arXiv subjects

Leandro Aurichi

Publications and source records attributed to Leandro Aurichi.

9 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

A topological characterization of end space of infinite graphs via games, subspaces and products

In 1992, Diestel asked which topological spaces could be represented as the end space of some graph. In 2023, Pitz provided a solution to this question by giving a topological characterization of end spaces using a hereditarily complete special subbase. In this paper, we present an alternative topological characterization of end spaces, in which we employ a special subbase and a topological game. Furthermore, we provide several applications of this characterization: we show that every end space is hereditarily Baire, that $G_{\delta}$ subspaces of end spaces are also end spaces, and that the product of end spaces is not always an end space.

math.GN↗

arXiv2026

Edge-ends versus topological ends of graphs

Diestel and K\"uhn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and \v{S}ir\'a\v{n}, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and K\"uhn for edge-end spaces.

math.CO↗

arXiv2025

A density counterpart of the Scheepers covering property

We introduce a density counterpart of the Scheepers covering property $\bigcup_{\mathrm{fin}}(\mathcal O,\Omega)$ and study its relations to known combinatorial density property. In particular, we show that it is equivalent to the $M$-separability under the Near Coherence of Filters principle of Blass and Weiss.

math.GN↗

arXiv2025

On orientations preserving edge-connectivity in infinite graphs

We prove that every 2k-edge-connected graph with countably many edge-ends admits a k-arc-connected orientation, extending the previous result by Assem, Koloschin and Pitz that also assumed the hypothesis of the graph being locally finite. We prove that, if every locally finite graph has a well-balanced orientation, so does every graph. Lastly, we explore an alternative to the Nash-Williams Orientation Conjecture via topological paths, and prove that it is true for every finitely separated graph.

math.CO↗

arXiv2025

On cycle covers of infinite bipartite graphs

Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the double Hall property if for every subset $X$ of vertices of $A$ with size at least 2, $\vert N^2_G(X)\vert \geq \vert X\vert$. Salia conjectured that if $G$ is a bipartite graph that satisfies the double Hall property, then there exists a cycle in $G$ that covers all vertices of $A$. In this work, we study this conjecture restricted to infinite graphs. For this, we use the definition of ends and infinite cycles. It is simple to see that Salia's conjecture is false for infinite graphs in general. Consequently, all our results are partial. Under certain hypothesis it is possible to obtain a collection of pairwise disjoint 2-regular subgraphs that covers $A$. We show that if side $B$ is locally finite and side $A$ is countable, then the conjecture is true. Furthermore, assuming the conjecture holds for finite graphs, we show that it holds for infinite graphs with a restriction on the degree of the vertices of $B$. This result is inspired by the result obtained by Bar\'at, Grzesik, Jung, Nagy and P\'alv\"olgyi for finite graphs. Finally, we also show that if Salia's conjecture holds for some cases of infinite graphs, then the conjecture about finite graphs presented by Lavrov and Vandenbussche is true.

math.CO↗

arXiv2024

Ray inflations of $\omega_1$-trees and ends of degree $\aleph_1$

We prove the followings result for ray inflations of sparse graphs on $\omega_1$-trees. First, let $T$ and $S$ be pruned $\omega_1$-trees, let $G_T$ be a sparse $T$-graph, and let $G_S$ be a sparse $S$-graph with uniformly finite adhesion. If $T$ is not special, then no subdivision of $G_S\# \mathbb{N}$ is isomorphic to $G_T\# \mathbb{N}$. Second, if $T$ is almost-Suslin and $S$ is special, then $G_T\# \mathbb{N}$ contains no subgraph isomorphic to $G_S\# \mathbb{N}$, for arbitrary choices of the sparse graphs. Consequently, under $\diamondsuit_{\omega_1}$, this gives a counterexample to Halin's end degree conjecture that is not isomorphic to a subdivision of any ray inflation with uniformly finite adhesion. Under $\diamondsuit_{\omega_1}^{*}$, the underlying tree may in addition be chosen almost-Suslin, and the resulting graph contains no subgraph isomorphic to a ray inflation over a special $\omega_1$-tree.

math.LO↗

arXiv2019

Covering properties of $ω$-mad families

We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while $\mathfrak b=\mathfrak c$ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture.

math.LO↗

arXiv2015

Selective versions of chain condition-type properties

We study selective and game-theoretic versions of properties like the ccc, weak Lindelöfness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal invariants of the continuum.

math.GN↗