Infinite families of crossing-critical graphs with prescribed average degree and crossing number

TL;DR

This paper constructs infinite families of crossing-critical graphs with prescribed average degree and crossing number, unifying previous results and answering open questions for all rational degrees between 3 and 6.

Contribution

It introduces two new graph constructions and combines existing methods to prove the existence of infinite crossing-critical graphs with any rational average degree in (3,6).

Findings

Existence of infinite families for all rational degrees in (3,6).

Universal lower bounds on crossing number for given average degrees.

Unification of previous partial results into a comprehensive framework.

Abstract

Siran constructed infinite families of k-crossing-critical graphs for every k=>3 and Kochol constructed such families of simple graphs for every k=>2. Richter and Thomassen argued that, for any given k>=1 and r>=6, there are only finitely many simple k-crossing-critical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k-crossing-critical graphs of prescribed average degree r>6. He established existence of infinite families of simple k-crossing-critical graphs with any prescribed rational average degree r in [4,6) for infinitely many k and asked about their existence for r in (3,4). The question was partially settled by Pinontoan and Richter, who answered it positively for r in (7/2,4). The present contribution uses two new constructions of crossing critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar's question by the following statement: for every rational number r in (3,6) there exists an integer N_r, such that, for any k>N_r, there exists an infinite family of simple 3-connected crossing-critical graphs with average degree r and crossing number k. Moreover, a universal lower bound on k applies for rational numbers in any closed interval I in (3,6).