Contemporary Mathematics Three-Dimensional Grid Drawings with Sub-Quadratic Volume
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by Vida Dujmović, David R. Wood
http://www.scs.carleton.ca/~davidw/papers/DujWoo-SubQuadraticDrawings.pdf
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Abstract:
Abstract. A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n 3/2) volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n 2). These results (partially) solve open problems due to Pach, Thiele, and Tóth (1997) and Felsner, Liotta, and Wismath (2001). 1.
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