{"paper":{"title":"A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Florian Lehner, Joshua Erde, Konstantinos Stavropoulos, Martin Merker, Max Pitz, Nathan Bowler","submitted_at":"2017-02-03T11:35:44Z","abstract_excerpt":"A \\emph{dynamic colouring} of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The \\emph{dynamic colouring number} $\\chi_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$.\n  Montgomery conjectured that $\\chi_2(G) \\leq \\chi(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $\\chi_2(G) \\leq 2\\chi(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n \\geq 2$, a regular graph $G$ with $\\chi(G) = n$ but $\\chi_2("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.00973","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":""},"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"}