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

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

2009-09-14

arxiv.org/abs/0909.2561v1

Mathematics

Combinatorics

21 pages, 5 figures

Scientific paper

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).

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