{"paper":{"title":"The t-tone chromatic number of random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Andrzej Dudek, Deepak Bal, Patrick Bennett","submitted_at":"2012-10-02T02:38:41Z","abstract_excerpt":"A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph $G$, denoted $\\tau_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\\ge Cn^{-1/4}\\ln^{9/4}n$, $\\tau_2(G_{n,p})=(2+o(1))\\chi(G_{n,p})$ where $\\chi$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $\\tau_2(G_{n,p}) = \\lc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0635","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"}