A {\em restraint} on a (finite undirected) graph G=(V,E) is a function r on V such that r(v) is a finite subset of N; a proper vertex colouring c of G is {\em permitted} by r if c(v)∈r(v) for all vertices v of G (we think of r(v) as the set of colours {\em forbidden} at v). Given a large number of colors, for restraints r with exactly one colour forbidden at each vertex the smallest number of colorings is permitted when r is a constant function, but the problem of what restraints permit the largest number of colourings is more difficult. We determine such extremal restraints for complete graphs and trees.
@article{arxiv.1611.08920,
title = {Extremal Restraints for Graph Colourings},
author = {Jason I. Brown and Aysel Erey and Jian Li},
journal= {arXiv preprint arXiv:1611.08920},
year = {2016}
}