{"paper":{"title":"The gonality sequence of complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Filip Cools, Marta Panizzut","submitted_at":"2016-05-12T10:38:54Z","abstract_excerpt":"The gonality sequence $(\\gamma_r)_{r\\geq1}$ of a finite graph / metric graph / algebraic curve comprises the minimal degrees $\\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\\gamma_r = kd - h$ if $r<g=\\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \\frac{k(k+3)}{2} - h$ with $1\\leq k\\leq d-3$ and $0 \\leq h \\leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03749","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}