Guillermo Pineda-Villavicencio

activity

20102026

most citedLower bound theorems for general polytopes

14 citations · 46 across the 26 of their papers we have counts for

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30 papers · 1 filter

math.CO2026

Equifacetted polytopes whose facets are products of simplices: The simplicial prism case

Federico Castillo, Guillermo Piñeda-Villavicencio, José Alejandro Samper

We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism $Δ^{d - 2} \times Δ^{1}$ . In dimension three these are exactly the…

math.CO2026

Connectivity, toughness, and hamiltonicity of simplicial polytopes

Guillermo Pineda-Villavicencio

For every $d \geq 4$ , every prescribed connectivity $r \geq d$ , and every $ε > 0$ , we construct a simplicial $d$ -polytope whose graph has connectivity exactly $r$ , is nontraceab…

math.CO2025

A lower bound theorem for $d$ -polytopes with at most $3 d - 1$ vertices

Guillermo Pineda-Villavicencio, Jie Wang

We prove a lower bound theorem for the number of $k$ -faces ( $1 \leq k \leq d - 2$ ) in a $d$ -dimensional polytope $P$ (or $d$ -polytope) with up to $3 d - 1$ vertices. Previous lower bound th…

math.CO2025

A refined lower bound theorem for $d$ -polytopes with at most $2 d$ vertices

Guillermo Pineda-Villavicencio, Jie Wang, David Yost

In 1967, Grünbaum conjectured that the function $ϕ_{k} (d + s, d) := (k + 1 d + 1) + (k + 1 d) - (k + 1 d + 1 - s), for 2 \leq s \leq d$ provides the minimum number of $k$ -…

math.CO2024

A lower bound theorem for $d$ -polytopes with $2 d + 2$ vertices

Guillermo Pineda-Villavicencio, Aholiab Tritama, Jie Wang +1

We establish a lower bound theorem for the number of $k$ -faces ( $1 \leq k \leq d - 2$ ) in a $d$ -dimensional polytope $P$ (abbreviated as a $d$ -polytope) with $2 d + 2$ vertices, extending t…

math.CO2024

Polytopes with low excess degree

Guillermo Pineda-Villavicencio, Jie Wang, David Yost

We study the existence and structure of $d$ -polytopes for which the number $f_{1}$ of edges is small compared to the number $f_{0}$ of vertices. Our results are more elegantly expresse…

math.CO2022

Edge connectivity of simplicial polytopes

Vincent Pilaud, Guillermo Pineda-Villavicencio, Julien Ugon

We show that the graph of a simplicial polytope of dimension $d \geq 3$ has no nontrivial minimum edge cut with fewer than $d (d + 1) /2$ edges, hence the graph is $min {δ, d (d + 1) /2}$ …

math.CO2022

Cycle space of graphs of polytopes

Guillermo Pineda-Villavicencio

It is folklore that the cycle space of graphs of polytopes is generated by the cycles bounding the 2-faces. We provide a proof of this result that bypass homological arguments, whi…

math.CO2021

Edge connectivity of simplicial polytopes

Guillermo Pineda-Villavicencio, Julien Ugon

A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We…

math.CO2021

A lower bound theorem for $d$ -polytopes with $2 d + 1$ vertices

Guillermo Pineda-Villavicencio, David Yost

The problem of calculating exact lower bounds for the number of $k$ -faces of $d$ -polytopes with $n$ vertices, for each value of $k$ , and characterising the minimisers, has recently…

math.CO2020

Linkedness of Cartesian products of complete graphs

Leif K. Jorgensen, Guillermo Pineda-Villavicencio, Julien Ugon

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2 k$ vertices is {\it $k$ -linked} if, for every set of $2 k$ distinct ver…

math.CO2020★ 2 cited

Reconstructibility of matroid polytopes

Guillermo Pineda-Villavicencio, Benjamin Schröter

We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the…

math.CO2020

The linkedness of cubical polytopes: The cube

Hoa T. Bui, Guillermo Pineda-Villavicencio, Julien Ugon

The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2 k$ vertices is \textit{ $k$ -linked} if, for every set of $k$ disjoint pairs of…

