The crossing number of a graph G is the least number of pairwise crossings of edges among all the drawings of G in the plane. The pancake graph is an important topology for interconnecting processors in parallel computers. In this paper, we prove the exact value of the crossing number of pancake graph P4 is six.

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr(Kn1,n2) ≤ Z(n1, n2) := ⌊ n1 2 ⌋ ⌊ n1−1 2 ⌋ ⌊ n2 2 ⌋ ⌊ n2−1 2 ⌋ . We define an ...

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e/n. Apart from the value of the constant, this bound cannot be improved. We establish some...

The exact values of crossing numbers of the Cartesian products of four special graphs of order five with cycles are given and, in addition, all known crossing numbers of Cartesian products of cycles with connected graphs on five vertices are summarized.

It has been long conjectured that the crossing numbers of the complete bipartite graph Km,n and of the complete graph Kn equal Z(m,n) := ⌊

The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = Kn if n ≤ 9. Using a combinatorial argument we show that for n = 10 the number is 62.

A tile T is a connected graph together with two specified sequences of vertices, the left and right walls. The crossing number tcr(T ) of a tile T is the minimum number of crossings among all drawings of T in the unit square with the left wall in order down the left hand side and the right wall in order down the right hand side. The tile T n is obtained by gluing n copies of T in a linear fashi...

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 (

We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of speci c families of graphs, in particular, of complete bipartite graphs. We nd a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We nd the exact biplanar crossing number of K5;q for every q.