arXiv subjects

Ron Aharoni

Publications and source records attributed to Ron Aharoni.

At least 37 records · Page 2Linked to original sources

arXiv2019

Rainbow fractional matchings

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a set of edges from distinct $E_i$'s which supports such a fractional matching). When the hypergraph is $r$-partite and $n$ is an integer, the number of sets needed goes down from $rn$ to $rn-r+1$. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for $r$-partite hypergraphs with $r>2$. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

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arXiv2018

Large rainbow matchings in general graphs

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Barát, Gyárfás and Sárközy, we conjecture that if $n$ is odd then the same is true also in general graphs, and that if $n$ is even then $2n$ matchings of size $n$ suffice. We prove that any $3n-2$ matchings of size $n$ have a partial rainbow matching of size $n$.

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arXiv2018

Finding a best approximation pair of points for two polyhedra

Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge to a best approximation pair. We propose a process based on projections onto the half-spaces defining the two polyhedra, which are more negotiable than projections on the polyhedra themselves. A central component in the proposed process is the Halpern--Lions--Wittmann--Bauschke algorithm for approaching the projection of a given point onto a convex set.

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arXiv2018

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $χ(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.

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arXiv2018

Rainbow triangles and the Caccetta-Häggkvist conjecture

A famous conjecture of Caccetta and Häggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ there is a directed cycle of length $r$ or less. We consider the following generalization: in an undirected graph on $n$ vertices, any collection of $n$ disjoint sets of edges, each of size at least $\frac{n}{r}$, has a rainbow cycle of length $r$ or less. We focus on the case $r=3$, and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. For any fixed $k$ and large enough $n$, we determine the maximum number of edges in an $n$-vertex edge-coloured graph where all colour classes have size at most $k$ and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.

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arXiv2017

Weighted domination of independent sets

The {\em independent domination number} $γ^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $γ^i$ is equal to $γ$, the ordinary domination number. The weighted version of this result is not true, but we show that it does hold for interval graphs, and for the intersection (that is, line) graphs of subtrees of a given tree, where each subtree is a single edge.

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arXiv2017

Independence-Domination duality in weighted graphs

Given a partition ${\mathcal V}=(V_1, \ldots,V_m)$ of the vertex set of a graph $G$, an {\em independent transversal} (IT) is an independent set in $G$ that contains one vertex from each $V_i$. A {\em fractional IT} is a non-negative real valued function on $V(G)$ that represents each part with total weight at least $1$, and belongs as a vector to the convex hull of the incidence vectors of independent sets in the graph. It is known that if the domination number of the graph induced on the union of every $k$ parts $V_i$ is at least $k$, then there is a fractional IT. We prove a weighted version of this result. This is a special case of a general conjecture, on the weighted version of a duality phenomenon, between independence and domination in pairs of graphs.

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arXiv2017

Representation of large matchings in bipartite graphs

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenm{ü}ller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \le \lceil \frac{3}{2}n \rceil+1$.

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arXiv2017

Fair representation in the intersection of two matroids

For a simplicial complex ${\mathcal C}$ denote by $β({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in at least $\frac{|A_i|}{β({\mathcal C})}$ elements. We conjecture that a slightly weaker result is true for the intersections of two matroids: if ${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcal Q}$ are matroids on the same ground set $V$ and $β({\mathcal P}), β({\mathcal P}) \le k$, then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal D}$ meeting each $A_i$ in at least $(\frac{1}{k}-\frac{1}{|V|})|A_i|-1$ elements. We prove this for a partition into two sets.

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arXiv2016

Fair representation by independent sets

For a hypergraph $H$ let $β(H)$ denote the minimal number of edges from $H$ covering $V(H)$. An edge $S$ of $H$ is said to represent {\em fairly} (resp. {\em almost fairly}) a partition $(V_1,V_2, \ldots, V_m)$ of $V(H)$ if $|S\cap V_i|\ge \lfloor\frac{|V_i|}{β(H)}\rfloor$ (resp. $|S\cap V_i|\ge \lfloor\frac{|V_i|}{β(H)}\rfloor-1$) for all $i \le m$. In matroids any partition of $V(H)$ can be represented fairly by some independent set. We look for classes of hypergraphs $H$ in which any partition of $V(H)$ can be represented almost fairly by some edge. We show that this is true when $H$ is the set of independent sets in a path, and conjecture that it is true when $H$ is the set of matchings in $K_{n,n}$. We prove that partitions of $E(K_{n,n})$ into three sets can be represented almost fairly. The methods of proofs are topological.

