The n crossing number of a graph G, denoted crn(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a > b > 0, there exists a graph G for which cr0(G) = a, cr1(G) = b, and cr2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.

We show that under suitable non-degeneracy conditions, m points and n 2–dimensional algebraic surfaces in R satisfying certain “pseudoflat” requirements can have at most O ( mn + m + n ) incidences, provided that m ≤ n2− for any > 0 (where the implicit constant in the above bound depends on ), or m ≥ n. As a special case, we obtain the Szemerédi-Trotter theorem for 2–planes in R, again provided...

4 We recall that a book with k pages consists of a straight line (the spine) and k half5 planes (the pages), such that the boundary of each page is the spine. If a graph is drawn 6 on a book with k pages in such a way that the vertices lie on the spine, and each edge 7 is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). 8 The pagenumber of a graph G is the ...

This paper investigates on those smallest regular graphs with given girths and having small crossing numbers.

Let Pm Pn be the strong product of two paths Pm and Pn. In 2013, Klešč et al. conjectured that the crossing number of Pm Pn is equal to (m − 1)(n − 1) − 4 for m ≥ 4 and n ≥ 4. In this paper we show that the above conjecture is true except when m = 4 and n = 4, and that the crossing number of P4 P4 is four.

There are three general lower bound techniques for the crossing numbers of graphs, all of which can be traced back to Leighton’s work on applications of crossing number in VLSI: the Crossing Lemma, the Bisection Method, and the Embedding Method. In this contribution, we sketch their adaptations to the minor crossing number.

We prove that for every k > 0 there is an integer n0(k) such that, for every n > n0, there exists a hypohamiltonian graph which has order n and crossing number k.

We prove that the crossing number of a graph decays in a " continuous fashion " in the following sense. For any ε > 0 there is a δ > 0 such that for a sufficiently large n, every graph G with n vertices and m ≥ n 1+ε edges, has a subgraph G ′ of at most (1 − δ)m edges and crossing number at least (1 − ε)cr(G). This generalizes the result of J. Fox and Cs. Tóth.

Résumé We prove that two disjoint graphs must always be drawn separately on the Klein bottle in order to minimize the crossing number of the whole drawing.