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Brown, Jian Li","submitted_at":"2016-11-27T22:07:34Z","abstract_excerpt":"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 ${\\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\\em permitted} by $r$ if $c(v) \\not\\in r(v)$ 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. 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