Abstract. We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes, {0, π/4, π/2, 3π/4}. We also prove that four slopes have this property if and only if we can draw K4 with them.

We show that every connected cubic graph can be can be drawn in the plane with straight line edges using only four distinct slopes.

Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It is shown that there are graphs on n vertices with obstacle number at least Ω (n/log n).

Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least Ω ( √ log n). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω (n/log 2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most 2 o(n 2 ) . Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides.

We show that there is an m = 2n + o(n), such that, in the Maker-Breaker game played on Z d where Maker needs to put at least m of his marks consecutively in one of n given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg [12] who showed that such a pairing strategy exits if m ≥ 3n. A simple argument shows that m has to be at least 2n + 1 if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.