91 research outputs found
A Note on the -colored Crossing Ratio of Dense Geometric Graphs
A \emph{geometric graph} is a graph whose vertex set is a set of points in
general position in the plane, and its edges are straight line segments joining
these points. We show that for every integer , there exists a constat
such that the following holds. The edges of every dense geometric graph
can be colored with colors, such that the number of pairs of edges of the
same color that cross is at most times the total number of pairs of
edges that cross. The case when and is a complete geometric graph,
was proved by Aichholzer et al.[\emph{GD} 2019]
Colouring the Triangles Determined by a Point Set
Let P be a set of n points in general position in the plane. We study the
chromatic number of the intersection graph of the open triangles determined by
P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P
is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary
P, the chromatic number is at most n^3/19.259+O(n^2)
Carathodory's theorem in depth
Let X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and tX(q)tX(q) ) and pairwise disjoint sets X1,…,Xd+1¿XX1,…,Xd+1¿X such that the following holds. Each XiXi has at least c|X| points, and for every choice of points xixi in XiXi , q is a convex combination of x1,…,xd+1x1,…,xd+1 . We also prove depth versions of Helly’s and Kirchberger’s theorems.Postprint (updated version
On the Connectivity of Token Graphs of Trees
Let and be integers such that , and let be a
simple graph of order . The -token graph of is the graph
whose vertices are the -subsets of , where two vertices are adjacent
in whenever their symmetric difference is an edge of . In this
paper we show that if is a tree, then the connectivity of is equal
to the minimum degree of
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