arXiv subjects

Eran Nevo

Publications and source records attributed to Eran Nevo.

At least 19 recordsLinked to original sources

arXiv2026

Forbidden Subgraphs of Graphs with Low Bandwidth

A layout of a graph G is an injective function $f : V(G) \rightarrow Z$, and the bandwidth of a layout f is $bw(G,f) = max_{uv \in E(G)} |f(u) - f(v)|$. The bandwidth bw(G) of G is the minimum bandwidth of a layout of G. Computing the bandwidth of a graph is a notoriously hard problem: assuming P != NP, there is no polynomial time algorithm, even on very restricted classes of trees [Monien, SIAM Journal on Algebraic Discrete Methods, 1986], and no constant factor approximation, even on trees [Dubey et al., JCSS 2011]. Assuming the Exponential Time Hypothesis, there is no algorithm with running time $f(k)n^{o(k)}$ to determine whether an input graph has bandwidth at most k, even on very restricted classes of trees [Dregi and Lokshtanov, ICALP 2014]. In this paper we show that {\sc Bandwidth} on general graphs is FPT-approximable. In particular we give an algorithm that takes as input a graph G and an integer k, runs in time $2^{O(9^k)}n^{O(1)}$, and outputs a subtree T of G such that $bw(T) \geq k$ or a layout of G of bandwidth at most $(10^{85} k^{28})^{4^k}$. This resolves in the affirmative an open problem of Chung and Seymour [Discrete Mathematics, 1989], who asked whether the bandwidth of every graph G is upper bounded in terms of the maximum bandwidth of a subtree of G. Our theorem leads to a forbidden subgraph characterization for graphs of bounded bandwidth, and can be seen as an analog for bandwidth of the classic grid minor theorem for treewidth, the forbidden subtree theorem for pathwidth, and the forbidden subpath theorem for treedepth.

cs.DS↗

arXiv2026

Analytic Spread via Linear Matroids

We give a systematic analysis of the analytic spread of the determinantal ideal $J_{G,H}$ arising from a pair of graphs $(G, H)$. We give sharp bounds for this analytic spread, and combinatorial conditions and obstructions for its maximality via a linear matroid. When $H$ is a single edge, $J_{G,H}$ is isomorphic to the binomial edge ideal $J_G$ and its analytic spread is shown to equal the rank of $G$ in Kalai's $2$-hyperconnectivity matroid.

math.AC↗

arXiv2025

Matroids arising from algebraic shifting

We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment. There are several examples of known matroids arising in this way: the simplicial matroid, the hyperconnectivity matroid and the area-rigidity matroid. For $k\ge 4$, we provide a similar characterization for shifted $k$-uniform hypergraphs satisfying an additional combinatorial condition. For symmetric shifting, we prove an analogous characterization for shifted simple graphs, where the classical generic rigidity matroid is an example of a matroid arising in this way.

math.CO↗

arXiv2025

On an extension problem on the moment curve

We show that for $2\le d\le 4$, every finite geometric simplicial complex $\Delta$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $\Delta, T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $\Delta$ on $n$ vertices for which no such triangulations $T$ exist. Our result for $d=4$ has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in $\mathcal{O}_{A_n^{2}}$ is cluster tilting, where $\mathcal{O}_{A_n^{\delta}}$ denotes a higher dimensional cluster category introduced by Oppermann and Thomas for $A_n^\delta$, where $A_n^\delta$ denotes a higher Auslander algebra of linearly oriented type $A$.

math.CO↗

arXiv2025

The typical algebraic shifting of a surface

We initiate a statistical study of Kalai's exterior algebraic shifting, focusing on concentration phenomena for random triangulations of a fixed space. First, for a uniform $n$-vertex refinement of any given graph $G$, we show that asymptotically almost-surely (a.a.s.) its exterior algebraic shifting is an explicit shifted graph depending only on $n$ and the Betti numbers of $G$. Next, for any given compact connected Riemannian surface $S$, sample $n$ points independently at random according to the volume measure, and consider the resulted a.a.s. unique Delaunay triangulation. We prove that a.a.s. its exterior algebraic shifting is an explicit shifted complex depending only on $n$ and the genus of $S$. In both results the expected shifted complex is a homology lex-segment complex, a notion we define combinatorially and characterize numerically a l\'{a} Bj\"{o}rner-Kalai. As a tool to prove the result on surfaces, we prove a universality result on edge contractions: for every fixed surface triangulation $K$, every dense enough point set in the surface yields a Delaunay triangulation that edge contracts to $K$.

math.CO↗

arXiv2025

On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$

We define a generic rigidity matroid for $k$-volumes of a simplicial complex in $\mathbb{R}^d$, and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic $d$-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of $d$-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in $k$-faces.

math.CO↗

arXiv2024

Exterior Shifting of Low Genus Surfaces

We characterize the possible exterior shiftings of $K$, where $K$ runs over all triangulation of the torus, or the projective plane, or the Klein bottle. Further, we give a deterministic polynomial-time algorithm for computing the exterior shifting of a given triangulation $K$ as above.

