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Grid representations of graphs and the chromatic number
Balko, Martin ; Valtr, Pavel (advisor) ; Kratochvíl, Jan (referee)
Grid Representations and the Chromatic Number Martin Balko August 2, 2012 Department: Department of Applied Mathematics Supervisor: doc. RNDr. Pavel Valtr Dr. Supervisor's email address: valtr@kam.mff.cuni.cz Abstract In the thesis we study grid drawings of graphs and their connections with graph colorings. A grid drawing of a graph maps vertices to distinct points of the grid Zd and edges to line segments that avoid grid points representing other vertices. We show that a graph G is qd -colorable, d, q ≥ 2, if and only if there is a grid drawing of G in Zd in which no line segment intersects more than q grid points. Second, we study grid drawings with bounded number of columns, introducing some new NP- complete problems. We also show a sharp lower bound on the area of plane grid drawings of balanced complete k-partite graphs, proving a conjecture of David R. Wood. 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 of D. Flores Pe˝naloza and F. J. Zaragoza Martinez. Keywords: graph representations, grid, chromatic number, plane
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