If a graph embeds in a surface with k crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the degenerate crossing number, dcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps in which every edge passes throu...

Let Pn,k be the number of permutations π on [n] = {1, 2, . . . , n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx . We propose some conjectures on the log-concavity and q-log-convexity of the polynomials Pn(x) and M2n(x).

Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87–97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numb...

In this paper we measured the soft-handover (SHO) probability at and near street crossings in UMTS. Literature quotes two main propagation paths in urban macro-cell scenarios: along street-canyons and over the rooftops. Due to the fact that propagation along street canyons is one of the main propagation paths we expected high SHO probability at street crossings. To evaluate this effect we measu...

The circular graph C(n, m) is such a graph that whose vertex set is {v0, v1, v2, · · · , vn−1} and edge set is {vivi+1, vivi+m | i = 0, 1, · · · , n − 1}, where m,n are natural numbers, addition is modulo n, and 2 ≤ m ≤ b2 c. This paper shows the crossing number of the circular graph C(2m + 2,m)(m ≥ 3) is m + 1.

In this article we determine the crossing numbers of the Cartesian products of given three graphs on five vertices with paths.

The odd crossing number of G is the smallest number of pairs of edges that cross an odd number of times in any drawing of G. We show that there always is a drawing realizing the odd crossing number of G whose crossing number is at most 9, where k is the odd crossing number of G. As a consequence of this and a result of Grohe we can show that the odd crossing number is fixed-parameter tractable.

The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph Kn is H(n) = 1 4 ⌊ n 2 ⌋⌊ n− 1 2 ⌋⌊ n− 2 2 ⌋⌊ n− 3 2 ⌋ . Ábrego et al. [3] introduced the notion of shellability of a drawing D of Kn. They proved that if D is s-shellable for some s ≥ b 2 c, then D has at least H(n) crossings. This is the first combinatorial condition on a dr...

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊ 2 ⌋ the number of (≤ k)-edges is at least Ek(S) ≥ 3 (

There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor crossing number to strin...