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Normas de factorización y un teorema inverso para MaxCut

Igor Balla1, Lianna Hambardzumyan2, István Tomon3

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Las matrices booleanas con norma de traza normalizada o norma gamma_2 acotada contienen grandes submatrices de unos/ceros. Esto verifica una conjetura y produce un teorema inverso para MaxCut, mostrando que los grafos con cortes casi máximos deben contener grandes cliques.

Palabras clave:
normas de factorizaciónteorema inversoMaxCutmatrices booleanassubmatricescombinatoriateoría de grafos

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Área de la Ciencia:

  • Combinatoria
  • Álgebra lineal
  • Teoría de grafos

Sus antecedentes:

  • Las matrices booleanas son fundamentales en matemáticas discretas y ciencias de la computación.
  • La norma gamma_2 y la norma de traza normalizada son medidas clave de las propiedades de las matrices.
  • El problema de MaxCut busca particionar los vértices de un grafo para maximizar los cortes de aristas.

Objetivo del estudio:

  • Demostrar que las matrices booleanas con norma de traza normalizada o norma gamma_2 acotada contienen submatrices de unos o ceros de tamaño lineal.
  • Verificar una conjetura de Hambardzumyan, Hatami y Hatami.
  • Establecer un teorema inverso para el problema de MaxCut.

Principales métodos:

  • Utilización de la teoría espectral de grafos y la combinatoria extremal.
  • Desarrollo de resultados estructurales para matrices booleanas.
  • Aplicación de propiedades de normas matriciales a problemas de corte de grafos.

Principales resultados:

  • Las matrices booleanas con norma de traza normalizada o norma gamma_2 acotada contienen necesariamente una submatriz de unos o ceros de tamaño lineal.
  • Se establece un teorema inverso para MaxCut: los grafos con MaxCut en a lo sumo m/2 + O(sqrt(m)) deben contener una clique de tamaño Omega(sqrt(m)).
  • El estudio proporciona información estructural adicional sobre matrices booleanas y sus aplicaciones.

Conclusiones:

  • Los hallazgos confirman una conjetura importante en la teoría de matrices booleanas.
  • El teorema inverso para MaxCut ofrece una nueva perspectiva sobre las estructuras de grafos con propiedades de corte específicas.
  • La investigación une conceptos de álgebra lineal, combinatoria y ciencias de la computación teóricas.