Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆-regular n-vertex graph with slope-number at least n1− 8+ε ∆+4 . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most O(log n). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered. †Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada ([email protected]). Supported by NSERC. §McGill Centre for Bioinformatics, School of Computer Science, McGill University, Montréal, Québec, Canada ([email protected]). Supported by NSERC. ¶Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Catalunya, Spain ([email protected]). Supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224. ∗A preliminary version of this paper was published as: “Really straight graph drawings.” Proceedings of the 12th International Symposium on Graph Drawing (GD ’04), Lecture Notes in Computer Science 3383:122–132, Springer, 2004.