Beyond-Planarity: Density Results for Bipartite Graphs

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter by edge crossings should be minimized (e.g., Huang et al. 2008) that could negatively affect the readability of the drawing. Different types of forbidden crossing configurations give rise to different families of nearly-planar graphs. Most of the literature focuses on Tur\'an-type problems, which ask for the maximum number of edges a nearly-planar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of nearly-planar graphs, i.e., 1-planar, 2-planar, fan-planar, and RAC graphs. We prove bounds on the number of edges that are tight up to small additive constants; some of them are surprising and not along the lines of the known results for non-bipartite graphs. Our findings lead to an improvement of the leading constant of the well-known Crossing Lemma for bipartite graphs, as well as to a number of interesting research questions on topological graphs.