Electronic Journal of Combinatorics
Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that a 0-1 matrix A is saturating for the forbidden 0-1 matrix P if A avoids P but changing any zero to a one in A creates a copy of P . Define sat(n, P ) to be the minimum possible number of ones in an n × n 0-1 matrix that is saturating for P . Fulek and Keszegh proved that for every 0-1 matrix P, either sat(n, P ) = O(1) or sat(n, P ) = Θ(n). They found two 0-1 matrices P for which sat(n, P ) = O(1), as well as infinite families of 0-1 matrices P for which sat(n, P ) = Θ(n). Their results imply that sat(n, P ) = Θ(n) for almost all k × k 0-1 matrices P . Fulek and Keszegh conjectured that there are many more 0-1 matrices P such that sat(n, P ) = O(1) besides the ones they found, and they asked for a characterization of all permutation matrices P such that sat(n, P ) = O(1). We affirm their conjecture by proving that almost all k × k permutation matrices P have sat(n, P ) = O(1). We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.
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Mathematics and Statistics
Jesse Geneson. "Almost all permutation matrices have bounded saturation functions" Electronic Journal of Combinatorics (2021). https://doi.org/10.37236/10124