We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $binom{2k}{k}$ steps with generalized rectangles there is a row or
a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{v{r}}{i}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))log_2log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.
Let ${cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The emph{$k$-level} of ${cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${cal L}$ passing below $v$. The complexity (the maximum size) of th
e $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(ncdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $mathbb{S}^d$ which are orthogonal to a set of random points on $mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Theta((k+1)^{d-1})$.
A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d in {2,3}$) is calle
d the emph{$d$-dimensional weak line cover number} and denoted by $pi^1_d(G)$. In 3D, the minimum number of emph{planes} needed to cover all vertices of~$G$ is denoted by $pi^2_3(G)$. When edges are also required to be covered, the corresponding numbers $rho^1_d(G)$ and $rho^2_3(G)$ are called the emph{(strong) line cover number} and the emph{(strong) plane cover number}. Computing any of these cover numbers -- except $pi^1_2(G)$ -- is known to be NP-hard. The complexity of computing $pi^1_2(G)$ was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph~$G$, whether $pi^1_2(G)=2$. We further show that the universal stacked triangulation of depth~$d$, $G_d$, has $pi^1_2(G_d)=d+1$. Concerning~3D, we show that any $n$-vertex graph~$G$ with $rho^2_3(G)=2$ has at most $5n-19$ edges, which is tight.
Research about crossings is typically about minimization. In this paper, we consider emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any g
raph has a emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.