Amanda Montejano

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.

Transitive Oriented 3 Hypergraphs of Cyclic Orders

In this paper we introduce the definition of transitivity for oriented 3-hypergraphs in order to study partial and complete cyclic orders. This definition allows us to give sufficient conditions on a partial cyclic order to be totally extendable. Furthermore, we introduce the 3-hypergraph associated to a cyclic permutation and characterize it in terms of cyclic comparability 3-hypergraphs.

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 .

Counting patterns in colored orthogonal arrays

Let S be an orthogonal array OA(d,k) and let c be an r-coloring of its ground set X. We give a combinatorial identity which relates the number of vectors in S with given color patterns under c with the cardinalities of the color classes. Several applications of the identity are considered. Among them it is shown that every coloring of an orthogonal array OA(d,d-1) contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color.

Rainbow-free 3-colorings in abelian groups

A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural descrip

Zero-sum subsequences in bounded-sum {−1,1}-sequences

Abstract The following result gives the flavor of this paper: Let t, k and q be integers such that q ≥ 0 , 0 ≤ t k and t ≡ k ( mod 2 ) , and let s ∈ [ 0 , t + 1 ] be the unique integer satisfying s ≡ q + k − t − 2 2 ( mod ( t + 2 ) ) . Then for any integer n such that n ≥ max ⁡ { k , 1 2 ( t + 2 ) k 2 + q − s t + 2 k − t 2 + s } and any function f : [ n ] → { − 1 , 1 } with | ∑ i = 1 n f ( i ) | ≤ q , there is a set B ⊆ [ n ] of k consecutive integers with | ∑ y ∈ B f ( y ) | ≤ t . Moreover, this bound is sharp for all the parameters involved and a characterization of the extremal sequences is given. This and other similar results involving different subsequences are presented, including decompositions of sequences into subsequences of bounded weight.

A rainbow Ramsey analogue of Rado's theorem

We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature.

About an Erdős---Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erdős and Grünbaum. Namely, if in an infinite family of convex sets in

Rainbow linear equations on three variables in Zp

\mathbb {R}^d