Maximising $H$-Colourings of Graphs

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 $\hom(G,H)$. 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. We also disprove a conjecture of Engbers on the graph $G$ that maximises the number of $H$-colourings when the assumption of the connectivity of $G$ is dropped. Finally, let $H$ be a graph with maximum degree $k$. We show that, if $H$ does not contain the complete looped graph on $k$ vertices or $K_{k,k}$ as a component and $\delta\geq\delta_0(H)$, then the following holds: for $n$ sufficiently large, the graph $K_{\delta,n-\delta}$ maximises the number of $H$-colourings among all graphs on $n$ vertices with minimum degree $\delta$. This partially answers another question of Engbers.