{"paper":{"title":"Distance-Uniform Graphs with Large Diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Arnau Messegu\\'e, Mikhail Lavrov, Po-Shen Loh","submitted_at":"2017-03-04T15:20:58Z","abstract_excerpt":"An $\\epsilon$-distance-uniform graph is one in which from every vertex, all but an $\\epsilon$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in an $\\epsilon$-distance-uniform graph with $n$ vertices. We show that for $\\frac1n \\le \\epsilon \\le \\frac1{\\log n}$, there exist $\\epsilon$-distance-uniform graphs with critical distance $2^{\\Omega(\\frac{\\log n}{\\log \\epsilon^{-1}})}$, disproving a conjecture of Alon et al. that $d$ can be at most logarithmic in $n$. We also show that our construction is best"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01477","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"}