The game chromatic number and the game colouring number of classes of oriented cactuses

We prove that the game chromatic and the game colouring number of the class of orientations of cactuses with girth of 2 or 3 is 4. We improve this bound for the class of orientations of certain forest-like cactuses to the value of 3. These results generalise theorems on the game colouring number of undirected forests (Faigle et al., 1993 [3]) resp. orientations of forests (Andres, 2009 [1]). For certain undirected cactuses with girth 4 we also obtain the tight bound 4, thus improving a result of Sidorowicz (2007) [8].

Research highlights► We consider colouring games for digraphs and their game colouring numbers (gcn). ► We examine the gcn of classes of orientations of cactuses. ► The gcn of the class of orientations of cactuses of girth 2 or 3 is 4. ► The gcn of directed cactus trees is at most 3. ► The gcn of undirected forests with thin 4-cycles is at most 4.