Bounded maximum degree conjecture holds precisely for c-crossing-critical graphs with c ≤ 12

03/13/2019
by   Drago Bokal, et al.
0

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c> 13 and d> 1, we give first explicit constructions of c-crossing-critical graphs containing a vertex of degree greater than d. We also show that such unbounded degree constructions do not exist for c< 12, precisely, that there exists a constant D such that every c-crossing-critical graph with c< 12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c< 12.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/05/2018

Structure and generation of crossing-critical graphs

We study c-crossing-critical graphs, which are the minimal graphs that r...
research
12/09/2021

Properties of Large 2-Crossing-Critical Graphs

A c-crossing-critical graph is one that has crossing number at least c b...
research
06/06/2023

Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs

In this paper we deal with the problem of computing the exact crossing n...
research
07/31/2018

Tight Upper Bounds on the Crossing Number in a Minor-Closed Class

The crossing number of a graph is the minimum number of crossings in a d...
research
12/15/2019

Meyniel's conjecture on graphs of bounded degree

The game of Cops and Robbers is a well known pursuit-evasion game played...
research
06/29/2022

RAC Drawings of Graphs with Low Degree

Motivated by cognitive experiments providing evidence that large crossin...
research
12/01/2021

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

A graph G is called self-ordered (a.k.a asymmetric) if the identity perm...

Please sign up or login with your details

Forgot password? Click here to reset