5 research outputs found
On the Minimum Size of a Point Set Containing a 5-Hole and a Disjoint 4-Hole
Let H(k; l), k β€ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117β122, 2004] proved that 12 β€ H(4, 5) β€ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481β507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90β100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12
Disjoint Empty Convex Pentagons in Planar Point Sets
Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved
that every set of 10 points in the plane, no three on a line, contains an empty
convex pentagon. From this it follows that the number of disjoint empty convex
pentagons in any set of points in the plane is least
. In this paper we prove that every set of 19
points in the plane, no three on a line, contains two disjoint empty convex
pentagons. We also show that any set of points in the plane, where
is a positive integer, can be subdivided into three disjoint convex regions,
two of which contains points each, and another contains a set of 9 points
containing an empty convex pentagon. Combining these two results, we obtain
non-trivial lower bounds on the number of disjoint empty convex pentagons in
planar points sets. We show that the number of disjoint empty convex pentagons
in any set of points in the plane, no three on a line, is at least
. This bound has been further improved to
for infinitely many .Comment: 23 pages, 28 figure
Disjoint Empty Convex Pentagons in Planar Point Sets
In this paper we obtain the first non-trivial lower bound on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of n points in the plane, no three on a line, is at least β5n/47β. This bound can be further improved to (3nβ1)/28 for infinitely many n
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group