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Guillermo Pineda-Villavicencio

Publications and source records attributed to Guillermo Pineda-Villavicencio.

At least 19 recordsLinked to original sources
arXiv2026

Connectivity, toughness, and hamiltonicity of simplicial polytopes

For every $d\ge4$, every prescribed connectivity $r\ge d$, and every $\varepsilon>0$, we construct a simplicial $d$-polytope whose graph has connectivity exactly $r$, is nontraceable, and has toughness less than $\varepsilon$. Thus the classical theorems of Whitney (1931) and Tutte (1956), which guarantee hamiltonicity under sufficiently high connectivity in dimension three, have no higher-dimensional counterpart. This answers a question posed by Bojan Mohar in 2001 on whether sufficiently high connectivity guarantees hamiltonicity of $4$-polytopal graphs and the connectivity questions raised by Ikegami, Maezawa, and Zamfirescu in 2021. For fixed $d\ge4$, $r\ge d$, and integer $k\ge2$, the examples can be chosen to have no closed spanning walk with at most $k$ visits per vertex, no spanning tree with degrees at most $k$, and nontraceable prisms. For fixed $d$ and $r$, we obtain graphs on $n$ vertices whose longest paths and cycles contain $O(\sqrt n)$ vertices. Consequently, the family of simplicial $d$-polytopes with connectivity exactly $r$ has Grünbaum--Walther shortness coefficient zero and shortness exponent at most $1/2$.

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arXiv2025

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

We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. Two distinct lower bounds emerge, depending on the number of facets of $P$. When $P$ has precisely $d+2$ facets, the lower bound is tight when $d$ is odd. If $P$ has at least $d+3$ facets, the lower bound is always tight, and equality holds for some $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets. Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets.

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arXiv2025

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

We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those with at most $2d$ vertices, $2d+1$ vertices, and $2d+2$ vertices. If $P$ has exactly $d+2$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound is tight for certain combinations of $d$ and $\ell$. When $P$ has at least $d+3$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound remains tight up to $\ell=d-1$, and equality for some $1\le k\le d-2$ is attained only when $P$ has precisely $d+3$ facets. We exhibit at least one minimiser for each number of vertices between $2d+1$ and $3d-1$, including two distinct minimisers with $2d+2$ vertices and three with $3d-2$ vertices.

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arXiv2025

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

In 1967, Grünbaum conjectured that the function $$ ϕ_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in[1\ldots d-2]$ and characterised the unique minimisers, each having $d+2$ facets. In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $ϕ_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes.

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arXiv2024

Polytopes with low excess degree

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 expressed in terms of the excess degree of the polytope, defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are not realised by any polytope. For $d$-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$.

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arXiv2023

The linkedness of cubical polytopes: beyond the cube

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 graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its graph is $k$-linked. In a previous paper \cite{BuiPinUgo20a} we proved that every cubical $d$-polytope is $\floor{d/2}$-linked. Here we strengthen this result by establishing the $\floor{(d+1)/2}$-linkedness of cubical $d$-polytopes, for every $d\ne 3$. A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We say that a polytope is (strongly) \textit{$k$-linked} if its graph is (strongly) $k$-linked. In this paper, we also prove that every cubical $d$-polytope is strongly $\floor{d/2}$-linked, for every $d\ne 3$. These results are best possible for this class of polytopes.

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arXiv2023

Edge connectivity of simplicial polytopes

We show that the graph of a simplicial polytope of dimension $d \ge 3$ has no nontrivial minimum edge cut with fewer than $d(d+1)/2$ edges, hence the graph is $\min\{δ, d(d+1)/2\}$-edge-connected where $δ$ denotes the minimum degree. When $d = 3$, this implies that every minimum edge cut in a plane triangulation is trivial. When $d \ge 4$, we construct a simplicial $d$-polytope whose graph has a nontrivial minimum edge cut of cardinality $d(d+1)/2$, proving that the aforementioned result is best possible.

