{"paper":{"title":"New results in $t$-tone coloring of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel W. Cranston, Jaehoon Kim, William B. Kinnersley","submitted_at":"2011-08-24T06:10:56Z","abstract_excerpt":"A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\\{1,..., k\\}$ so that vertices at distance $d$ share fewer than $d$ common colors. The {\\it $t$-tone chromatic number} of $G$, denoted $\\tau_t(G)$, is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring. Bickle and Phillips showed that always $\\tau_2(G) \\le [\\Delta(G)]^2 + \\Delta(G)$, but conjectured that in fact $\\tau_2(G) \\le 2\\Delta(G) + 2$; we confirm this conjecture when $\\Delta(G) \\le 3$ and also show that always $\\tau_2(G) \\le \\ceil{(2 + \\sqrt{2})\\Delta(G)}$. For general $t$ we prove that $\\tau_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4751","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"}