Solution to a problem of Grünbaum on the edge density of 4-critical planar graphs (Q6607843)

In this paper, the authors show that lim sup \(\vert E(G)\vert /\vert V(G)\vert \) = 2.5 over all 4-critical planar graphs \(G\), by constructing, for each integer \( m\) sufficiently large, a 4-critical planar graph \(Q_m\) on at most \(m\) vertices such that \(L=\limsup_{m}\vert E(Q_m)\vert /m= 2.5\).\par This answers a question of \textit{B. Grünbaum} [Combinatorica 8, No. 1, 137--139 (1988; Zbl 0810.05028)]. Note that Koester \textit{G. Koester} [Discrete Math. 98, No. 2, 147--151 (1991; Zbl 0756.05070)] showed that \(L\leq 2.5\).