arXiv subjects

Rodrigo Santos Monteiro

Publications and source records attributed to Rodrigo Santos Monteiro.

5 recordsLinked to original sources

arXiv2026

Kriesell's conjecture for infinite graphs

Let $G$ be a graph and $S\subseteq V(G)$ be a subset of vertices. An $S$-Steiner tree $T$ of $G$ is a tree of $G$ which contains $S$ in its vertex set $V(T)$. Kriesell conjectured that for every $2k$-edge-connected subset $S\subseteq V(G)$ in a finite connected graph $G$, there exist $k$ pairwise edge-disjoint $S$-Steiner trees. This conjecture is false for infinite graphs. We present a version of Kriesell's conjecture with topological $S$-Steiner trees for countable finitely edge-separable graphs and a version with $F$-limits of trees for rayless graphs. We show that if Kriesell's conjecture holds for finite graphs, then it holds for every connected, rayless and finitely edge-separable graph. We also show that every $2k$-edge-connected rayless and finitely edge-separable graph contains $k$ pairwise edge-disjoint spanning trees.

math.CO↗

arXiv2026

Menger and Rothberger games on convergence spaces

We introduce Menger and Rothberger selection principles and games for convergence spaces. Alice plays families that meet every convergent filter, and Bob's selections are required either to retain this property or merely to cover the underlying set. When the convergence is topological, both games recover the classical games. The winning condition requiring an $L$-cover satisfies analogues of the Hurewicz and Pawlikowski characterizations, the condition requiring a cover of $X$ is represented by the weak Menger and Rothberger games. We also show that, for a regular convergence space, a winning strategy for Bob in $\mathsf G_{\fin}(\mathcal C_L,\Cov(X))$ implies an Alster-type covering property. Under hereditary Lindelöfness the space is moreover a countable union of compactoid subsets, which are compact in the pretopological case.

math.GN↗

arXiv2026

On convergence structures in graphs

A closure operator on a set $X$ is a function $\operatorname{cl}: \wp(X) \to \wp(X)$ satisfying, for all $A, B \subseteq X$, the following properties: extensivity, $A \subseteq \operatorname{cl}(A)$; monotonicity, which states that if $A \subseteq B$ then $\operatorname{cl}(A) \subseteq \operatorname{cl}(B)$; and preservation of unions, $\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)$. Every graph $G$ naturally carries such an operator on its vertex set by assigning to each subset $A \subseteq V(G)$ the set $\operatorname{cl}(A) = A \cup N(A)$, where $N(A)$ denotes the vertices adjacent to a vertex in $A$. Since closure operators and pretopological spaces are equivalent notions, this operator induces a canonical convergence structure on $V(G)$. We describe this convergence in terms of nets and relate combinatorial properties of the graph to convergence-theoretic ones.

math.CO↗

arXiv2024

Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces

Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this work, we use a net-theoretic approach to convergence spaces. The goal is to simplify the description of continuous convergence and apply it to problems related to homotopy theory. We present methods to develop the basis of homotopy theory in limit spaces, define the fundamental groupoid, and prove the groupoid version of the Seifert-van Kampen Theorem for limit spaces.

math.AT↗

arXiv2024

A net theoretic approach to homotopy theory

This paper uses a net-theoretic approach to convergence spaces, aimed to simplify the description of continuous convergence in order to apply it in problems concerning Homotopy Theory. We present methods for handling homotopies of limit spaces, define fundamental groupoids, and prove a generalized version of the Seifert-van Kampen Theorem for limit spaces.

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