arXiv subjects

Alan Lew

Publications and source records attributed to Alan Lew.

23 records · Page 2Linked to original sources

arXiv2020

Representability and boxicity of simplicial complexes

Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $τ\subset V$ such that $τ\notin X$ but $σ\in X$ for any $σ\subsetneq τ$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ is at most $\left\lfloor\frac{1}{d+1}\binom{n}{d}\right\rfloor$. The bound is sharp: the $d$-boxicity of a simplicial complex whose set of missing faces form a Steiner $(d,d+1,n)$-system is exactly $\frac{1}{d+1}\binom{n}{d}$.

math.CO↗

arXiv2019

Complexes of graphs with bounded independence number

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $Δ$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $Δ$. Then, every collection of $\left\lfloor\left(\fracΔ{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.

math.CO↗

arXiv2019

Collapsibility of simplicial complexes of hypergraphs

Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.

math.CO↗

arXiv2018

Spectral gaps, missing faces and minimal degrees

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $σ\notin X$ such that $τ\in X$ for any $τ\subsetneq σ$. For a $k$-dimensional simplex $σ$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $δ_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $μ_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $μ_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: \[ μ_k(X)\geq (d+1)(δ_k+k+1)-d n. \] As a consequence we obtain a new proof of a vanishing result for the homology of simplicial complexes without large missing faces. We present a family of examples achieving equality at all dimensions, showing that the bound is tight. For $d=1$ we characterize the equality case.

math.CO↗

arXiv2017

Spectral gaps of simplicial complexes without large missing faces

Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $μ_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $μ_{k}(X)$ for $k\geq d$ and $μ_{d-1}(X)$. In particular, we establish the following vanishing result: If $μ_{d-1}(X)>(1-\binom{k+1}{d}^{-1})n$, then $\tilde{H}^{j}(X;\mathbb{R})=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval and Montejano for general position sets in matroids.

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