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Abstract order type extension and new results on the rectilinear crossing number
We extend the order type data base of all realizable order types in the plane to point sets of
cardinality 11. More precisely, we provide a complete data base of all combinatorial different …
cardinality 11. More precisely, we provide a complete data base of all combinatorial different …
Improved bounds for the crossing numbers of Km, n and Kn
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):=m-12m2n-12n2. Another …
graph K_m,n equals the Zarankiewicz number Z(m,n):=m-12m2n-12n2. Another …
Crossing Numbers and Combinatorial Characterization of Monotone Drawings of
In 1958, Hill conjectured that the minimum number of crossings in a drawing of K_n K n is
exactly Z (n)= 1 4\big ⌊ n 2\big ⌋\big ⌊ n-1 2\big ⌋\big ⌊ n-2 2\big ⌋\big ⌊ n-3 2\big ⌋ Z (n) …
exactly Z (n)= 1 4\big ⌊ n 2\big ⌋\big ⌊ n-1 2\big ⌋\big ⌊ n-2 2\big ⌋\big ⌊ n-3 2\big ⌋ Z (n) …
The Rectilinear Crossing Number of K n : Closing in (or Are We?)
BM Ábrego, S Fernández-Merchant… - Thirty essays on geometric …, 2013 - Springer
The calculation of the rectilinear crossing number of complete graphs is an important open
problem in combinatorial geometry, with important and fruitful connections to other classical …
problem in combinatorial geometry, with important and fruitful connections to other classical …
New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn
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≦⌊(n-2)/2⌋ the number of (≤ k)-edges is at …
plane in general position. We show that for 0≦k≦⌊(n-2)/2⌋ the number of (≤ k)-edges is at …
On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of Kn
BM Ábrego, M Cetina, S Fernández-Merchant… - Discrete & …, 2012 - Springer
Let P be a set of points in general position in the plane. Join all pairs of points in P with
straight line segments. The number of segment-crossings in such a drawing, denoted by …
straight line segments. The number of segment-crossings in such a drawing, denoted by …
[PDF][PDF] Crossing numbers of graphs: A bibliography
I Vrt'o - Available electronically at ftp://ifi. savba. sk/pub/imrich …, 2008 - Citeseer
62] Turan, P., A note of welcome, J. Graph Theory 1 (1977) 7-9. 63] Dambitis, J., An
algorithm for superimposing a nonplanar graph onto the plane with nearly minimal number …
algorithm for superimposing a nonplanar graph onto the plane with nearly minimal number …
[HTML][HTML] 3-symmetric and 3-decomposable geometric drawings of Kn
BM Ábrego, M Cetina, S Fernández-Merchant… - Discrete Applied …, 2010 - Elsevier
Even the most superficial glance at the vast majority of crossing-minimal geometric drawings
of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold …
of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold …
Geometric drawings of Kn with few crossings
BM Ábrego, S Fernández-Merchant - Journal of Combinatorial Theory …, 2007 - Elsevier
We give a new upper bound for the rectilinear crossing number cr¯(n) of the complete
geometric graph Kn. We prove that cr¯(n)⩽ 0.380559 (n4)+ Θ (n3) by means of a new …
geometric graph Kn. We prove that cr¯(n)⩽ 0.380559 (n4)+ Θ (n3) by means of a new …
On the crossing number of complete graphs
Let (G) denote the rectilinear crossing number of a graph G. We determine (K 11)= 102 and
(K 12)= 153. Despite the remarkable hunt for crossing numbers of the complete graph K n …
(K 12)= 153. Despite the remarkable hunt for crossing numbers of the complete graph K n …