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تسجيل مستخدم جديدMidrange crossing constants for graphs classes
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For positive integers $n$ and $e$, let $kappa(n,e)$ be the minimum crossing number (the standard planar crossing number) taken over all graphs with $n$ vertices and at least $e$ edges. Pach, Spencer and Toth [Discrete and Computational Geometry 24 623--644, (2000)] showed that $kappa(n,e) n^2/e^3$ tends to a positive constant (called midrange crossing constant) as $nto infty$ and $n ll e ll n^2$, proving a conjecture of ErdH{o}s and Guy. In this note, we extend their proof to show that the midrange crossing constant exists for graph classes that satisfy a certain set of graph properties. As a corollary, we show that the the midrange crossing constant exists for the family of bipartite graphs. All these results have their analogues for rectilinear crossing numbers.
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Sunil Kumar R (Research Scholar
, Department of Computer Applications
, n Cochin University of Science
2014
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