math.CO2020

The linkedness of cubical polytopes: beyond the cube

Hoa T. Bui, Guillermo Pineda-Villavicencio, Julien Ugon

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A gra…

math.CO2020

A new proof of Balinski's theorem on the connectivity of polytopes

Guillermo Pineda-Villavicencio

Balinski (1961) proved that the graph of a $d$ -dimensional convex polytope is $d$ -connected. We provide a new proof of this result. Our proof provides details on the nature of a se…

math.CO2020

Minimum number of edges of polytopes with 2d + 2 vertices

Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

We define an analogue of the cube and an analogue of the 5-wedge in higher dimensions, each with $2 d + 2$ vertices and $d^{2} + 2 d - 3$ edges. We show that these two are the only minimiser…

math.CO2018★ 2 cited

The linkedness of cubical polytopes

Hoa Thi Bui, Guillermo Pineda-Villavicencio, Julien Ugon

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A gra…

math.CO2018

Connectivity of cubical polytopes

Hoa T. Bui, Guillermo Pineda-Villavicencio, Julien Ugon

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish…

math.CO2017★ 1 cited

Polytopes close to being simple

Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$ -polytopes with at most $d - 2$ nonsimple vertices are reconstructib…

math.CO2017★ 2 cited

The excess degree of a polytope

Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

We define the excess degree $ξ (P)$ of a $d$ -polytope $P$ as $2 f_{1} - d f_{0}$ , where $f_{0}$ and $f_{1}$ denote the number of vertices and edges, respectively. This parameter measures how mu…

math.CO2017

On the reconstruction of polytopes

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio +2

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$ -skeleton. Call a vertex of a $d$ -polytope \emph{nonsim…

math.CO2015

Almost simplicial polytopes I. The lower and upper bound theorems

Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon +1

We study $n$ -vertex $d$ -dimensional polytopes with at most one nonsimplex facet with, say, $d + s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and uppe…

math.CO2015★ 14 cited

Lower bound theorems for general polytopes

Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

For a $d$ -dimensional polytope with $v$ vertices, $d + 1 \leq v \leq 2 d$ , we calculate precisely the minimum possible number of $m$ -dimensional faces, when $m = 1$ or $m \geq 0.62 d$ . This conf…

math.CO2013★ 2 cited

On the maximum order of graphs embedded in surfaces

Eran Nevo, Guillermo Pineda-Villavicencio, David R. Wood

The maximum number of vertices in a graph of maximum degree $Δ \geq 3$ and fixed diameter $k \geq 2$ is upper bounded by $(1 + o (1)) (Δ - 1)^{k}$ . If we restrict our graphs to certain class…

Equifacetted polytopes whose facets are products of simplices: The simplicial prism case

Connectivity, toughness, and hamiltonicity of simplicial polytopes

A lower bound theorem for d-polytopes with at most 3d−1 vertices

A refined lower bound theorem for d-polytopes with at most 2d vertices

A lower bound theorem for d-polytopes with 2d+2 vertices

Polytopes with low excess degree

Edge connectivity of simplicial polytopes

Cycle space of graphs of polytopes

Edge connectivity of simplicial polytopes

A lower bound theorem for d-polytopes with 2d+1 vertices

Linkedness of Cartesian products of complete graphs

Reconstructibility of matroid polytopes

The linkedness of cubical polytopes: The cube

The linkedness of cubical polytopes: beyond the cube

A new proof of Balinski's theorem on the connectivity of polytopes

Minimum number of edges of polytopes with 2d + 2 vertices

The linkedness of cubical polytopes

Connectivity of cubical polytopes

Polytopes close to being simple

The excess degree of a polytope

On the reconstruction of polytopes

Almost simplicial polytopes I. The lower and upper bound theorems

Lower bound theorems for general polytopes

On the maximum order of graphs embedded in surfaces

A lower bound theorem for $d$ -polytopes with at most $3 d - 1$ vertices

A refined lower bound theorem for $d$ -polytopes with at most $2 d$ vertices

A lower bound theorem for $d$ -polytopes with $2 d + 2$ vertices

A lower bound theorem for $d$ -polytopes with $2 d + 1$ vertices