Now showing items 21-38 of 38

• #### Note on the number of obtuse angles in point sets ﻿

(2014-09)
Article
Open Access
In $1979$ Conway, Croft, Erd\H{o}s and Guy proved that every set $S$ of $n$ points in general position in the plane determines at least $\frac{n^3}{18}-O(n^2)$ obtuse angles and also presented a special set of $n$ points ...
• #### On cyclic Kautz digraphs ﻿

(2015)
External research report
Open Access
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given ...
• #### On k-gons and k-holes in point sets ﻿

(2015-08-01)
Article
Open Access
We consider a variation of the classical Erdos-Szekeres problems on the existence and number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane. Allowing the k-gons to be non-convex, we show ...
• #### On some partitioning problems for two-colored point sets ﻿

(2009)
Conference report
Open Access
Let S be a two-colored set of n points in general position in the plane. We show that S admits at least 2 n 17 pairwise disjoint monochromatic triangles with vertices in S and empty of points of S. We further show ...
• #### On the disks with diameters the sides of a convex 5-gon ﻿

(Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II (MA2), 2015)
Conference report
Restricted access - publisher's policy
We prove that for any convex pentagon there are two disks, among the five disks having a side of the pentagon as diameter and the midpoint of the side as its center, that do not intersect. This shows that K5 is never the ...
• #### Optimal grid drawings of complete multipartite graphs and an integer variant of the algebraic connectivity ﻿

(Springer, 2018)
Conference lecture
Restricted access - publisher's policy
How to draw the vertices of a complete multipartite graph G on different points of a bounded d-dimensional integer grid, such that the sum of squared distances between vertices of G is (i) minimized or (ii) maximized? For ...
• #### Order types of random point sets can be realized with small integer coordinates ﻿

(2017)
Conference report
Open Access
Let S := {p1, . . . , pn} be a set of n points chosen independently and uniformly at random from the unit square and let M be a positive integer. For every point pi = (xi , yi) in S, let p 0 i = (bMxic, bMyic). Let S 0 := ...
• #### Production matrices for geometric graphs ﻿

(2016)
Article
Open Access
We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot ...
• #### Ramsey numbers for empty convex polygons ﻿

(University of Ljubljana, 2015)
Conference report
Restricted access - publisher's policy
We study a geometric Ramsey type problem where the vertices of the complete graph Kn are placed on a set S of n points in general position in the plane, and edges are drawn as straight-line segments. We define the empty ...
• #### Recoloring directed graphs ﻿

(Prensas Universitarias de Zaragoza, 2009)
Conference report
Open Access
Let G be a directed graph and k a positive integer. We define the k-color graph of G (Dk(G) for short) as the directed graph having all k-colorings of G as node set, and where two k-colorings and ' are joined by a ...
• #### The connectivity of the flip graph of Hamiltonian paths of the grid graph ﻿

(2017)
Conference report
Open Access
Let Gn,m be the grid graph with n columns and m rows. Let Hn,m be the graph whose vertices are the Hamiltonian paths in Gn,m, where two vertices P1 and P2 are adjacent if we can obtain P2 from P1 by deleting an edge in P1 ...
• #### The degree-diameter problem in maximal bipartite planar graphs ﻿

(2014)
Conference lecture
Open Access
The (A ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree and diameter D. We consider the (A ,D) problem for maximal planar bipartite ...
• #### The degree/diameter problem in maximal planar bipartite graphs ﻿

(2014)
Article
Open Access
The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, ...
• #### The degree/diameter problem in maximal planar bipartite graphs ﻿

(2016)
Article
Open Access
The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar ...
• #### The diameter of cyclic Kautz digraphs ﻿

(2015-11-12)
Article
Open Access
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given ...
• #### The diameter of cyclic Kautz digraphs ﻿

(2017)
Article
Open Access
We present a new kind of digraphs, called cyclic Kautz digraphs CK(d, ), which are subdigraphs of the well-known Kautz digraphs K(d, ). The latter have the smallest diameter among all digraphs with their number of vertices ...
• #### The number of empty four-gons in random point sets ﻿

(2014)
Article
Open Access
Let S be a set of n points distributed uniformly and independently in the unit square. Then the expected number of empty four-gons with vertices from S is T(n^2 log¿ n). A four-gon is empty if it contains no points of S ...
• #### The rotation graph of k-ary trees is Hamiltonian ﻿

(Crete University Press, 2006)
Article
Restricted access - publisher's policy
In this paper we show that the graph of k-ary trees, connected by rotations, contains a Hamilton cycle. Our proof is constructive and thus provides a cyclic Gray code for k-ary trees. Furthermore, we identify a basic ...