TY - GEN
T1 - On the Edge-Vertex Ratio of Maximal Thrackles
AU - Aichholzer, Oswin
AU - Kleist, Linda
AU - Klemz, Boris
AU - Schröder, Felix
AU - Vogtenhuber, Birgit
PY - 2019
Y1 - 2019
N2 - A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway\â\\s conjecture states that a thrackle has at most as many edges as vertices. In this paper, we investigate the edge-vertex ratio of maximal thrackles, that is, thrackles in which no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles, we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to the natural lower bound of $12$. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio arbitrary close to~$ 45$.
AB - A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway\â\\s conjecture states that a thrackle has at most as many edges as vertices. In this paper, we investigate the edge-vertex ratio of maximal thrackles, that is, thrackles in which no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles, we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to the natural lower bound of $12$. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio arbitrary close to~$ 45$.
U2 - https://doi.org/10.1007/978-3-030-35802-0_37
DO - https://doi.org/10.1007/978-3-030-35802-0_37
M3 - Conference paper
VL - 11904
T3 - Lecture Notes in Computer Science (LNCS)
SP - 482
EP - 495
BT - Graph Drawing and Network Visualization. GD 2019
CY - Prague, Czechia
ER -