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Gerardo L. Maldonado

Publications and source records attributed to Gerardo L. Maldonado.

5 recordsLinked to original sources
arXiv2026

Disconnected graphs and extremal bounds for realizable distance orders

Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hernández and Martínez-Sandoval proved that every total order on the edges of the complete graph $K_n$ is realizable in $\mathbb{R}^{n-2}$. We show that the same is not true for the disjoint union of two complete graphs: for every $n\geq 3$ there is a total order on the edges of $K_n\sqcup K_n$ that is not realizable in $\mathbb{R}^{n-2}$, but is in $\mathbb{R}^{n-1}$. Surprisingly, the realizability of an order on a disconnected graph is not determined by its restrictions to the connected components. We also study realizability on the real line: we characterize which disjoint unions of two cycles are realizable, and estimate the largest number of edges an $n$-vertex graph can have while all of its edge-orders remain realizable on the line. In general dimension, we show that the largest number of edges of an $n$-vertex graph all of whose edge-orders are realizable in $\mathbb{R}^d$ is $dn+O\!\left(dn/\ln(dn)\right)$.

math.CO↗
arXiv2026

Extinction thresholds in a graph-based model of HIV infection dynamics

We study a graph-based cellular automaton for HIV infection dynamics in lymph-node networks, originally introduced by Mukwembi. Each vertex represents a cell site that may be healthy, infected, or dead, and the evolution is controlled by a replacement parameter $R$, which determines whether a dead cell is replaced by an infected or a healthy cell according to the number of its infected neighbors. For a graph $G$, we introduce two extinction parameters. The parameter $\mathbf{hiv}(G)$ is the smallest value of $R$ for which extinction occurs for every admissible initial configuration, whereas $\mathbf{HIV}(G)$ is the smallest threshold such that extinction occurs for every replacement parameter greater than or equal to it. We prove the general bounds $2\leq \mathbf{hiv}(G)\leq \mathbf{HIV}(G)\leq Δ(G)+1$ and characterize the extremal case $\mathbf{HIV}(G)=Δ(G)+1$. We also show that the gap $\mathbf{HIV}(G)-\mathbf{hiv}(G)$ is unbounded and determine both parameters for some classical families of graphs. Finally, we study the dynamics of the model using the state-transition digraph of the system and the configurations whose trajectories converge to nontrivial periodic orbits. The results show that extinction depends not only on the replacement parameter but also on the structural properties of the underlying graph.

math.DS↗
arXiv2025

Total orders realizable as the distances between two sets of points

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let $n\le m$ be positive integers and let $X$ and $Y$ be sets of sizes $n$ and $m$ in $\mathbb{R}^{n-1}$ such that every pair of points in $X\cup Y$ defines a unique distance. There is a natural order on $X\times Y$ induced by the distances between the corresponding points. The question is if all possible orders on $X\times Y$ can be obtained in this way. We show that the answer is negative when $n<m$. The case $n=m$ remains open.

math.CO↗
arXiv2024

On the orthogonal Grünbaum partition problem in dimension three

Grünbaum's equipartition problem asked if for any measure $μ$ on $\mathbb{R}^d$ there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$ $μ$-equal parts. This problem is known to have a positive answer for $d\le 3$ and a negative one for $d\ge 5$. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for $d\le 2$ and there is reason to expect it to have a negative answer for $d\ge 3$. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of $8n$ in $\mathbb{R}^3$ can be split evenly by $3$ mutually orthogonal planes. To our surprise, it seems the probability that a random set of $8$ points chosen uniformly and independently in the unit cube does not admit such a partition is less than $0.001$.

math.CO↗