On the Edge-Vertex Ratio of Maximal Thrackles

Oswin Aichholzer, Linda Kleist, Boris Klemz, Felix Schröder, Birgit Vogtenhuber

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Abstract

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$.
Original languageEnglish
Title of host publicationGraph Drawing and Network Visualization. GD 2019
Place of PublicationPrague, Czechia
Pages482-495
Number of pages14
Volume11904
DOIs
Publication statusPublished - 2019

Publication series

NameLecture Notes in Computer Science (LNCS)

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Fields of Expertise

  • Information, Communication & Computing

Cite this

Aichholzer, O., Kleist, L., Klemz, B., Schröder, F., & Vogtenhuber, B. (2019). On the Edge-Vertex Ratio of Maximal Thrackles. In Graph Drawing and Network Visualization. GD 2019 (Vol. 11904, pp. 482-495). (Lecture Notes in Computer Science (LNCS)). Prague, Czechia. https://doi.org/10.1007/978-3-030-35802-0_37, https://doi.org/10.1007/978-3-030-35802-0_37