ω-searchlight obedient graph drawings

Abstract

A drawing of a graph in the plane is ω-searchlight obedient if every vertex of the graph is located on the centerline of some strip of width ω, which does not contain any other vertex of the graph. We estimate the maximum possible value ω(n) of an ω-searchlight obedient drawing of a graph with n vertices, which is contained in the unit square. We show a lower bound and an upper bound on ω(n), namely, ω(n) =ωΩ (log n/n) and ω(n) = O(1=n4\7−ε), for an arbitrarily small ε > 0. Any improvement for either bound will also carry on to the famous Heilbronn's triangle problem.