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

We show that \(\limsup |E(G)|/|V(G)| = 2.5\) $lim\; sup|E(G)|/|V(G)|=2.5$ over all 4-critical planar graphs G, answering a question of Grünbaum from 1988.