Minimal Color Assignments on Pure Simplicial Complexes

Abstract

Given a pure simplicial complex C of dimension d-1, assign a subset of {1,...,d} to each vertex, in such a way that each maximal face has all d colors. What is the minimum total number NC(C) of color assignments needed? The number of vertices is a lower bound on NC(C), which is attained if and only if the simplicial complex is completely balanced. Thus for any given pure simplicial complex, we have a measure of the extent to which it fails to be completely balanced. We obtain complete solutions for NC(C) in the case where C has four or less facets.