Gr\"unbaum coloring and its generalization to arbitrary dimension

TL;DR

This paper reviews and discusses recent research on Gr"unbaum colorings, their significance, and extends the concept from 2D to higher dimensions, highlighting its relation to the 4-Color Theorem.

Contribution

It introduces a generalization of Gr"unbaum colorings to arbitrary dimensions, connecting it to existing conjectures and theorems.

Findings

Gr"unbaum's Conjecture is equivalent to the 4-Color Theorem in 2D

Extension of Gr"unbaum coloring to higher dimensions proposed

Survey of recent developments in the field

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.