Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists -regular graphs with arbitrarily large geometric thickness. In particular, for all 9 and for all large n, there exists a -regular graph with geometric thickness at least c n1/2-4/. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmovi

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: János Barát, Jirí Matousek, David R. Wood
• Keywords: -regular Graph, COMBINATORICS 2006, Geometric Thickness, Large Geometric Thickness