Searcharxiv⌕ Search

arXiv subjects

Attila Joó

Publications and source records attributed to Attila Joó.

At least 37 records · Page 2Linked to original sources
arXiv2022

On generalisations of the Aharoni-Pouzet base exchange theorem

The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $ is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite dimensional vector spaces. Later he and Aharoni answered this conjecture affirmatively not just for vector spaces but for infinite matroids. We prove two generalisations of their result. On the one hand, we show that `being a singleton' can be relaxed to `being finite' and this is sharp in the sense the exclusion of infinite sets is really necessary. On the other hand, we prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $ between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.

math.CO↗
arXiv2022

Cutting a cake for infinitely many guests

Fair division with unequal shares is an intensively studied recourse allocation problem. For $ i\in [n] $, let $ μ_i $ be an atomless probability measure on the measurable space $(C,\mathcal{S}) $ and let $ t_i $ be positive numbers (entitlements) with $ \sum_{i=1}^{n}t_i=1 $. A fair division is a partition of $ C $ into sets $ S_i\in \mathcal{S} $ with $ μ_i(S_i)\geq t_i$ for every $ i\in [n] $. We introduce new algorithms to solve the fair division problem with irrational entitlements. They are based on the classical Last diminisher technique and we believe that they are simpler than the known methods. Then we show that a fair division always exists even for infinitely many players.

math.CO↗
arXiv2021

The Lovász-Cherkassky theorem in countable graphs

Lovász and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $ T $ is $ \sum_{t\in T}λ(t, T-t) $ where $λ$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $ T $-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs.

math.CO↗
arXiv2021

Enlarging vertex-flames in countable digraphs

A rooted digraph is a vertex-flame if for every vertex $v$ there is a set of internally disjoint directed paths from the root to $v$ whose set of terminal edges covers all ingoing edges of $v$. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by the third author and the analogue of Lovász' result for countable digraphs was shown. We strengthen this result by proving that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.

math.CO↗
arXiv2021

Intersection of a partitional and a general infinite matroid

Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $ I_M\sqcup I_N $ such that $ \mathsf{span}_M(I_M)\cup \mathsf{span}_N(I_N)=E $. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special case where one of the matroids is a direct sum of uniform matroids, i.e., is a partitional matroid. We show that if $ M $ is an arbitrary matroid and $ N $ is the direct sum of finitely many uniform matroids, then $ \{ M, N \} $ has the Intersection property.

math.CO↗
arXiv2021

Proof of Nash-Williams' Intersection Conjecture for countable matroids

We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $ and $ I\cap E_N $ spans $ E_N $ in $ N $. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.

math.CO↗
arXiv2021

Greedoids from flames

A digraph $ D $ with $ r\in V(D) $ is an $ r $-flame if for every $ {v\in V(D)-r} $, the in-degree of $ v $ is equal to the local edge-connectivity $ λ_D(r,v) $. We show that for every digraph $ D $ and $ r\in V(D) $, the edge sets of the $ r $-flame subgraphs of $ D $ form a greedoid. Our method yields a new proof of Lovász' theorem stating: for every digraph $ D $ and $ r\in V(D) $, there is an $ r $-flame subdigraph $ F $ of $ D $ such that $ λ_F(r,v) =λ_D(r,v) $ for $ v\in V(D)-r $. We also give a strongly polynomial algorithm to find such an $ F $ working with a fractional generalization of Lovász' theorem.

math.CO↗
arXiv2021

On the Packing/Covering Conjecture of Infinite Matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family $ (M_i: i\inΘ) $ of matroids on the common edge set $ E $ is a system $ (S_i: i\inΘ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is panning in $ M_i $. Similarly, a covering is a system $ (I_i: i\inΘ) $ with $\bigcup_{i\inΘ} I_i=E $ where $ I_i $ is independent in $ M_i $. The conjecture states that for every matroid family on $ E $ there is a partition $E=E_p \sqcup E_c$ such that $ (M_i \upharpoonright E_p: i\in Θ) $ admits a packing and $ (M_i. E_c: i\in Θ) $ admits a covering. We prove the special case where $ E $ is countable and each $ M_i $ is either finitary or cofinitary. The connection between packing/covering and matroid intersection problems discovered by Bowler and Carmesin can be established for every well-behaved matroid class. This makes possible to approach the problem from the direction of matroid intersection. We show that the generalized version of Nash-Williams' Matroid Intersection Conjecture holds for countable matroids having only finitary and cofinitary components.

math.CO↗
arXiv2021

On partitioning the edges of an infinite digraph into directed cycles

Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.

math.CO↗
arXiv2020

Reducing the dichromatic number via cycle reversions in infinite digraphs

We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.

math.CO↗
arXiv2020

Base partition for mixed families of finitary and cofinitary matroids

Let ${\mathcal{M} = (M_i \colon i\in K)}$ be a finite or infinite family consisting of matroids on a common ground set $E$ each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each $M_i$, which covers the set $E$, and also a collection of bases which is pairwise disjoint, then there is a collection of bases which partitions $E$. We also show that the failure of this Cantor-Bernstein-type statement for arbitrary matroid families is consistent relative to the axioms of set theory ZFC.

