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In this paper, we consider the following problem due to Erd˝ os: for each m ∈ N, is there a (least) positive integer f (m) so that every finite m-colored tournament contains an absorbent set S by monochromatic directed paths of f (m) vertices? In particular, is f (3) = 3? We prove several bounds for absorbent sets of m-colored tournaments under certain… (More)

In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let D be an m-colored digraph. We prove that if every chromatic class of D is quasi-transitive, every cycle is quasi-transitive in the rim and D does not contain polychromatic triangles, then D has a… (More)

- Bernardo Llano, Juan José Montellano-Ballesteros, Eduardo Rivera-Campo, Ricardo Strausz
- Journal of Graph Theory
- 2008

- Jorge L. Arocha, Bernardo Llano
- Discussiones Mathematicae Graph Theory
- 2000

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… (More)

- Bernardo Llano, Victor Neumann-Lara
- Discrete Mathematics
- 2008

- Susan A. van Aardt, Alewyn P. Burger, Marietjie Frick, Bernardo Llano, Rita Zuazua
- Graphs and Combinatorics
- 2014

- Bernardo M. Ábrego, Silvia Fernández-Merchant, Bernardo Llano
- Discrete & Computational Geometry
- 2010

- Ana Paulina Figueroa, Bernardo Llano, Rita Zuazua
- Discrete Mathematics
- 2010

Given a finite set P ⊆ R d , called a pattern, t P (n) denotes the maximum number of translated copies of P determined by n points in R d. We give the exact value of t P (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that t P (n) = n − m r (n), where r is the rational affine dimension… (More)

- Bernardo Llano, Mika Olsen
- Discussiones Mathematicae Graph Theory
- 2007

In this paper, we construct infinite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an infinite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show… (More)