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arXiv2016

Independent sets in the union of two Hamiltonian cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c 12$ vertices is the union of two Hamiltonian cycles and $α(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

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arXiv2016

A rainbow $r$-partite version of the Erdős-Ko-Rado theorem

Let $f(n,r,k)$ be the minimal number such that every hypergraph larger than $f(n,r,k)$ contained in $\binom{[n]}{r}$ contains a matching of size $k$, and let $g(n,r,k)$ be the minimal number such that every hypergraph larger than $g(n,r,k)$ contained in the $r$-partite $r$-graph $[n]^{r}$ contains a matching of size $k$. The Erdős-Ko-Rado theorem states that $f(n,r,2)=\binom{n-1}{r-1}$~~($r \le \frac{n}{2}$) and it is easy to show that $g(n,r,k)=(k-1)n^{r-1}$. The conjecture inspiring this paper is that if $F_1,F_2,\ldots,F_k\subseteq \binom{[n]}{r}$ are of size larger than $f(n,r,k)$ or $F_1,F_2,\ldots,F_k\subseteq [n]^{r}$ are of size larger than $g(n,r,k)$ then there exists a rainbow matching, i.e. a choice of disjoint edges $f_i \in F_i$. In this paper we deal mainly with the second part of the conjecture, and prove it for $r\le 3$. \vspace{.1cm} We also prove that for every $r$ and $k$ there exists $n_0=n_0(r,k)$ such that the $r$-partite version of the conjecture is true for $n>n_0$.

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arXiv2016

Cross-intersecting pairs of hypergraphs

Two hypergraphs $H_1,\ H_2$ are called {\em cross-intersecting} if $e_1 \cap e_2 \neq \emptyset$ for every pair of edges $e_1 \in H_1,~e_2 \in H_2$. Each of the hypergraphs is then said to {\em block} the other. Given parameters $n,r,m$ we determine the maximal size of a sub-hypergraph of $[n]^r$ (meaning that it is $r$-partite, with all sides of size $n$) for which there exists a blocking sub-hypergraph of $[n]^r$ of size $m$. The answer involves a fractal-like (that is, self-similar) sequence, first studied by Knuth. We also study the same question with $\binom{n}{r}$ replacing $[n]^r$.

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arXiv2016

Degree conditions for matchability in $3$-partite hypergraphs

We study conjectures relating degree conditions in $3$-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko \cite{drisko} (as generalized by the first two authors \cite{ab}), that every family of $2n-1$ matchings of size $n$ in a bipartite graph has a partial rainbow matching of size $n$. We show that milder restrictions on the sizes of the matchings suffice. Another result that is strengthened is a theorem of Cameron and Wanless \cite{CamWan}, that every Latin square has a diagonal (permutation submatrix) in which no symbol appears more than twice. We show that the same is true under the weaker condition that the square is row-Latin.

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arXiv2016

Edge-covers in d-interval hypergraphs

A d-interval hypergraph has d disjoint copies of the unit interval as its vertex set, and each edge is the union of d subintervals, one on each copy. Extending a classical result of Gallai on the case d = 1, Tardos and Kaiser used topological tools to bound the ratio between the transversal number and the matching number in such hypergraphs. We take a dual point of view, and bound the edge-covering number (namely the minimal number of edges covering the entire vertex set) in terms of a parameter expressing independence of systems of partitions of the d unit intervals. The main tool we use is an extension of the KKM theorem to products of simplices, due to Peleg. Our approach also yields a new proof of the Tardos-Kaiser result.

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arXiv2016

On a conjecture of Stein

Stein proposed the following conjecture: if the edge set of $K_{n,n}$ is partitioned into $n$ sets, each of size $n$, then there is a partial rainbow matching of size $n-1$. He proved that there is a partial rainbow matching of size $n(1-\frac{D_n}{n!})$, where $D_n$ is the number of derangements of $[n]$. This means that there is a partial rainbow matching of size about $(1- \frac{1}{e})n$. Using a topological version of Hall's theorem we improve this bound to $\frac{2}{3}n$.

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arXiv2015

Uniqueness of the extreme cases in theorems of Drisko and Erdős-Ginzburg-Ziv

Drisko \cite{drisko} proved (essentially) that every family of $2n-1$ matchings of size $n$ in a bipartite graph possesses a partial rainbow matching of size $n$. In \cite{bgs} this was generalized as follows: Any $\lfloor \frac{k+2}{k+1} n \rfloor -(k+1)$ matchings of size $n$ in a bipartite graph have a rainbow matching of size $n-k$. We extend this latter result to matchings of not necessarily equal cardinalities. Settling a conjecture of Drisko, we characterize those families of $2n-2$ matchings of size $n$ in a bipartite graph that do not possess a rainbow matching of size $n$. Combining this with an idea of Alon \cite{alon}, we re-prove a characterization of the extreme case in a well-known theorem of Erdős-Ginzburg-Ziv in additive number theory.

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arXiv2015

Rainbow sets in the intersection of two matroids

Given sets $F_1, \ldots ,F_n$, a {\em partial rainbow function} is a partial choice function of the sets $F_i$. A {\em partial rainbow set} is the range of a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\sqrt{n}$.

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