{"paper":{"title":"On the generalized coloring numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LO","math.CO"],"primary_cat":"cs.DM","authors_text":"Sebastian Siebertz","submitted_at":"2025-01-15T10:17:52Z","abstract_excerpt":"The \\emph{coloring number} $\\mathrm{col}(G)$ of a graph $G$, which is equal to the \\emph{degeneracy} of $G$ plus one, provides a very useful measure for the uniform sparsity of $G$. The coloring number is generalized by three series of measures, the \\emph{generalized coloring numbers}. These are the \\emph{$r$-admissibility} $\\mathrm{adm}_r(G)$, the \\emph{strong $r$-coloring number} $\\mathrm{col}_r(G)$ and the \\emph{weak $r$-coloring number} $\\mathrm{wcol}_r(G)$, where $r$ is an integer parameter. The generalized coloring numbers measure the edge density of bounded-depth minors and thereby prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.08698","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.08698/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"}