Restraints Permitting the Largest Number of Colourings

A \textit{restraint} $r$ on $G$ is a function which assigns each vertex $v$ of $G$ a finite set of forbidden colours $r(v)$. A proper colouring $c$ of $G$ is said to be \textit{permitted by the restraint r} if $c(v)\notin r(v)$ for every vertex $v$ of $G$. A restraint $r$ on a graph $G$ with $n$ vertices is called a \textit{$k$-restraint} if $|r(v)|=k$ and $r(v) \subseteq \{1,2,\dots ,kn\}$ for every vertex $v$ of $G$. In this article we discuss the following problem: among all $k$-restraints $r$ on $G$, which restraints permit the largest number of $x$-colourings for all large enough $x$? We determine such extremal restraints for all bipartite graphs.