arXiv subjects

Érika Roldán

Publications and source records attributed to Érika Roldán.

12 recordsLinked to original sources

arXiv2026

Extremal fences with polyforms

We present results around an isoperimetric problem built on polyforms: What is the biggest enclosed area one can build using polyforms in each of the three plane tessellations? We give challenges to the readers and present Shimauchi's proof of the biggest area a fence made of pentominoes can enclose. A translation of the instance using integer linear programming is also given. A companion web app is available to test some of the challenges we propose and for activities, and we included extra pages with a cutout handout of the board and pieces of the puzzles, so that you can print and cut them to read this paper hands on.

math.HO↗

arXiv2026

Holey Hyperbolic Polyforms

A polyform is a planar figure formed by gluing congruent regular polygons along entire edges. We study polyforms in hyperbolic ${p,q}$-tessellations and the extremal problem of minimizing the number of tiles needed to realize exactly $h$ holes. Denoting this minimum by $g_{p,q}(h)$, we establish general lower and upper bounds, compute exact values in several small cases, and give a sufficient structural condition for a polyform to have $h$ holes and $g_{p,q}(h)$ tiles.

math.CO↗

arXiv2024

Approximately Optimal Search on a Higher-dimensional Sliding Puzzle

Higher-dimensional sliding puzzles are constructed on the vertices of a $d$-dimensional hypercube, where $2^d-l$ vertices are distinctly coloured. Rings with the same colours are initially set randomly on the vertices of the hypercube. The goal of the puzzle is to move each of the $2^d-l$ rings to pre-defined target vertices on the cube. In this setting, the $k$-rule constraint represents a generalisation of edge collision for the movement of colours between vertices, allowing movement only when a hypercube face of dimension $k$ containing a ring is completely free of other rings. Starting from an initial configuration, what is the minimum number of moves needed to make ring colours match the vertex colours? An algorithm that provides us with such a number is called God's algorithm. When such an algorithm exists, it does not have a polynomial time complexity, at least in the case of the 15-puzzle corresponding to $k=1$ in the cubical puzzle. This paper presents a comprehensive computational study of different scenarios of the higher-dimensional puzzle. A benchmark of three computational techniques, an exact algorithm (the A* search) and two approximately optimal search techniques (an evolutionary algorithm (EA) and reinforcement learning (RL)) is presented in this work. The experiments show that all three methods can successfully solve the puzzle of dimension three for different face dimensions and across various difficulty levels. When the dimension increases, the A* search fails, and RL and EA methods can still provide a generally acceptable solution, i.e. a distribution of a number of moves with a median value of less than $30$. Overall, the EA method consistently requires less computational time, while failing in most cases to minimise the number of moves for the puzzle dimensions $d=4$ and $d=5$.

cs.AI↗

arXiv2023

There is no perfect Mondrian partition for squares of side lengths less than 1001

In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length $n$ square for $n\leq 1000$. Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a $n \times m$ rectangle for $n,m \leq 400$.

math.CO↗

arXiv2023

Tverberg Partition Graphs

Given a finite set of points in $\mathbb{R}^d$, Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given set of points, whose edges describe closeness between different Tverberg partitions. We prove bounds on the minimum and maximum degree of this graph, the number of vertices of maximal degree, its clique number, and its connectedness.

math.CO↗

arXiv2023

Higher-dimensional cubical sliding puzzles

We introduce higher-dimensional cubical sliding puzzles that are inspired by the classical 15 Puzzle from the 1880s. In our puzzles, on a $d$-dimensional cube, a labeled token can be slid from one vertex to another if it is topologically free to move on lower-dimensional faces. We analyze the solvability of these puzzles by studying how the puzzle graph changes with the number of labeled tokens vs empty vertices. We give characterizations of the different regimes ranging from being completely stuck (and thus all puzzles unsolvable) to having only one giant component where almost all puzzles can be solved. For the Cube, the Tesseract, and the Penteract ($5$-dimensional cube) we have implemented an algorithm to completely analyze their solvability and we provide specific puzzles for which we know the minimum number of moves needed to solve them.

math.CO↗

arXiv2021

Polyiamonds Attaining Extremal Topological Properties, Part II

In Part II of this work, we construct crystallized polyiamonds with $h$ holes for every $h\ge1$, that is polyiamonds which use the fewest possible tiles necessary to enclose $h$ holes. Furthermore, we prove that crystallized polyiamonds satisfy a set of structural conditions, and for every $h\ge 3$ there are multiple distinct crystallized polyiamonds with $h$ holes.

math.CO↗

arXiv2021

Extremal $\{p, q\}$-Animals

An animal is a planar shape formed by attaching congruent regular polygons, known as tiles, along their edges. In this paper, we study extremal animals defined on regular tessellations of the plane. In 1976, Harary and Harborth studied animals in the Euclidean cases, finding extremal values for their vertices, edges, and tiles, when any one of these parameters is fixed. Here, we generalize their results to hyperbolic animals. For each hyperbolic tessellation, we exhibit a sequence of spiral animals and prove that they attain the minimal numbers of edges and vertices within the class of animals with $n$ tiles. In their conclusions, Harary and Harborth also proposed the question of enumerating extremal animals with a fixed number of tiles. This question has previously only been considered for Euclidean animals. As a first step in solving this problem, we find special sequences of extremal animals that are unique extremal animals, in the sense that any animal with the same number of tiles which is distinct up to isometries can't be extremal.

math.CO↗

arXiv2020

Topology of random 2-dimensional cubical complexes

We study a natural model of random 2-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result is to exhibit a sharp threshold p=1/2 for homology vanishing as $n \to \infty$. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing the connectivity threshold for random cubical graphs. Our main result can also be seen as a cubical counterpart to the Linial--Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhibit strikingly different behaviors. We show that if $p > 1 - \sqrt{1/2} \approx 0.2929$, then with high probability the fundamental group is a free group with one generator for every maximal $1$-dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong "hitting time" sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles -- we show that with high probability the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

math.CO↗

arXiv2019

Topology and Geometry of Crystallized Polyominoes

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystallized polyominoes and characterize their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. We also develop a new dynamical method of creating sequences of polyominoes which is invariant with respect to crystallization and efficient structure. Using this technique, we prove that crystallized polyominoes with $h_l=(2^{2l}-1)/3$ holes are unique.

math.CO↗

arXiv2018

Art gallery problem with rook and queen vision

How many chess rooks or queens does it take to guard all the squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can "see" whichever squares the rook or queen attacks. We show that floor(n/2) rooks or floor(n/3) queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We also prove that finding the minimum number of rooks or the minimum number of queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. We also use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

cs.CC↗

arXiv2018

Polyominoes with maximally many holes

What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$ for $k \ge 1$. We also give nearly matching upper and lower bounds for large $n$, showing as a corollary that $f(n) \approx n/2$.

math.CO↗