2010/02/20 by Radoslav Fulek, Fulek, Radoslav, Janos Pach +1 · 2 citations
Mathematics · #05C10 #68R10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C10 #msc:68R10
paper · pdf · doi:10.48550/arxiv.1002.3904
16 pages, 7 figures
arxiv created 2010/02/20 · arxiv updated 2010/02/26
A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any eps>0, we give an algorithm terminating in eO((1/eps2)ln(1/eps)) steps to decide whether t(n)<(1+eps)n for all n>2. Using this approach, we improve the best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to 167/117n<1.428n.