega(G_n)$ as $n \\to \\infty$. Both McDiarmid and Penrose considered the range of $r$ when $r \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.6065","kind":"arxiv","version":3},"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"}