Restraints Permitting the Largest Number of Colourings
Jason Brown, Aysel Erey

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} on is a function which assigns each vertex of a finite set of forbidden colours . A proper colouring of is said to be \textit{permitted by the restraint r} if for every vertex of . A restraint on a graph with vertices is called a \textit{-restraint} if and for every vertex of . In this article we discuss the following problem: among all -restraints on , which restraints permit the largest number of -colourings for all large enough ? We determine such extremal restraints for all bipartite graphs.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
