Crossing numbers of imbalanced graphs

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Crossing numbers of imbalanced graphs
The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of Ajtai, Chv´atal, Newborn, Szemer´edi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e3 /n2 . Apart from the value of the constant, this bound cannot be improved. We establish some stronger lower bounds, under the assumption that the distribution of the degrees of the vertices is irregular. In particular, we show that if the degrees of the vertices are d1 ≥ d2 ≥ . . . ≥ dn, then the crossing number satisfies cr(G) ≥ c1 n n i=1 id3 i − c2n2 , and that this bound is tight apart from the values of the constants c1, c2 > 0. Some applications are also presented.
János Pach, József Solymosi, G&aacut
Added 28 Jan 2011
Updated 28 Jan 2011
Type Journal
Year 2010
Where JGT
Authors János Pach, József Solymosi, Gábor Tardos
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