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Infinite families of crossingcritical graphs with prescribed average degree and crossing number
, 2006
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Crossing Numbers and Cutwidths
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2003
"... The crossing number of a graph G =(V,E), denoted by cr(G), is the smallest number of edge crossings in any drawing of G in the plane. ..."
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The crossing number of a graph G =(V,E), denoted by cr(G), is the smallest number of edge crossings in any drawing of G in the plane.
with prescribed average degree and crossing number
, 2006
"... iráň constructed infinite families of kcrossingcritical 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 kcrossingcritical graphs with minimum degree ..."
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iráň constructed infinite families of kcrossingcritical 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 kcrossingcritical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple kcrossingcritical graphs of prescribed average degree r> 6. He established existence of infinite families of simple kcrossingcritical 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 r ∈ (31 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 ∈ (3, 6) there exists an integer Nr, such that, for any k> Nr, there exists an infinite family of simple 3connected crossingcritical 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 ⊂ (3, 6).