Restraints Permitting the Largest Number of Colourings

TL;DR

This paper investigates how to choose vertex restrictions on a graph to maximize the number of proper colourings, identifying extremal restrictions for bipartite graphs when the restrictions are uniform and bounded.

Contribution

It characterizes the extremal restraints that allow the maximum number of colourings in bipartite graphs for large colour sets.

Findings

Identifies extremal restraints for bipartite graphs.

Determines maximum permitted colourings under uniform restraints.

Provides a solution for large colour sets.

Abstract

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.