List $3$-coloring $C_4$-free graphs of diameter-$2$ in polynomial-time

We show that list $3$-coloring a~$C_4$-free graph of diameter-$2$ can be done in polynomial-time. Our algorithm is based on a structural characterization showing that many such graphs are not~$3$-colorable. In particular, we show that~$C_4$-free graphs of diameter-$2$ without universal vertices, where the maximum degree is at least~$17$, are not~$3$-colorable.