Combinatorics: The Probabilistic Method

On planar graphs, orientations, and contact graphs: for a planar graph where no two vertices in R 2 have the same y-coordinate, orient any edge uv so that u → v where the y-coordinate of v is greater than that of u. In the dual, orient any edge so that they run clockwise from the edges it crosses in the original graph. (Ex 1) A planar 2-connected graph has a drawing such that its inherited orientation has a single source/sink. (Ex 2) The dual orientation of a graph is acyclic. A graph's vertices can be represented by segments in R 2 , where intersections of segments corresponds to adjacency of vertices. It is NP-complete to decide whether a partial representation of segments corresponding to a subgraph of V (G) can be completed. (Ex 3) A planar bipartite graph is a contact graph of vertical and horizontal segments. (Ex 4) Find some non-trivial condition on the parital exponentiation to make the above problem P-time solvable. A tournament is a directed graph which without orientation is equivalent to K n. (Ex 1) Does there exist a tournament such that for every v 1 , v 2 ∈ V (G), there exists u such that u beats v 1 , v 2 ?