Abstract. A [Formula: see text]- page book drawing of a graph [Formula: see text] is a drawing of [Formula: see text] on [Formula: see text] halfplanes with a line [Formula: see text] as a common boundary such that the vertices are located on [Formula: see text] and the edges cannot cross [Formula: see text]. The [Formula: see text]- page book crossing number of the graph [Formula: see text], denoted by [Formula: see text], is the minimum number of edge-crossings over all [Formula: see text]-page book drawings of [Formula: see text]. This paper improves previous results on [Formula: see text]-page book crossing numbers of the complete graph [Formula: see text]. We determine [Formula: see text] whenever [Formula: see text] and improve the lower bounds on [Formula: see text] for all [Formula: see text]. Our proofs rely on bounding the number of edges in convex geometric graphs with few crossings per edge.

We determine the minimum number of crossings in an outerplanar drawing of the complete bipartite graph Km,n for any values of m and n, which was known only for the specific case when m|n. Moreover, we provide a one-to-one correspondence (up to equivalence) between the outerplanar and cylindrical drawings of the complete bipartite graph that preserves the number of crossings but not the crossings themselves.

The crossing number of a graph G is the minimum number of edge-crossings over all drawings of G in the plane. Determining the crossing number of a graph is a well-known problem in combinatorics. One of its most studied variants is to restrict the minimum to all drawings of G whose edges are straight line segments. This minimum is known as the geometric crossing number of G. We determine the exact crossing number of centrally symmetric geometric drawings of the complete graph.

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is \(H(n) := \frac 1 4 łfloor\fracn2\rfloor łfloor\fracn-12\rfloor łfloor\fracn-22\rfloor łfloor\fracn-32\rfloor.\) Ábrego et al. [ Discrete Comput. Geom., 52 (2014), pp. 743--753] introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor-2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s\geq5$. This shows that we have properly extended the class of drawings for which the Harary--Hill conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(\lfloor \frac{n}{2} \rfloor-2)$-bishellable, but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.

Given a large weighted graph G = (V, E) and a subset U of V , we define several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most common proximity graphs in Euclidean spaces. We prove basic properties of the defined graphs and provide algorithms for their computation.

Around 1958, Hill conjectured that the crossing number CRg(Kn) of the complete graph KKn is Z(n):=1/4 ⌊ n/2 ⌋ ⌊(n-1)/2⌋ ⌊ (n-2)/2 ⌋ ⌊ (n-3)/2 ⌋ and provided drawings of Kn with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of Kn with Z(n) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line l (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by l. The 2-page crossing number of Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2-page book drawing of Kn. Since CRG(Kn) ≤ ν2(Kn) and ν2(Kn) ≤ Z(n), a natural step towards Hill's Conjecture is the weaker conjecture ν2(Kn) = Z(n), that was popularized by Vrt'o. In this paper we develop a novel and innovative technique to investigate crossings in drawings of Kn, and use it to prove that ν2(Kn) = Z(n). To this end, we extend the inherent geometric definition of k-edges for finite sets of points in the plane to topological drawings of Kn. We also introduce the concept of ≤≤k-edges as a useful generalization of ≤k-edges. Finally, we extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of Kn in terms of its number of k-edges to the topological setting.