Gr\"unbaum coloring and its generalization to arbitrary dimension

Abstract

This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Gr\"unbaum colorings, their existence and usage. In particular, one of the most striking significances of Gr\"unbaum's Conjecture in the 2-dimensional case is its equivalence to the 4-Color Theorem. The notion of Gr\"unbaum coloring is extended from the 2-dimensional case to the case of arbitrary finite hyper-dimensions.