Using a refinement of the methods of Erdös et al. [6] we prove that the game chromatic index of forests of maximum node degree 5 is at most 6. This improves the previously known upper bound 7 for this parameter. The bound 6 is tight [6].

The clique number ω(D) of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number χ(H) defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number ω(H). Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐scriptNP‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is NP‐complete to decide whether a perfect digraph has a kernel.

The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k, and without 4-cycles, which can be embedded in a surface S. (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k ≥ 5, these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [11] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game coloring number, and thus for the game chromatic number of (di)graphs with girth k and without 4-cycles embeddable in S. The game chromatic resp. game coloring number were introduced by Bodlaender [3] resp. Zhu [22] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game coloring number of a directed simple forest is at most 3.

We introduce the incidence game chromatic number which unifies the ideas of game chromatic number and incidence coloring number of an undirected graph. For k-degenerate graphs with maximum degree ∆, the upper bound 2∆ + 4k − 2 for the incidence game chromatic number is given. If ∆ ≥ 5k, we improve this bound to the value 2∆ + 3k − 1. We also determine the exact incidence game chromatic number of cycles, stars and sufficiently large wheels and obtain the lower bound 3 2 ∆ for the incidence game chromatic number of graphs of maximum degree ∆.