math.CO↗
arXiv2019

Vertex-flames in countable rooted digraphs preserving an Erdős-Menger separation for each vertex

It follows from a theorem of Lovász that if $ D $ is a finite digraph with $ r\in V(D) $ then there is a spanning subdigraph $ E $ of $ D $ such that for every vertex $ v\neq r $ the following quantities are equal: the local connectivity from $ r $ to $ v $ in $ D $, the local connectivity from $ r $ to $ v $ in $ E $ and the indegree of $ v $ in $ E $. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a result is the generalization of Menger's theorem to infinite digraphs. We generalize the result of Lovász above in this spirit. Our main result is that every countable $ r $-rooted digraph $ D $ has a spanning subdigraph $ E $ with the following property. For every $ v\neq r $, $ E $ contains a system $ \mathcal{R}_v $ of internally disjoint $ r\rightarrow v $ paths such that the ingoing edges of $ v $ in $ E $ are exactly the last edges of the paths in $ \mathcal{R}_v $. Furthermore, the path-system $ \mathcal{R}_v $ is `big' in $ D $ in the Erdős-Menger sense, i.e., one can choose from each path in $ \mathcal{R}_{v} $ either an edge or an internal vertex in such a way that a resulting set separates $ v $ from $ r $ in $ D $.

math.CO↗
arXiv2019

On the growth rate of dichromatic numbers of finite subdigraphs

Chris Lambie-Hanson proved recently that for every function $ f:\mathbb{N}\rightarrow \mathbb{N} $ there is an $ \aleph_1 $-chromatic graph $ G $ of size $ 2^{\aleph_1} $ such that every $ (n+3) $-chromatic subgraph of $ G $ has at least $ f(n) $ vertices. Previously, this fact was just known to be consistently true due to P. Komjáth and S. Shelah. We investigate the analogue of this question for directed graphs. In the first part of the paper we give a simple method to construct for an arbitrary $ f:\mathbb{N}\rightarrow \mathbb{N} $ an uncountably dichromatic digraph $ D $ of size $ 2^{\aleph_0} $ such that every $ (n+2) $-dichromatic subgraph of $ D $ has at least $ f(n) $ vertices. In the second part we show that it is consistent with arbitrary large continuum that in the previous theorem "uncountably dichromatic" and "of size $ 2^{\aleph_0} $" can be replaced by "$κ$-dichromatic" and "of size $ κ$" respectively where $ κ$ is universally quantified with bounds $ \aleph_0 \leq κ\leq 2^{\aleph_0}$.

math.CO↗
arXiv2019

Uncountable dichromatic number without short directed cycles

A. Hajnal and P. Erdős proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain for example $ C_4 $ (among other obligatory subgraphs). It was shown recently by D. T. Soukup that, in contrast of the undirected case, it is consistent that for any $ n<ω$ there exists an uncountably dichromatic digraph without directed cycles shorter than $ n $. He asked if it is provable already in ZFC. We answer his question positively by constructing for every infinite cardinal $ κ$ and $ n<ω$ a digraph of size $ 2^κ $ with dichromatic number at least $ κ^{+} $ which does not contain directed cycles of length less than $ n $ as a subdigraph.

math.CO↗
arXiv2019

The Complete Lattice of Erdős-Menger Separations

F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal (i.e. minimal sized) separations form a finite distributive lattice with respect to a natural partial order. Furthermore, any finite distributive lattice can be represented this way. If there is no finite separation then cardinality is a too rough measure to capture being 'optimal'. Menger's theorem provides a structural characterization of optimality if there is a finite separation. We use this characterization to define Erdős-Menger separations even if there is no finite separation. The generalization of Menger's theorem to infinite graphs (which was not available until 2009) ensures that Erdős-Menger separations always exist. We show that they form a complete lattice with respect to the partial order given by Escalante and every complete lattice can be represented this way.

math.CO↗
arXiv2017

King-serf duo by monochromatic paths in k-edge-coloured tournaments

An open conjecture of Erdős states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $κ>0$ there is a cardinal $ λ_κ$ such that in every $κ$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ λ_κ$ so that every vertex $ v $ has a monochromatic path of length at most two from $K$ to $v$ or from $v$ to $S$.

math.CO↗
arXiv2017

Edmonds' Branching Theorem in Digraphs without Forward-infinite Paths

Let $ D $ be a finite digraph, and let $ V_0,\dots,V_{k-1} $ be nonempty subsets of $ V(D) $. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $ \mathcal{B}_0,\dots, \mathcal{B}_{k-1} $ in $ D $ such that the root set of $ \mathcal{B}_i $ is $ V_i\ (i=0,\dots,k-1) $ if and only if for all $ \varnothing \neq X\subseteq V(D) $ the number of ingoing edges of $ X $ is greater than or equal to the number of sets $ V_i $ disjoint from $ X $. As was shown by R. Aharoni and C. Thomassen in \cite{aharoni1989infinite}, this theorem does not remain true for infinite digraphs. Thomassen also proved that for the class of digraphs without backward-infinite paths, the above theorem of Edmonds remains true. Our main result is that for digraphs without forward-infinite paths, Edmonds' branching theorem remains true as well.

math.CO↗