arXiv subjects

Marta Pavelka

Publications and source records attributed to Marta Pavelka.

7 recordsLinked to original sources

arXiv2026

Skeleton Chordalities

We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) $Δ$ skeleton-E-chordal $\Rightarrow$ $Δ^\vee$ vertex-decomposable $\Rightarrow$ $Δ$ skeleton-clique-chordal. Moreover, for subflag complexes, $Δ$ skeleton-E-chordal $\Longleftrightarrow$ $Δ^\vee$ vertex-decomposable. (For $d=1$ this boils down to ``$G$ chordal $\Longleftrightarrow$ $G^\vee$ vertex-decomposable'', a result closely related to Fröberg's theorem.) (2) For subflag complexes, $Δ$ is skeleton-E-chordal $\Longleftrightarrow$ it splits as $Δ= Δ_1 \cup Δ_2$, with each $Δ_i$ a skeleton-E-chordal induced subcomplex of $Δ$, and with $Δ_1 \cap Δ_2$ a complex whose $1$-skeleton is a clique. (This generalizes ``$G$ chordal $\Longleftrightarrow$ $G$ splits as a union of chordal graphs that intersect in a common clique''). (3) $Δ$ skeleton-E-chordal $\Longleftrightarrow$ every nonempty induced subcomplex of $Δ$ has a skeleton-E-simplicial vertex. (Generalizes ``$G$ chordal $\Leftrightarrow$ every nonempty induced subgraph has a simplicial vertex''.) (4) $Δ$ underclosed $\Rightarrow$ $Δ$ skeleton-weakly-chordal and weakly-closed. (Generalizes ``$G$ interval $\Rightarrow$ $G$ chordal and co-comparability''.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)

math.CO↗

arXiv2026

Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fröberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic analogues of interval and chordal graphs have been proposed, in part as an effort to extend Fröberg's theorem to ideals generated in higher degree. In this paper, we study two such classes from the literature, cointerval hypergraphs and underclosed complexes, and show that they are in fact equivalent up to complementation. We then consider their place in the broader theory of higher-dimensional chordality, proving that an underclosed clutter is chordal in the sense of Woodroofe. As a consequence, we answer a question of Dochtermann and Engström by showing that the associated Alexander dual complexes are vertex decomposable, implying that the corresponding circuit ideals have linear quotients. We furthermore show that these dual complexes have shellings induced by their underclosed vertex orders.

math.CO↗

arXiv2025

Higher-dimensional counterexamples to Hamiltonicity

For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Grünbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10 n^α$, for some constant $α<1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.

math.CO↗

arXiv2024

Vertex orders in higher dimensions

Unit interval and interval complexes are higher-dimensional generalizations of unit interval and interval graphs, respectively. We show that strongly connected unit interval complexes are shellable with shellings induced by their unit interval orders. We also show that these complexes are vertex decomposable and hence shelling completable. On the other hand, we give simple examples of strongly connected interval complexes that are not shellable in dimensions two and higher.

math.CO↗

arXiv2024

A conditional lower bound for the Turán number of spheres

We consider the hypergraph Turán problem of determining $\mathrm{ex}(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then $\mathrm{ex}(n, S^d) \geq Ω(n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$. Furthermore, this lower bound holds unconditionally for 2-LC spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on $\mathrm{ex}(n, S^d)$ of $O(n^{d + 1 - 1/2^{d - 1}})$ using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound.

math.CO↗

arXiv2022

2-LC triangulated manifolds are exponentially many

We introduce "$t$-LC triangulated manifolds" as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion interpolates between the class of LC manifolds introduced by Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most $2^{d^2 \, N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove that there are at most $2^{\frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds with $N$ facets. This extends to all dimensions an intuition by Mogami for $d=3$. We also introduce "$t$-constructible complexes", interpolating between constructible complexes (the case $t=1$) and all complexes (case $t=d$). We show that all $t$-constructible pseudomanifolds are $t$-LC, and that all $t$-constructible complexes have (homotopical) depth larger than $d-t$. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen--Macaulay.

math.CO↗

arXiv2019

On-line algorithms for multiplication and division in real and complex numeration systems

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $β$ such that $|β|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $β$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(β, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $β$ and a digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > |β|$. For a complex base $β$ and symmetric digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > β\overlineβ + |β+ \overlineβ|$. Provided that addition and subtraction are realizable in parallel in the system $(β, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $β=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $β=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $β= -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + ω$, where $ω= \exp{\frac{2iπ}{3}}$, with digits $A = \{0, \pm 1, \pm ω, \pm ω^2 \}$.

cs.DS↗