Waiter–Client and Client–Waiter planarity, colorability and minor games

Planarity, Colorability, and Minor Games

Letm and b be positive integers and let F be a hypergraph. In an (m, b) Maker-Breaker game F two players, called Maker and Breaker, take turns selecting previously unclaimed vertices of F . Maker selects m vertices per move and Breaker selects b vertices per move. The game ends when every vertex has been claimed by one of the players. Maker wins if he claims all the vertices of some hyperedge o...

Waiter-Client and Client-Waiter planarity, colorability and minor games

For a finite set X, a family of sets F ⊆ 2X and a positive integer q, we consider two types of two player, perfect information games with no chance moves. In each round of the (1 : q) Waiter-Client game (X,F), the first player, called Waiter, offers the second player, called Client, q + 1 elements of the board X which have not been offered previously. Client then chooses one of these elements w...

On planarity and colorability of circulant graphs

For given positive integers n; a1; : : : ; am, we consider the undirected circulant graph G=(V; E) with set of vertices V = {0; : : : ; n − 1} and set of edges E = {[i; j]: i − j ≡ ±ak (mod n) for some 16 k6m}. We prove that G is planar if m = 1 and non-planar if m¿ 3. For m = 2 we completely characterize planarity. It is shown that G is bipartite if and only if there is an l such that 2 divide...

Colorability Saturation Games

We consider the following two-player game: Maxi and Mini start with the empty graph on n vertices and take turns, always adding one additional edge to the graph such that the chromatic number is at most k, where k ∈ N is a given parameter. The game is over when the graph is saturated and no further edge can be inserted. Maxi wants to maximize the length of the game while Mini wants to minimize ...

On Free Minor Closed Classes of Graphs Outside Planarity