Graphs drawn with few crossings per edge Article Received: 09 July 1996 Revised: 09 January 1998 DOI:
Cite this article as: Pach, J. & Tóth, G. Combinatorica (1997) 17: 427. doi:10.1007/BF01215922 Abstract
We show that if a graph of
v vertices can be drawn in the plane so that every edge crosses at most k>0 others, then its number of edges cannot exceed 4.108√ kv. For k≤4, we establish a better bound, ( k+3)( v−2), which is tight for k=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges. Mathematics Subject Classification (1991) 05C10
Supported by NSF grant CCR-94-24398 and PSC-CUNY Research Award 667339.
Supported by OTKA-T-020914, OTKA-F-22234 and the Margaret and Herman Sokol Postdoctoral Fellowship Award.
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© János Bolyai mathematical Society 1997