7 research outputs found
A new upper bound on the game chromatic index of graphs
We study the two-player game where Maker and Breaker alternately color the
edges of a given graph with colors such that adjacent edges never get
the same color. Maker's goal is to play such that at the end of the game, all
edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored
edge where every color is blocked. The game chromatic index
denotes the smallest for which Maker has a winning strategy.
The trivial bounds hold for every
graph , where is the maximum degree of . In 2008, Beveridge,
Bohman, Frieze, and Pikhurko proved that for every there exists a
constant such that holds for any graph
with , and conjectured that the same
holds for every graph . In this paper, we show that is true for all graphs with . In
addition, we consider a biased version of the game where Breaker is allowed to
color edges per turn and give bounds on the number of colors needed for
Maker to win this biased game.Comment: 17 page
The game of arboricity
Using a fixed set of colors , Ann and Ben color the edges of a graph so that no monochromatic cycle may appear. Ann wins if all edges of have been colored, while Ben wins if completing a coloring is not possible. The minimum size of for which Ann has a winning strategy is called the of , denoted by . We prove that for any graph of arboricity , and that there are graphs such that . The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction
On characterizing game-perfect graphs by forbidden induced subgraphs
A graph is called -perfect if, for any induced subgraph of , the game chromatic number of equals the clique number of . A graph is called -col-perfect if, for any induced subgraph of , the game coloring number of equals the clique number of . In this paper we characterize the classes of -perfect resp. -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 -perfect and -perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs
Recommended from our members
Mini-Workshop: Positional Games
Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions