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Extremal hypercuts and shadows of simplicial complexes

  • Nati Linial
  • Ilan Newman
  • Yuval Peled
  • Yuri Rabinovich
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Abstract

Let F be an n-vertex forest. An edge eF is said to be in F’s shadow if F ∪ {e} contains a cycle. It is easy to see that if F is an “almost tree”, i.e., a forest that contains two components, then its shadow contains at least \(\left\lfloor {\frac{{{{(n - 3)}^2}}}{4}} \right\rfloor \) edges and this is tight. Equivalently, the largest number of edges in an n-vertex cut is \(\left\lfloor {\frac{{{n^2}}}{4}} \right\rfloor \). These notions have natural analogs in higher d-dimensional simplicial complexes which played a key role in several recent studies of random complexes. The higher-dimensional situation differs remarkably from the one-dimensional graph-theoretic case. In particular, the corresponding bounds depend on the underlying field of coefficients. In dimension d = 2 we derive the (tight) analogous theorems. We construct 2-dimensional “ℚ-almost-hypertrees” (defined below) with an empty shadow. We prove that an “\(\mathbb{F}_2\) -almost-hypertree” cannot have an empty shadow, and we determine its least possible size. We also construct large hyperforests whose shadow is empty over every field. For d ≥ 4 even, we construct a d-dimensional \(\mathbb{F}_2\) -almost-hypertree whose shadow has vanishing density.

Several intriguing open questions are mentioned as well.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Nati Linial
    • 1
  • Ilan Newman
    • 2
  • Yuval Peled
    • 1
  • Yuri Rabinovich
    • 2
  1. 1.School of Computer Science and EngineeringThe Hebrew University of Jerusalem, Givat RamJerusalemIsrael
  2. 2.Department of Computer ScienceUniversity of Haifa, Mount CarmelHaifaIsrael

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