{"paper":{"title":"Borodin-Kostochka conjecture and Partitioning a graph into classes with no clique of specified size","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yaser Rowshan","submitted_at":"2023-11-15T08:35:34Z","abstract_excerpt":"For a given graph $H$ and the graphical properties $P_1, P_2,\\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\\ldots,V_k$, such that for each $i\\in[k]$, the subgraph induced by $V_i$ has the property $P_i$. In $1979$, Bollob\\'{a}s and Manvel showed that for a graph $H$ with maximum degree $\\Delta(H)\\geq 3$ and clique number $\\omega(H)\\leq \\Delta(H)$, if $\\Delta(H)= p+q$, then there exists a $(V_1,V_2)$-partition of $V(H)$, such that $\\Delta(H[V_1])\\leq p$, $\\Delta(H[V_2])\\leq q$, $H[V_1]$ is $(p-1)$-degene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.08772","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/2311.08772/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"}