It has been long-conjectured that the crossing number cr(K m,n ) of the complete bipartite graph K m,n equals the Zarankiewicz Number Z(m, n) :Another long-standing conjecture states that the crossing number cr(K n ) of the complete graph K n equals Z(n) := 1 4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values:The previous best known lower bounds were 0.8m/(m − 1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K 7,n ) ≥ 2.1796n 2 − 4.5n. To obtain this improved lower bound for cr(K 7,n ), we use some elementary topological facts on drawings of K 2,7 to set up a quadratic program on 6! variables whose minimum p satisfies cr(K 7,n ) ≥ (p/2)n 2 − 4.5n, and then use state-ofthe-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593.

A c-crossing-critical graph is one that has crossing number at least c but each of its proper subgraphs has crossing number less than c. Recently, a set of explicit construction rules was identified by Bokal, Oporowski, Richter, and Salazar to generate all large 2-crossing-critical graphs (i.e., all apart from a finite set of small sporadic graphs). They share the property of containing a generalized Wagner graph V10 as a subdivision.In this paper, we study these graphs and establish their order, simple crossing number, edge cover number, clique number, maximum degree, chromatic number, chromatic index, and treewidth. We also show that the graphs are linear-time recognizable and that all our proofs lead to efficient algorithms for the above measures.

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z (n) := 1

There are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of Kn have n2+O(nlogn) and O(n2), respectively, empty triangles. All the arguments are elementary, algorithmic, and self‐contained.

Around 1958, Hill described how to draw the complete graph K n with Z (n) := 1 4 n 2 n − 1 2 n − 2 2 n − 3 2 K n in terms of its number of (≤ k)-edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of K n and show that, up to equivalence, they are unique for n even, but that there exist an exponential number of non-homeomorphic such drawings for n odd.