{"paper":{"title":"A Domatic Analogue of $\\chi$-Bounded Graph Classes and the Gy\\'{a}rf\\'{a}s-Sumner Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fran\\c{c}ois Pirot, Hoang La, Hossein Zaredehabadi, Quentin Chuet, Selma Djelloul","submitted_at":"2026-06-01T10:18:06Z","abstract_excerpt":"Given a graph $G$, a dominating set is a subset $X\\subseteq V(G)$ such that $N[X]=V(G)$. The \\emph{domatic number} of $G$, denoted ${\\rm dom}(G)$, is the maximum size of a partition of $V(G)$ into dominating sets. In analogy with the lower bound of the chromatic number by the clique number, the domatic number satisfies the upper bound ${\\rm dom}(G)\\le \\delta(G)+1$ where $\\delta(G)$ is the minimum degree of $G$. Therefore, as an analogue of the notion of $\\chi$-bounded graph classes, we say that a class of graphs $\\mathscr{G}$ is \\emph{DOM-bounded} if there exists a positive unbounded function "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02030","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/2606.02030/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"}