{"paper":{"title":"$\\chi_D(G)$, $|Aut(G)|$, and a variant of the Motion Lemma","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Niranjan Balachandran, Sajith Padinhatteeri","submitted_at":"2015-05-13T14:17:21Z","abstract_excerpt":"The \\textit{Distinguishing Chromatic Number} of a graph $G$, denoted $\\chi_D(G)$, was first defined in \\cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $\\phi$ of the graph $G$ fixes each color class of $G$. In this paper, 1. We prove a lemma that may be considered a variant of the Motion lemma of \\cite{RS} and use this to give examples of several families of graphs which satisfy $\\chi_D(G)=\\chi(G)+1$. 2.We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03396","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"}