On the Uncrossed Number of Graphs

Visualizing a graph G in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masar\'ik and Hlinen\'y GD 2023 recently asked for each edge of G to be drawn without crossings while allowing multiple different drawings of G. More formally, a collection D of drawings of G is uncrossed if, for each edge e of G, there is a drawing in D such that e is uncrossed. The uncrossed number unc (G) of G is then the minimum number of drawings in some uncrossed collection of G. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hlinen\'y and Masar\'ik. We also present a strong general lower bound on unc (G) in terms of the number of vertices and edges of G. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph G, which is the smallest number of edges of G taken over all drawings of G that participate in a crossing. This problem was posed as open by Schaefer in his book Crossing Numbers of Graphs 2018.