{"paper":{"title":"Tree-partitions of graphs with bounded tree-depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huayue Liu, Rong chen","submitted_at":"2026-08-13T02:09:50Z","abstract_excerpt":"Wood~ recently showed that every graph $G$ of pathwidth $h$ and $\\Delta(G)\\ge1$ admits a $T$-partition of width at most $4(h+1)^2\\Delta(G)$ for some tree $T$ with $pw(T)\\leq2h+1$. In this paper, we establish an analogous result for tree-depth, which is a stronger parameter than pathwidth. We prove that every connected graph with tree-depth $h$ admits a $T$-partition of width at most $\\mathrm{max}\\{1, (4h-10)\\Delta(G)+1\\}$ for some tree $T$ with $\\operatorname{rad}(T)\\leq h-1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12723","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/2608.12723/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"}