{"paper":{"title":"Maximising $H$-Colourings of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Scott, Hannah Guggiari","submitted_at":"2016-11-09T13:11:06Z","abstract_excerpt":"For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\\psi:V(G)\\rightarrow V(H)$ such that $ij\\in E(G)\\Rightarrow\\psi(i)\\psi(j)\\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\\hom(G,H)$.\n  We prove the following: for all graphs $H$ and $\\delta\\geq3$, there is a constant $\\kappa(\\delta,H)$ such that, if $n\\geq\\kappa(\\delta,H)$, the graph $K_{\\delta,n-\\delta}$ maximises the number of $H$-colourings among all connected graphs with $n$ vertices and minimum degree $\\delta$. This answers a question of Engbers.\n  We also disprove a conjecture of Engbers on the graph $G$ that maximis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.02911","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"}