Faces in rectilinear drawings of complete graphs

Martin Balko, Anna Brötzner, Fabian Klute, Josef Tkadlec

Abstract: We initiate the study of extremal problems about faces in convex rectilinear drawings of Kn<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of Kn<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex k<math><mi is="true">k</mi></math>-gon with k≥6<math><mrow is="true"><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">6</mn></mrow></math>.A convex rectilinear drawing of Kn<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> is regular if its vertices correspond to vertices of a regular convex n<math><mi is="true">n</mi></math>-gon. We characterize positive integers n<math><mi is="true">n</mi></math> for which regular drawings of Kn<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> contain a face forming a convex 5-gon.To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.