math.CO↗

arXiv2023

On flag-no-square $4$-manifolds

Which $4$-manifolds admit a flag-no-square (fns) triangulation? We introduce the "star-connected-sum" operation on such triangulations, which preserves the fns property, from which we derive new constructions of fns $4$-manifolds. In particular, we show the following: (i) there exist non-aspherical fns $4$-manifolds, answering in the negative a question by Przytycki and Swiatkowski; (ii) for every large enough integer $k$ there exists a fns $4$-manifold $M_{2k}$ of Euler characteristic $2k$, and further, (iii) $M_{2k}$ admits a super-exponential number (in $k$) of fns triangulations - at least $2^{Ω(k \log k)}$ and at most $2^{O(k^{1.5} \log k)}$.

math.GT↗

arXiv2023

On colorings of hypergraphs embeddable in $\mathbb{R}^d$

The (weak) chromatic number of a hypergraph $H$, denoted by $\chi(H)$, is the smallest number of colors required to color the vertices of $H$ so that no hyperedge of $H$ is monochromatic. For every $2\le k\le d+1$, denote by $\chi_L(k,d)$ (resp. $\chi_{PL}(k,d)$) the supremum $\sup_H \chi(H)$ where $H$ runs over all finite $k$-uniform hypergraphs such that $H$ forms the collection of maximal faces of a simplicial complex that is linearly (resp. PL) embeddable in $\mathbb{R}^d$. Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For $d \geq 3$, we show that A. $\chi_L(k,d)=\infty$ for all $2\le k\le d$, B. $\chi_{PL}(d+1,d)=\infty$ and C. $\chi_L(d+1,d)\ge 3$ for all odd $d\ge 3$. As an application, we extend the results by Lutz and M{\o}ller on the weak chromatic number of the $s$-dimensional faces in the triangulations of a fixed triangulable $d$-manifold $M$: D. $\chi_s(M)=\infty$ for $1\leq s \leq d$.

math.CO↗

arXiv2023

Volume rigidity and algebraic shifting

We study the generic volume rigidity of $(d-1)$-dimensional simplicial complexes in $\mathbb R^{d-1}$, and show that the volume rigidity of a complex can be identified in terms of its exterior shifting. In addition, we establish the volume rigidity of triangulations of several $2$-dimensional surfaces and prove that, in all dimensions $>1$, volume rigidity is {\em not} characterized by a corresponding hypergraph sparsity property.

math.CO↗

arXiv2023

Rigidity expander graphs

Jordán and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for $a_d(G)$ defined in terms of the spectral expansion of certain subgraphs of $G$ associated with a partition of its vertices into $d$ parts. In particular, we obtain a new sufficient condition for the rigidity of a graph $G$. As a first application, we prove the existence of an infinite family of $k$-regular $d$-rigidity-expander graphs for every $d\ge 2$ and $k\ge 2d+1$. Conjecturally, no such family of $2d$-regular graphs exists. Second, we show that $a_d(K_n)\geq \frac{1}{2}\left\lfloor\frac{n}{d}\right\rfloor$, which we conjecture to be essentially tight. In addition, we study the extremal values $a_d(G)$ attained if $G$ is a minimally $d$-rigid graph.

math.CO↗

arXiv2022

Sharp threshold for rigidity of random graphs

We consider the Erdős-Rényi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d$, and it becomes globally rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d+1$.

math.CO↗

arXiv2022

On the $d$-dimensional algebraic connectivity of graphs

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$, introduced by Jordán and Tanigawa, is a quantitative measure of the $d$-dimensional rigidity of $G$ that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set $V$ into $\mathbb{R}^d$. Here, we analyze the $d$-dimensional algebraic connectivity of complete graphs. In particular, we show that, for $d\geq 3$, $a_d(K_{d+1})=1$, and for $n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq \frac{2n}{3(d-1)}+\frac{1}{3}. \]

math.CO↗

arXiv2022

Vertex spanning planar Laman graphs in triangulated surfaces

We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Kir{á}ly, we conclude that every $1$-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices.

math.CO↗

arXiv2022

Stable sets in flag spheres

We provide lower and upper bounds on the minimum size of a maximum stable set over graphs of flag spheres, as a function of the dimension of the sphere and the number of vertices. Further, we use stable sets to obtain an improved Lower Bound Theorem for the face numbers of flag spheres.

math.CO↗

arXiv2021

Embedding Divisor and Semi-Prime Testability in f-vectors of polytopes

We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.

math.CO↗

arXiv2019

On the realization space of the cube

We consider the realization space of the $d$-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical $d$-polytopes, and apply this construction to certain cubical $d$-polytopes to conclude that the rays spanned by $f$-vectors of cubical $d$-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.

math.CO↗

arXiv2019

Rigidity with few locations

Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to make the resulted embedding of $G$ infinitesimally rigid. This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of $A$ increases with the genus. The assertion fails, namely no such finite $A$ exists, for the larger family of all graphs that are generically rigid in $3$-space and even in the plane.

math.CO↗