For graphs G and H, an H-colouring of G is a map ψ : V (G) → V (H) such that ij ∈ E(G) ⇒ ψ(i)ψ(j) ∈ E(H). The number of H-colourings of G is denoted by hom(G,H).

We prove the following: for all graphs H and δ ≥ 3, there is a constant k(δ;H) such that, if n ≥ k(δ;H), the graph Kδ;n-δ max- imises the number of H-colourings among all connected graphs with n vertices and minimum degree δ. This answers a question of Engbers.

We also disprove a conjecture of Engbers on...