Game colouring directed graphs (Q2380443)

Summary: A colouring game and two versions of marking games (the weak and the strong) on digraphs are studied. We introduce the weak game chromatic number \(\chi_{\text{wg}}(D)\) and the weak game colouring number \(\text{wgcol}(D)\) of digraphs \(D\). It is proved that if \(D\) is an oriented planar graph, then \(\chi_{\text{wg}}(D)\leq\text{wgcol}(D)\leq 9\), and if \(D\) is an oriented outerplanar graph, then \(\chi_{\text{wg}}(D)\leq\text{wgcol}(D)\leq 4\). Then we study the strong game colouring number \(\text{sgcol}(D)\) (which was first introduced by Andres as game colouring number) of digraphs \(D\). It is proved that if \(D\) is an oriented planar graph, then \(\text{sgcol}(D)\leq 16\). The asymmetric versions of the colouring and marking games of digraphs are also studied. Upper and lower bounds of related parameters for various classes of digraphs are obtained.