# Bounded maximum degree conjecture holds precisely for $c$-crossing-critical graphs with $c \leq 12$

## TL;DR

This paper proves that the bounded maximum degree conjecture for $c$-crossing-critical graphs is true precisely for $c \,\leq\, 12$, providing explicit constructions for $c \,\geq\, 13$ and establishing degree bounds.

## Contribution

It explicitly constructs $c$-crossing-critical graphs with unbounded degree for $c \,\geq\, 13$ and proves the conjecture holds exactly for $c \,\leq\, 12$, clarifying the conjecture's validity range.

## Key findings

- Explicit constructions for $c \,\geq\, 13$ with unbounded degree.
- Existence of a constant $D$ bounding degree for $c \,\leq\, 12$.
- Confirmation that the conjecture holds exactly for $c \,\leq\, 12$.

## Abstract

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 $c\ge 13$ and $d\ge 1$, 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 $c\le 12$, precisely, that there exists a constant $D$ such that every $c$-crossing-critical graph with $c\le 12$ has maximum degree at most $D$. Hence, the bounded maximum degree conjecture of $c$-crossing-critical graphs, which was generally disproved in 2010 by Dvo\v{r}\'ak and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values $c\le 12.$