{"paper":{"title":"$\\delta^{(k)}$-Colouring of Cycle Related Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Johan Kok, Sudev Naduvath","submitted_at":"2018-07-05T09:32:09Z","abstract_excerpt":"With respect to a proper colouring of a graph $G$, we know that $\\delta(G) \\leq \\chi(G) \\leq \\Delta(G)+1$. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of each of $k$, $1\\leq k < \\chi(G)$ technology types together with the minimum adjacency between similar technology types. In an improper colouring an edge $uv$ such that $c(u)=c(v)$ is called a bad edge. In this paper, we introduce the notion of $\\delta^{(k)}$-colouring which is a near proper colouring of $G$ with exactly $1\\leq k < \\chi(G)$ distinct colours "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.01915","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"}