Searcharxiv⌕ Search

arXiv subjects

Ron Aharoni

Publications and source records attributed to Ron Aharoni.

47 records · Page 3Linked to original sources
arXiv2015

Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.

math.CO↗
arXiv2015

Covers in Partitioned Intersecting Hypergraphs

Given an integer $r$ and a vector $\vec{a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{i \le p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be $\vec{a}$-partitioned if $V(H)=\bigcup_{i \le p}V_i$, where the sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i \le p$. A $\vec{1}$-partitioned hypergraph is said to be $r$-partite. Let $t(\vec{a})$ be the maximum, over all intersecting $\vec{a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t(\vec{1})\le r-1$. Tuza conjectured that if $r>2$ then $t(\vec{a})=r$ for every two components vector $\vec{a}=(a,b)$. We prove this conjecture whenever $a\neq b$, and also for $\vec{a}=(2,2)$ and $\vec{a}=(4,4)$.

math.CO↗
arXiv2014

Fractional covers and matchings in families of weighted $d$-intervals

A $d$-{\em interval} is a union of at most $d$ disjoint closed intervals on a fixed line. Tardos [Combinatorica 15 (1995), 123-134] and the second author [Disc. Comput. Geom. 18 (1997), 195-203] used topological tools to bound the transversal number $τ$ of a family $H$ of $d$-intervals in terms of $d$ and the matching number $ν$ of $H$. We investigate the weighted and fractional versions of this problem and prove upper bounds that are tight up to constant factors. We apply both the topological method and an approach of Alon [Disc. Comput. Geom. 19 (1998), 333-334]. For the use of the latter, we prove a weighted version of Turán's theorem. We also provide a proof of the second author's upper bound that is more direct than the original proof.

math.CO↗
arXiv2013

On a Generalization of the Ryser-Brualdi-Stein Conjecture

A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to this concept (some of them first presented here), and to present some partial results on one of these conjectures, that seems central among them.

math.CO↗
arXiv2013

A weak version of Rota's basis conjecture for odd dimensions

The Alon-Tarsi Latin square conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column). A modified version of Onn's colorful determinantal identity is used to show how the validity of this conjecture implies a weak version of Rota's basis conjecture for odd dimensions, namely that a set of $n$ bases in $\mathbb{R}^n$ has $n-1$ disjoint independent transversals.

math.CO↗
arXiv2009

The Max-Flow Min-Cut Theorem for Countable Networks

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are "orthogonal" to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends.

math.CO↗
arXiv2009

Strongly maximal matchings in infinite weighted graphs

Given an assignment of weights w to the edges of a graph G, a matching M in G is called strongly w-maximal if for any matching N the sum of weights of the edges in N\M is at most the sum of weights of the edges in M\N. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching.

math.CO↗
arXiv2009

Perfect matchings in r-partite r-graphs

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal (r-1)-tuple contained in the complement of this side is strictly larger than n/2. We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus.

math.CO↗
arXiv2007

Menger's theorem for infinite graphs

We prove that Menger's theorem is valid for infinite graphs, in the following strong form: let $A$ and $B$ be two sets of vertices in a possibly infinite digraph. Then there exist a set $\cp$ of disjoint $A$-$B$ paths, and a set $S$ of vertices separating $A$ from $B$, such that $S$ consists of a choice of precisely one vertex from each path in $\cp$. This settles an old conjecture of Erdős.

math.CO↗
arXiv2000

A tree version of Konig's theorem

Konig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation of this result, in which each point in one side of the graph is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell.

math.CO↗