A Note on Choosability with Separation for Planar Graphs

A graph \(G\) is \((p, q, r)\)-choosable if for every list assignment \(L\) with \(|L(v)| \geq p\) for each \(v \in V(G)\) and \(|L(u) \cap L(v)| < p – r\) whenever \(u, v\) are adjacent vertices, \(G\) is \(q\)-tuple \(L\)-colorable. We give an alternative proof of \((4t, t, 3t)\)-choosability for the planar graphs and construct a triangle-free planar graph on \(119\) vertices which is not \((3, 1, 1)\)-choosable (and so neither \(3\)-choosable). We also propose some problems.