{"paper":{"title":"Extremal graphs with no subgraph admitting $k+1$ edge-disjoint spanning trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Qinglin Wang, Yingzhi Tian","submitted_at":"2026-06-26T15:48:16Z","abstract_excerpt":"A graph $G$ is $\\tau_k$-maximal if $G$ contains no subgraph admitting $k+1$ edge-disjoint spanning trees, while the addition of any edge in the complement of $G$ yields a subgraph that admits $k+1$ edge-disjoint spanning trees. In this paper, we prove that for any integers $k\\geq 1$ and $n\\geq 2k+2$, every $\\tau_k$-maximal graph of order $n$ satisfies $|E(G)|\\leq (k+1)(n-1)-1$. Furthermore, we construct a family of $\\tau_k$-maximal graphs on $n\\ge 2k+2$ vertices that have exactly $(k+1)(n-1)-1$ edges, which establishes the tightness of the upper bound. Then we conjecture that every $\\tau_k$-ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28198","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/2606.28198/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"}