Some planar graphs with star chromatic number between three and four (Q2774163)

Let \(x\) be an integer, \(k>1\) be a positive integer, and \(\mathcal Z_k=\{0,1,2, \dots, k-1\}\). Let \(|x|_k\) be the distance from \(x\) to the nearest multiple of \(k\) and \(d\) be a positive integer coprime with \(k\) such that \(2d<k\). A \((k,d)\)-coloring of a graph \(G\) is a function \(c: V(G)\to \mathcal Z_k\) such that for any edge \(uv\) of \(E(G), |c(u)-c(v)|_k\geq d\). \textit{A. Vince} [J. Graph Theory 12, No. 4, 551-559 (1988; Zbl 0658.05028)] has defined the star chromatic number of \(G\) as \(\chi_c(G)=\inf \{k/d: G\) has a \((k,d)\)-coloring\(\}\). The author provides a few infinite families of planar graphs with star chromatic number between three and four. This answers a question of Vince in part.