Grid Drawings and the Chromatic Number

Balko, Martin (2013) Grid Drawings and the Chromatic Number. In: 20th International Symposium, GD 2012, September 19-21, 2012 , pp. 315-326(Official URL:

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A grid drawing of a graph maps vertices to the grid $ℤ^d$ and edges to line segments that avoid grid points representing other vertices. We show that a graph G is $q^d$ -colorable, $d, q ≥ 2$, if and only if there is a grid drawing of G in $ℤ^d$ in which no line segment intersects more than q grid points. This strengthens the result of D. Flores Pen̋aloza and F. J. Zaragoza Martinez. Second, we study grid drawings with a bounded number of columns, introducing some new NP-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by D. Flores Pen̋aloza and F. J. Zaragoza Martinez.

Item Type: Conference Paper
Additional Information: 10.1007/978-3-642-36763-2_28
Classifications: G Algorithms and Complexity > G.070 Area / Edge Length
P Styles > P.720 Straight-line

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