Dr. Bernardo Abrego

Teaching (Spring 2010)

Problem Solving

I am the new Problems Editor of Mathematics Magazine! If you like challenging problems try the problems in the column.

Research

Combinatorics, Discrete and Computational Geometry, Graph Theory.

Publications

• B. M. Ábrego, S. Fernández-Merchant, and B. Llano. On the maximum number of translates. Discrete and Computational Geometry, 43 (2010), 1-20.
• B. M. Ábrego and S. Fernández-Merchant. Point-sets with many similar copies of a pattern: without a fixed number of collinear points or in parallelogram-free position. To appear in Geombinatorics (2010).
• B. M. Ábrego and S. Fernández-Merchant. Point-sets in general position with many similar copies of a pattern. To appear in Geombinatorics (2010).
• B. M. Ábrego, E. Arkin, S. Fernández-Merchant, F. Hurtado, M. Kano, J. Mitchell, J. Urrutia. Matching points with squares.Discrete and Computational Geometry 41 (2009), 77-95.
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Sum of squares of degrees in a graph. Journal of Inequalities in Pure and Applied Mathematics, 10(3) (2009), Art. 64, 34 pp.
• B. M. Ábrego, M. Cetina, S. Fernández-Merchant, J. Leaños, and G. Salazar. 3-symmetric and 3-decomposable drawings of Kn. To appear in Discrete Applied Mathematics. Available online as article in press, doi:10.1016/j.dam.2009.09.020.
• B. M. Ábrego, S. Fernández-Merchant, and B. Llano. An inequality for Kruskal-Macaulay Functions. Submitted (2008).
• B. M. Ábrego, S. Fernández-Merchant, J. Leaños, and G. Salazar. A central approach to bound the number of crossings in a generalized configuration. Electronic Notes in Discrete Mathematics 30 (2008) 273-278.
• B. M. Ábrego, S. Fernández-Merchant, J. Leaños, and G. Salazar. The maximum number of halving lines and the rectilinear crossing number of Kn for n<=27 . Electronic Notes in Discrete Mathematics 30 (2008), 261-266.
• B. M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, and G. Salazar. An extended lower bound on the number of (<=k)-edges to generalized configurations of points and the pseudolinear crossing number of Kn. Journal of Combinatorial Theory, Series A. 115 (2008), 1257-1264.
• B. M. Ábrego, S. Fernández-Merchant, J. Leaños, and G. Salazar. Recent developments on the number of (<=k)-sets, halving-lines, and the rectilinear crossing number of Kn. Proceedings of XII Encuentros de Geometría Computacional, Universidad de Valladolid, Spain, June 25-27, (2007), ISBN 978-84-690-6900-4.
• B. M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, and G. Salazar. On (<=k)–pseudoedges in generalized configurations and the pseudolinear crossing number of Kn. Preprint (2006).
• B. M. Ábrego and S. Fernández-Merchant. Geometric drawings of Kn with few crossings. Journal of Combinatorial Theory, Series A 114 (2007) 373-379.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Trace-minimal graphs and D-optimal weighing designs, Linear Algebra and its Applications 412 (2006), 161-221.
  
• B. M. Ábrego and S. Fernández-Merchant. A lower bound for the rectilinear crossing number. Graphs and Combinatorics, Springer-Verlag Tokyo Inc., Vol. 21, Num 3, (2005), p. 293 - 300.
  
• B. M. Ábrego, E. Arkin, S. Fernández-Merchant, F. Hurtado, M. Kano, J. Mitchell, J. Urrutia. Matching points with geometric objects: combinatorial results. Discrete and Computational Geometry Japanese Conference, JCDCG 2004, Tokyo, Japan, October 8-11, 2004. Springer-Verlag, Lecture Notes in Computer Science Series, Vol. 3742, Akiyama, Jin; Kano, Mikio; Tan, Xuehou (Eds.) 2005, VIII, p. 1-15, ISBN: 3-540-30467-3.
  
• B. M. Ábrego, S. Fernández-Merchant, and Gy. Elekes. Structural results for planar sets with many similar subsets. Combinatorica 24 (2004), no. 4, 541-554.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. D-optimal weighing designs for n´ -1 (mod 4) objects and a large number of weighings, Linear Algebra and its Applications 374 (2003), 175-218.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Maximum volume simplices and trace-minimal graphs, Proceedings of X Encuentros de Geometria Computacional, University of Sevilla, Sevilla Spain, June 16-17, (2003), 51-56.
  
• B. M. Ábrego and S. Fernández-Merchant. The unit distance problem for centrally symmetric convex polygons. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). Discrete and Computational Geometry 28 (2002), no 4, 467-473.
  
• B. M. Ábrego and S. Fernández-Merchant. Convex Polyhedra in R^3 Spanning W(n^(4/3)) Congruent Triangles, Journal of Combinatorial Theory, Series A 98, 406-409 (2002).
  
• B. M. Ábrego and S. Fernández-Merchant. On the Maximum Number of Equilateral Triangles I. Discrete and Computational Geometry 23, (2000), no. 1, 129-115.
  
• B. M. Ábrego, J. L. Arocha, S. Fernández-Merchant, and V. Neumann-Lara. Tightness Problems in the Plane. Discrete Mathematics 194 (1999), no. 1-3, 1-11.
• B. M. Ábrego, Problemas Combinatorios Sobre Conjuntos Finitos de Puntos (Combinatorial Problems on Finite Sets). Aportaciones Matemáticas: Comunicaciones 19, SMM, Mexico (1997), ii+177pp, (1997) ISBN 968-36-5771-0.
• B. M. Ábrego, El Teorema de Sylvester (Sylvester's Theorem), Miscelánea Matemática, 23 (1996), 17-26.