Abstract
Širáň 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 the existence of infinite families of simple k‐crossing‐critical graphs with any prescribed rational average degree r∈[4, 6) for infinitely many k and asked about their existence for r∈(3, 4). The question was partially settled by Pinontoan and Richter, who answered
it positively for \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$r\in(3\frac{1}{2},4)$\end{document}.
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∈(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⊂(3, 6). © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 139–162, 2010