Jorge L. Arocha

On the minimum size of tight hypergraphs

A k-graph, H = (V, E), is tight if for every surjective mapping f: V {1,….k} there exists an edge α ϵ E sicj tjat f|α is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number ϕ of edges in a tight k-graph with n vertices are given. We conjecture that ϕ for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.

Very Colorful Theorems

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

Mean value for the matching and dominating polynomial

The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.

Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk—Ulam-type theorem which has as a corollary a generalization of Hadwiger’s theorem.

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line.

Long induced paths in 3-connected planar graphs

It is shown that every 3-connected planar graph with a large number of vertices has a long induced path.

Configurations of Flats, I: Manifolds of Points in the Projective Line

AbstractThe topological manifolds arising from configurations of points in the real and complex projective lines are classified. Their topology and combinatorics are described for the real case. A general setting for the study of the spaces of configurations of flats is established and a projective duality among them is proved in its full generality.

Tightness problems in the plane

Abstract A 3-uniform hypergraph is called tight if for any 3-coloring of its vertex set a heterochromatic edge can be found. In this paper we study tightness of 3-graphs with vertex set R 2 and edge sets arising from simple geometrical considerations. Basically, we show that sets of triangles with ‘fat shadows’ are tight and also that some interesting sets of triangles with ‘thin shadows’ are tight.

A quick proof of Höbinger-Burton-Larman's theorem

Through the notion of projective center of symmetry of a convex body we will give a quick proof and clarify the ideas surrounding Höbingers problem, originally proved by Burton and Larman in [1].

Null and non-rainbow colorings of projective plane and sphere triangulations

By considering graphs as topological spaces we introduce, at the level of homology, the notion of a null coloring, which provides new information on the task of clarifying the structure of cycles in a graph. We prove that for any graph G a maximal null coloring f is such that the quotient graph G / f is acyclic. As an application, for maximal planar graphs (sphere triangulations) of order n ź 4 , we prove that a vertex-coloring containing no rainbow faces uses at most ź 2 n - 1 3 ź colors, and this is best possible. For maximal graphs embedded on the projective plane we obtain the analogous best bound ź 2 n + 1 3 ź .