{"paper":{"title":"Shifting the Phase Transition Threshold for Random Graphs and 2-SAT using Degree Constraints","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.LO","math.PR"],"primary_cat":"math.CO","authors_text":"Sergey Dovgal, Vlady Ravelomanana","submitted_at":"2017-04-21T19:08:22Z","abstract_excerpt":"We show that by restricting the degrees of the vertices of a graph to an arbitrary set \\( \\Delta \\), the threshold point $ \\alpha(\\Delta) $ of the phase transition for a random graph with $ n $ vertices and $ m = \\alpha(\\Delta) n $ edges can be either accelerated (e.g., $ \\alpha(\\Delta) \\approx 0.381 $ for $ \\Delta = \\{0,1,4,5\\} $) or postponed (e.g., $ \\alpha(\\{ 2^0, 2^1, \\cdots, 2^k, \\cdots \\}) \\approx 0.795 $) compared to a classical Erd\\H{o}s--R\\'{e}nyi random graph with $ \\alpha(\\mathbb Z_{\\geq 0}) = \\tfrac12 $. In particular, we prove that the probability of graph being nonplanar and the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06683","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"}