{"paper":{"title":"Dynamic coloring parameters for graphs with given genus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin Reiniger, Jennifer Wise, Sarah Loeb, Thomas Mahoney","submitted_at":"2015-11-12T17:40:16Z","abstract_excerpt":"A proper vertex coloring of a graph $G$ is $r$-dynamic if for each $v\\in V(G)$, at least $\\min\\{r,d(v)\\}$ colors appear in $N_G(v)$. In this paper we investigate $r$-dynamic versions of coloring, list coloring, and paintability. We prove that planar and toroidal graphs are 3-dynamically 10-colorable, and this bound is sharp for toroidal graphs. We also give bounds on the minimum number of colors needed for any $r$ in terms of the genus of the graph: for sufficiently large $r$, every graph with genus $g$ is $r$-dynamically $((r+1)(g+5)+3)$-colorable when $g\\leq2$ and $r$-dynamically $((r+1)(2g+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03983","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"}