{"paper":{"title":"Digraphs with degree two and excess two are diregular","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"James Tuite","submitted_at":"2017-03-25T20:52:52Z","abstract_excerpt":"A $k$-geodetic digraph with minimum out-degree $d$ has excess $\\epsilon $ if it has order $M(d,k) + \\epsilon $, where $M(d,k)$ represents the Moore bound for out-degree $d$ and diameter $k$. For given $\\epsilon $, it is simple to show that any such digraph must be out-regular with degree $d$ for sufficiently large $d$ and $k$. However, proving in-regularity is in general non-trivial. It has recently been shown that any digraph with excess $\\epsilon = 1$ must be diregular. In this paper we prove that digraphs with minimum out-degree $d = 2$ and excess $\\epsilon = 2$ are diregular for $k \\geq 2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08739","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":""},"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"}