On Degree Properties of Crossing-Critical Families of Graphs

TL;DR

This paper constructs infinite crossing-critical graph families with prescribed odd degrees, answering a 2007 open question and exploring degree behaviors in such graphs.

Contribution

It introduces properties and operations on infinite graph families to control vertex degrees, solving a longstanding open problem in crossing-critical graph theory.

Findings

Existence of infinite $k$-crossing-critical families with prescribed odd degrees.

Ability to specify degree sets and average degrees in crossing-critical families.

Construction methods for preserving properties across graph families.

Abstract

Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large~$k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $\min(D)\geq 3$ and $3,4\in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.