Solution to a Problem of Grnbaum on the Edge Density of 4-Critical Planar Graphs

We show that lim sup|E(G)|/|V(G)|=2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup |E(G)|/|V(G)| = 2.5$$\end{document} over all 4-critical planar graphs G, answering a question of Grunbaum from 1988.