We settle a problem of Dujmović, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straight-line edges, using at most f (d) different slopes. If we allow the edges to be represented by polygonal paths with one bend, then 2d slopes suffice. Allowing two bends per edge, every planar graph with maximum degree d ≥ 3 can be drawn using segments of at most ⌈d/2⌉ different slopes. There is only one exception: the graph formed by the edges of an octahedron is 4-regular, yet it requires 3 slopes. These bounds cannot be improved.
As already stated, Keszegh et al. REF have proved that every planar graph of maximum degree k admits a planar drawing with at most k 2 different slopes in which each edge has at most two bends.
Drawing planar graphs of bounded degree with few slopes
ArXiv
ArXiv
Computer Science Mathematics