12 research outputs found
Decomposing graphs into edges and triangles
We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4
Algorithms for Colourful Simplicial Depth and Medians in the Plane
The colourful simplicial depth of a point x in the plane relative to a
configuration of n points in k colour classes is exactly the number of closed
simplices (triangles) with vertices from 3 different colour classes that
contain x in their convex hull. We consider the problems of efficiently
computing the colourful simplicial depth of a point x, and of finding a point,
called a median, that maximizes colourful simplicial depth.
For computing the colourful simplicial depth of x, our algorithm runs in time
O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For
finding the colourful median, we get a time of O(n^4). For comparison, the
running times of the best known algorithm for the monochrome version of these
problems are O(n log(n)) in general, improving to O(n) if the points are sorted
around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure
Inducibility of directed paths
A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles
Compactness and finite forcibility of graphons
Graphons are analytic objects associated with convergent sequences of graphs.
Problems from extremal combinatorics and theoretical computer science led to a
study of graphons determined by finitely many subgraph densities, which are
referred to as finitely forcible. Following the intuition that such graphons
should have finitary structure, Lovasz and Szegedy conjectured that the
topological space of typical vertices of a finitely forcible graphon is always
compact. We disprove the conjecture by constructing a finitely forcible graphon
such that the associated space is not compact. The construction method gives a
general framework for constructing finitely forcible graphons with non-trivial
properties
The tripartite-circle crossing number of graphs with two small partition classes
A tripartite-circle drawing of a tripartite graph is a drawing in the plane,
where each part of a vertex partition is placed on one of three disjoint
circles, and the edges do not cross the circles. The tripartite-circle crossing
number of a tripartite graph is the minimum number of edge crossings among all
its tripartite-circle drawings. We determine the exact value of the
tripartite-circle crossing number of , where .Comment: 22 pages, 11 figures. Added new results and revised throughout.
Originally appeared in arXiv:1910.06963v1, now removed from
arXiv:1910.06963v
Bounding the tripartite-circle crossing number of complete tripartite graphs
A tripartite-circle drawing of a tripartite graph is a drawing in the plane,
where each part of a vertex partition is placed on one of three disjoint
circles, and the edges do not cross the circles. We present upper and lower
bounds on the minimum number of crossings in tripartite-circle drawings of
and the exact value for . In contrast to 1- and 2-circle
drawings, which may attain the Harary-Hill bound, our results imply that
balanced restricted 3-circle drawings of the complete graph are not optimal
On Crossing Numbers of Complete Tripartite and Balanced Complete Multipartite Graphs
The crossing number cr(G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number (cr) over bar (G) of G is the minimum number of crossings in a such drawing of G with edges as straight line segments. Zarankiewicz proved in 1952 that (cr) over bar (K-n1,K- n2) A(n1, n2, n3) := [GRAPHICS] (left perpendicular n(j)/2 right perpendicular left perpendicular n(j)-1/2 right perpendicular left perpendicular n(k)/2 right perpendicular left perpendicular n(k)-1/2 right perpendicular + left perpendicular n(i)/2 right perpendicular left perpendicular n(i)-1/2 right perpendicular left perpendicular n(j)n(k)/2 right perpendicular), and prove (cr) over bar (K-n1,K- n2,K- n3) infinity of cr(K-n,K- n) over the maximum number of crossings in a drawing of K-n,K- n exists and is at most 1/4. We define zeta(r) := 3(r(2) - r)/8(r(2) + r-3) and show that for a fixed r and the balanced complete r- partite graph, zeta(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.This is the peer-reviewed version of the following article: Gethner, Ellen, Leslie Hogben, Bernard LidickĂ˝, Florian Pfender, Amanda Ruiz, and Michael Young. "On crossing numbers of complete tripartite and balanced complete multipartite graphs." Journal of Graph Theory 84, no. 4 (2017): 552-565, which has been published in final form at DOI: 10.1002/jgt.22041. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.</p