Digraph Coloring Games and Game-Perfectness

In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix

On characterizing game-perfect graphs by forbidden induced subgraphs

A graph $G$ is called $g$-perfect if, for any induced subgraph $H$ of $G$, the game chromatic number of $H$ equals the clique number of $H$. A graph $G$ is called $g$-col-perfect if, for any induced subgraph $H$ of $G$, the game coloring number of $H$ equals the clique number of $H$. In this paper we characterize the classes of $g$-perfect resp. $g$-col-perfect graphs by a set of forbidden induced subgraphs and explicitly. Moreover, we study similar notions for variants of the game chromatic number, namely $B$-perfect and $[A,B]$-perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

ATHENA Research Book, Volume 2

ATHENA European University is an association of nine higher education institutions with the mission of promoting excellence in research and innovation by enabling international cooperation. The acronym ATHENA stands for Association of Advanced Technologies in Higher Education. Partner institutions are from France, Germany, Greece, Italy, Lithuania, Portugal and Slovenia: University of Orléans, University of Siegen, Hellenic Mediterranean University, Niccolò Cusano University, Vilnius Gediminas Technical University, Polytechnic Institute of Porto and University of Maribor. In 2022, two institutions joined the alliance: the Maria Curie-Skłodowska University from Poland and the University of Vigo from Spain. Also in 2022, an institution from Austria joined the alliance as an associate member: Carinthia University of Applied Sciences. This research book presents a selection of the research activities of ATHENA University's partners. It contains an overview of the research activities of individual members, a selection of the most important bibliographic works of members, peer-reviewed student theses, a descriptive list of ATHENA lectures and reports from individual working sections of the ATHENA project. The ATHENA Research Book provides a platform that encourages collaborative and interdisciplinary research projects by advanced and early career researchers

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Lightness of digraphs in surfaces and directed game chromatic number

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