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Factorization norms and an inverse theorem for MaxCut.

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  • 1Faculty of Mathematics and Computer Science, Leipzig University, 04109 Leipzig, Germany.

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Summary

Boolean matrices with bounded gamma_2-norm or normalized trace norm contain large all-ones/all-zeros submatrices. This verifies a conjecture and yields an inverse theorem for MaxCut, showing graphs with near-maximal cuts must contain large cliques.

Keywords:
15A1815A60Primary 68Q11Secondary 05C50

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Area of Science:

  • Combinatorics
  • Linear Algebra
  • Graph Theory

Background:

  • Boolean matrices are fundamental in discrete mathematics and computer science.
  • The gamma_2-norm and normalized trace norm are key measures of matrix properties.
  • The MaxCut problem seeks to partition graph vertices to maximize edge cuts.

Purpose of the Study:

  • To prove that Boolean matrices with bounded gamma_2-norm or normalized trace norm contain linear-sized all-ones or all-zeros submatrices.
  • To verify a conjecture by Hambardzumyan, Hatami, and Hatami.
  • To establish an inverse theorem for the MaxCut problem.

Main Methods:

  • Utilizing spectral graph theory and extremal combinatorics.
  • Developing structural results for Boolean matrices.
  • Applying matrix norm properties to graph-cut problems.

Main Results:

  • Boolean matrices with bounded gamma_2-norm or normalized trace norm necessarily contain a linear-sized all-ones or all-zeros submatrix.
  • An inverse theorem for MaxCut is established: graphs with MaxCut at most m/2 + O(sqrt(m)) must contain a clique of size Omega(sqrt(m)).
  • The study provides further structural insights into Boolean matrices and their applications.

Conclusions:

  • The findings confirm a significant conjecture in the theory of Boolean matrices.
  • The inverse theorem for MaxCut offers a new perspective on graph structures with specific cut properties.
  • The research bridges concepts from linear algebra, combinatorics, and theoretical computer science.