Extremal hypercuts and shadows of simplicial complexes

Abstract

Let F be an n-vertex forest. An edge e ∉ F 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.