On semi-simple drawings of the complete graph

Oswin Aichholzer, Florian Ebenführer, Irene Parada, Alexander Pilz, Birgit Vogtenhuber

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandBegutachtung


In this work we study rotation systems and semi-simple drawings of $K_n$. A simple drawing of a graph is a drawing in which every pair of edges intersects in at most one point. In a semi-simple drawing, edge pairs might intersect in multiple points, but incident edges only intersect in their common endpoint. A rotation system is called (semi-)realizable if it can be realized with a (semi-)simple drawing. It is known that a rotation system is realizable if and only if all its 5-tuples are realizable. For the problem of characterizing semi-realizability, we present a rotation system with six vertices that is not semi-realizable, although all its 5-tuples are semi-realizable. Moreover, by an exhaustive computer search, we show that also for seven vertices there exist minimal not semi-realizable rotation systems (that is, rotation systems in which all proper sub-rotation systems are semi-realizable). This indicates that checking semi-realizability is harder than checking realizability. Finally we show that for semi-simple drawings, generalizations of Conway's Thrackle Conjecture and the conjecture on the existence of plane Hamiltonian cycles do not hold.
TitelProc. XVII Encuentros de Geometría Computacional
ErscheinungsortAlicante, Spain
PublikationsstatusVeröffentlicht - 2017
VeranstaltungXVII Spanish Meeting on Computational Geometry - , Spanien
Dauer: 26 Juni 201728 Juni 2017


KonferenzXVII Spanish Meeting on Computational Geometry
KurztitelEGC 2017

Dieses zitieren