{"paper":{"title":"Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huan Zhou, Luis Kuffner, Lujia Wang, Reza Naserasr, Xiaowei Yu, Xuding Zhu","submitted_at":"2024-12-28T03:57:07Z","abstract_excerpt":"The Kneser signed graph $\\KS(n,k)$, $k\\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\\{ \\pm 1, \\pm 2, \\ldots, \\pm n\\}$ such that $S\\cap (-S)=\\emptyset$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\\cap (-B)=\\emptyset$ and with a negative edge if $A\\cap B=\\emptyset$. We prove that the balanced chromatic number of $\\KS(n,k)$ is $n-k+1$. We then introduce the signed analogue of Schrijver graphs and show that they form vertex-critical subgraphs of $\\KS(n,k)$ with respect to balanced colouring. Further connection to topological me"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20001","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/2412.20001/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"}