{"paper":{"title":"The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Miriam Goetze, Torsten Ueckerdt","submitted_at":"2025-01-23T12:38:15Z","abstract_excerpt":"A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\\min(r, \\mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$ admits an $r$-dynamic coloring with no more than $(k-1)r+1$ colors. As a consequence, for every class of graphs of bounded expansion, the $r$-dynamic chromatic number is bounded by a linear function in $r$. We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.13617","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.13617/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}