Grid Drawings and the Chromatic NumberBalko, Martin (2013) Grid Drawings and the Chromatic Number. In: 20th International Symposium, GD 2012, September 1921, 2012 , pp. 315326(Official URL: http://link.springer.com/chapter/10.1007/9783642...). Full text not available from this repository.
Official URL: http://link.springer.com/chapter/10.1007/9783642...
AbstractA 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 NPcomplete 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.
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