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arXiv2023

Edge connectivity of simplicial polytopes

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 first establish that, for any $d\ge 3$, for any $d\ge 3$, every minimum edge cut of cardinality at most $4d-7$ in such a graph is \textit{trivial}, namely it consists of all the edges incident with some vertex. A consequence of this is that, for $d\ge 3$, the graph of a simplicial $d$-polytope with minimum degree $δ$ is $\min\{δ,4d-6\}$-edge-connected. In the particular case of $d=3$, we have that every minimum edge cut in a plane triangulation is trivial; this may be of interest to researchers in graph theory. Second, for every $d\ge 4$ we construct a simplicial $d$-polytope whose graph has a nontrivial minimum edge cut of cardinality $(d^{2}+d)/2$. This gives a simplicial 4-polytope with a nontrivial minimum edge cut that has ten edges. Thus, the aforementioned result is best possible for simplicial $4$-polytopes.

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arXiv2022

Cycle space of graphs of polytopes

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, which seem to be the most widely known proof. As a corollary, we obtain a result of Blind & Blind (1994) stating that graphs of polytopes are bipartite if and only if graphs of every 2-face are bipartite.

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arXiv2022

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

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 been solved for $n\le2d$. We establish the corresponding result for $n=2d+1$; the nature of the lower bounds and the minimising polytopes are quite different in this case. As a byproduct, we also characterise all $d$-polytopes with $d+3$ vertices, and only one or two edges more than the minimum.

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arXiv2021

Reconstructibility of matroid polytopes

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 polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present a $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterisation of all matroids with isomorphic basis exchange graphs.

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arXiv2021

The linkedness of cubical polytopes: The cube

The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its graph is $k$-linked. We establish that the $d$-dimensional cube is $\lfloor(d+1)/2\rfloor$-linked, for every $d\ne 3$; this is the maximum possible linkedness of a $d$-polytope. This result implies that, for every $d\ge 1$, a cubical $d$-polytope is $\lfloor{d/2}\rfloor$-linked, which answers a question of Wotzlaw \cite{Ron09}. Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We show that cubical 4-polytopes are strongly $2$-linked and that, for each $d\ge 1$, $d$-dimensional cubes are strongly $\lfloor{d/2}\rfloor$-linked.

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arXiv2020

Linkedness of Cartesian products of complete graphs

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best possible. %A polytope is said to be {\it $k$-linked} if its graph is $k$-linked. This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})\times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})\times T(d_{2})$ is a {\it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges. While not every $d$-polytope is $\floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

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arXiv2020

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

We define an analogue of the cube and an analogue of the 5-wedge in higher dimensions, each with $2d+2$ vertices and $d^2+2d-3$ edges. We show that these two are the only minimisers of the number of edges, amongst d-polytopes with $2d+2$ vertices, for all $d$ except 4, 5 and 7. We also show that there are four sporadic minimisers in these low dimensions. We announce a partial solution to the corresponding problem for polytopes with $2d + 3$ vertices.

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arXiv2019

The linkedness of cubical polytopes

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 graph with at least $2k$ vertices is $k$-linked if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. Larman and Mani in 1970 proved that simplicial $d$-polytopes, polytopes with all their facets being combinatorially equivalent to simplices, are $\floor{(d+1)/2}$-linked; this is the maximum possible linkedness given the facts that a $\floor{(d+1)/2}$-linked graph is at least $(2\floor{(d+1)/2}-1)$-connected and that some of these graphs are $d$-connected but not $(d+1)$-connected. Here we establish that cubical $d$-polytopes are also $\floor{(d+1)/2}$-linked for every $d\ne 3$; this is again the maximum possible linkedness for such a class of polytopes.

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arXiv2019

Connectivity of cubical polytopes

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 that, for any $d\ge 3$, the graph of a cubical $d$-polytope with minimum degree $δ$ is $\min\{δ,2d-2\}$-connected. Second, we show, for any $d\ge 4$, that every minimum separator of cardinality at most $2d-3$ in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself.

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arXiv2019

Lower bound theorems for general polytopes

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Grünbaum, for these values of $m$. For $v=2d+1$, we solve the same problem when $m=1$ or $d-2$; the solution was already known for $m= d-1$. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of $m$-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

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