Technical details regarding infinite families of crossing-critical graphs with prescribed average degree and crossing number

TL;DR

This paper presents new constructions of crossing-critical graphs with prescribed average degrees in (3,6), unifying previous results and answering Salazar's question about their existence for all rational degrees in that range.

Contribution

It introduces two new graph constructions and a unifying approach to establish infinite families of crossing-critical graphs with any rational average degree in (3,6).

Findings

Existence of infinite families with average degree in (3,6)

Universal lower bound on crossing number for given average degree

Supplementary Mathematica notebook for verification

Abstract

Siran constructed infinite families of k-crossing-critical graphs for every k > 2 and Kochol constructed such families of simple graphs for every k > 1. Richter and Thomassen argued that, for any given k > 0 and r > 5, 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 the 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 (3.5, 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: there exist infinite families of simple k-crossing-critical graphs with any prescribed average degree r in (3, 6), for any k greater than some lower bound N(r). Moreover, a universal lower bound N(I) on k applies for rational numbers in any closed interval I contained in (3, 6). This Mathematica notebook is presented as a supplement of the paper with the aforementioned results. It contains technical details omitted in the paper and can be used as a hint of how to rigorously verify the constraints that are imposed on the parameters in the main construction of the paper. The reader may either use Mathematica or some other software to verify the listed claims, or may derive them in a more clear, oldfashioned way. As the notebook is not self-contained, it is advisable to read the paper before.