On k-planar Graphs without Short Cycles
Michael A. Bekos, Prosenjit K. Bose, Aaron Büngener, Vida Dujmović, Michael M. Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, Alexandra Weinberger · Journal of Graph Algorithms and Applications · 2025
We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c\sqrt{k}n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c\sqrt{k}n, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2-- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.