[Paper Review] Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes

Rogers Mathew, Ilan Newman+2 more|arXiv (Cornell University)|Jul 16, 2015

Topological and Geometric Data Analysis15 references3 citations

TL;DR

This paper investigates the combinatorial structure of $d$-hypertrees and their boundaries in simplicial complexes, providing a full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq 2$ and constructing Hamiltonian $d$-cycles of size ${n-1 \choose d}+1$ for $d=2$ and ${n-1 \choose d} - O(n^{d-2})$ for $d \geq 3$. It further establishes that $d$-hypertrees can have average fundamental cycle size at least $c_d \cdot {n-1 \choose d}$, where $c_d$ depends only on $d$, resolving a key question on cycle dependency in higher-dimensional trees.

ABSTRACT

A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $ ilde{H}_{d-1}(T;\mathbb{F})=0$. The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycles defined by them. Our findings include: 1. A full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq 2$, and some partial results for $d \geq 3$. 2. Lower bounds on the maximum size of a largest simple $d$-cycle on $[n]$. In particular, for $d=2$, we construct a {\em Hamiltonian $d$-cycle} $H$ on $[n]$, i.e., a simple $d$-cycle of size ${{n-1} \choose d} + 1$. For $d\geq 3$, we construct a simple $d$-cycle of size ${{n-1} \choose d} - O(n^{d-2})$. 3. Observing that the maximum of the expected distance between two vertices chosen uniformly at random in a tree ($1$-hypertree) on $[n]$ is at most $ hicksim n/3$, attained on Hamiltonian paths, we ask a similar question about $d$-hypertrees. "How large can be the {\em average} size of a fundamental cycle of a $d$-hypertree $T$ (i.e., the expected size of the dependency created by adding a $d$-simplex on $[n]$, chosen uniformly at random, to $T$)?" For every $d \in \mathbb{N}$, we construct an infinite family of $d$-hypertrees $\{T\}$ with the average size of a fundamental cycle at least $c_d\, |T| \,=\, c_d\,{n-1 \choose d}$, where $c_d$ is a constant depending on the dimension $d$ alone.

Motivation & Objective

• To understand the structural properties of $d$-hypertrees and their boundaries in simplicial complexes, especially over finite fields like $\mathbb{F}_2$.
• To determine the maximum possible size of a simple $d$-cycle in $[n]$-vertex complexes, particularly Hamiltonian cycles.
• To analyze the average size of fundamental cycles formed by adding a random $d$-simplex to a $d$-hypertree, quantifying dependency in higher-dimensional trees.
• To extend results from $\mathbb{F}_2$ to $\mathbb{Q}$, showing that $\mathbb{Q}$-weighted trees can realize larger or equivalent cycle structures than over $\mathbb{F}_2$.

Proposed method

• Characterizes the boundary $\partial_d T$ of $d$-hypertrees over $\mathbb{F}_2$ using algebraic topology and homology vanishing conditions, focusing on $d \leq 2$.
• Uses combinatorial and extremal arguments to bound the size of the largest simple $d$-cycle, constructing explicit Hamiltonian cycles via recursive cone and link operators.
• Applies averaging and extremal counting techniques to show that $d$-hypertrees with large filling-volume must contain a $d$-face participating in many fillings.
• Leverages the $l_2$-norm of filling volumes and double-counting arguments to derive lower bounds on the maximum product of cut and filling sizes.
• Extends results from $\mathbb{F}_2$ to $\mathbb{Q}$ by showing that $\mathbb{F}_2$-acyclic simple cycles are also $\mathbb{Q}$-acyclic, enabling construction of $\mathbb{Q}$-weighted trees with desired boundaries.
• Employs conical extension and case analysis for $n=6$ to prove existence of $\mathbb{Q}$-weighted trees with any given $1$-cycle boundary, generalizing to higher dimensions.

Experimental results

Research questions

• RQ1 What is the structure of the boundary $\partial_d T$ of a $d$-hypertree over $\mathbb{F}_2$, particularly for $d \leq 2$?
• RQ2 What is the maximum possible size of a simple $d$-cycle in a $d$-dimensional simplicial complex on $n$ vertices, and can Hamiltonian cycles be constructed for $d \geq 2$?
• RQ3 How large can the average size of a fundamental cycle be in a $d$-hypertree, and what lower bounds can be established for this average?
• RQ4 Can results on $\mathbb{F}_2$-cycles be extended to $\mathbb{Q}$, and do $\mathbb{Q}$-weighted trees allow for larger or more flexible cycle structures than over $\mathbb{F}_2$?

Key findings

• For $d \leq 2$, the paper provides a complete characterization of the boundary $\partial_d T$ of a $d$-hypertree over $\mathbb{F}_2$, resolving a foundational structural question.
• A Hamiltonian $2$-cycle of size ${n-1 \choose 2} + 1$ is explicitly constructed on $[n]$ vertices, achieving the maximum possible size for a simple $2$-cycle.
• For $d \geq 3$, the paper constructs a simple $d$-cycle of size ${n-1 \choose d} - O(n^{d-2})$, demonstrating that nearly maximal cycles exist in higher dimensions.
• The paper constructs an infinite family of $d$-hypertrees with average fundamental cycle size at least $c_d \cdot {n-1 \choose d}$, where $c_d = 16(48)^{-3^{d-1}}$, showing that dependency in higher-dimensional trees can be substantial.
• Results over $\mathbb{F}_2$ extend to $\mathbb{Q}$: every simple $\mathbb{F}_2$-cycle is contained in a simple $\mathbb{Q}$-cycle, and $\mathbb{Q}$-weighted trees can realize all $1$-cycles on $n \geq 6$ vertices.
• For $d=2$, $n$-vertex Hamiltonian $2$-cycles exist over $\mathbb{Q}$ for all $n \geq 6$, extending beyond the $\mathbb{F}_2$-case where such cycles are restricted by parity.