arXiv subjects

Raphael Yuster

Publications and source records attributed to Raphael Yuster.

67 records · Page 4Linked to original sources

arXiv2007

A comment on Ryser's conjecture for intersecting hypergraphs

Let $τ(\mathcal{H})$ be the cover number and $ν(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $τ(\mathcal{H}) \leq (r-1) ν(\mathcal{H})$. This conjecture is open for all $r \ge 4$. For intersecting hypergraphs, namely those with $ν(\mathcal{H})=1$, Ryser's conjecture reduces to $τ(\mathcal{H}) \leq r-1$. Even this conjecture is extremely difficult and is open for all $ r \ge 6$. For infinitely many $r$ there are examples of intersecting $r$-partite hypergraphs with $τ(\mathcal{H})=r-1$, demonstrating the tightness of the conjecture for such $r$. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting $r$-partite hypergraph be, given that its cover number is as large as possible, namely $τ(\mathcal{H}) \ge r-1$? In this paper we solve this question for $r \le 5$, give an almost optimal construction for $r=6$, prove that any $r$-partite intersecting hypergraph with $τ(H) \ge r-1$ must have at least $(3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))$ edges, and conjecture that there exist constructions with $Θ(r)$ edges.

math.CO↗

arXiv2006

Finding heaviest H-subgraphs in real weighted graphs, with applications

For a graph G with real weights assigned to the vertices (edges), the MAX H-SUBGRAPH problem is to find an H-subgraph of G with maximum total weight, if one exists. The all-pairs MAX H-SUBGRAPH problem is to find for every pair of vertices u,v, a maximum H-subgraph containing both u and v, if one exists. Our main results are new strongly polynomial algorithms for the all-pairs MAX H-SUBGRAPH problem for vertex weighted graphs. We also give improved algorithms for the MAX-H SUBGRAPH problem for edge weighted graphs, and various related problems, including computing the first k most significant bits of the distance product of two matrices. Some of our algorithms are based, in part, on fast matrix multiplication.

cs.DS↗

arXiv2004

Mean Ramsey-Turán numbers

A $ρ$-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most $ρ$. For a graph $H$ and for $ρ\geq 1$, the {\em mean Ramsey-Turán number} $RT(n,H,ρ-mean)$ is the maximum number of edges a $ρ$-mean colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. It is conjectured that $RT(n,K_m,2-mean)=RT(n,K_m,2)$ where $RT(n,H,k)$ is the maximum number of edges a $k$ edge-colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. We prove the conjecture holds for $K_3$. We also prove that $RT(n,H,ρ-mean) \leq RT(n,K_{χ(H)},ρ-mean)+o(n^2)$. This result is tight for graphs $H$ whose clique number equals their chromatic number. In particular we get that if $H$ is a 3-chromatic graph having a triangle then $RT(n,H,2-mean) = RT(n,K_3,2-mean)+o(n^2)=RT(n,K_3,2)+o(n^2)=0.4n^2(1+o(1))$.

math.CO↗

arXiv2003

Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-packing} of a graph $G$ is a set of edge disjoint copies of $H$ in $G$. The following results are proved. For every fixed $k > 2$, every graph with $n$ vertices and minimum degree at least $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges.

math.CO↗

arXiv2003

Packing 4-cycles in Eulerian and bipartite Eulerian tournaments with an application to distances in interchange graphs

We prove that every Eulerian orientation of $K_{m,n}$ contains $\frac{1}{4+\sqrt{8}}mn(1-o(1))$ arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with $n$ vertices contains $\frac{1}{8+\sqrt{32}}n^2(1-o(1))$ arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between two antipodal vertices in interchange graphs.

math.CO↗

arXiv2003

Integer and fractional packing of families of graphs

Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $ν_{\cal F}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $ψ$ from the set of elements of ${\cal F}$ in $G$ to $[0,1]$ is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {ψ(H)} \leq 1$ for each $e \in E(G)$. The {\em fractional ${\cal F}$-packing number}, denoted $ν^*_{\cal F}(G)$, is defined to be the maximum value of $\sum_{H \in {{G} \choose {\cal F}}} ψ(H)$ over all fractional ${\cal F}$-packings $ψ$. Our main result is that $ν^*_{\cal F}(G)-ν_{\cal F}(G) = o(|V(G)|^2)$. Furthermore, a set of $ν_{\cal F}(G) -o(|V(G)|^2)$ edge-disjoint elements of ${\cal F}$ in $G$ can be found in randomized polynomial time. For the special case ${\cal F}=\{H_0\}$ we obtain a significantly simpler proof of a recent difficult result of Haxell and Rödl \cite{HaRo} that $ν^*_{H_0}(G)-ν_{H_0}(G) = o(|V(G)|^2)$.

math.CO↗

arXiv2003

The number of edge disjoint transitive triples in a tournament

We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and Rödl.

math.CO↗

arXiv2002

The order of monochromatic subgraphs with a given minimum degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. For $n > k > d$ let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

math.CO↗

arXiv2002

Tiling transitive tournaments and their blow-ups

Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$ and $c_k=1-2^{-k-\log k}$ for $k \geq 4$. For every $ε> 0$ there exists $N=N(ε,h,k)$ such that for every undirected graph $G$ with $n > N$ vertices and with $δ(G) \geq c_kn$, every orientation of $G$ contains vertex disjoint copies of $TT(h,k)$ that cover all but at most $εn$ vertices. In the cases $k=2$ and $k=3$ the result is asymptotically tight. For $k \geq 4$, $c_k$ cannot be improved to less than $1-2^{-0.5k(1+o(1))}$.

math.CO↗

arXiv2002

Families of trees decompose the random graph in any arbitrary way

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $α_1 e(H_1) + ... + α_k e(H_k) = m$ where each $α_i$ is a nonnegative integer, there is a coloring of the edges of $G$ with $α_1+...+α_k$ colors such that exactly $α_i$ color classes induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree, then $\log n/n$ is a sharp threshold function for the property that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$.

math.CO↗

arXiv2002

Equitable coloring of k-uniform hypergraphs

Let $H$ be a $k$-uniform hypergraph with $n$ vertices. A {\em strong $r$-coloring} is a partition of the vertices into $r$ parts, such that each edge of $H$ intersects each part. A strong $r$-coloring is called {\em equitable} if the size of each part is $\lceil n/r \rceil$ or $\lfloor n/r \rfloor$. We prove that for all $a \geq 1$, if the maximum degree of $H$ satisfies $Δ(H) \leq k^a$ then $H$ has an equitable coloring with $\frac{k}{a \ln k}(1-o_k(1))$ parts. In particular, every $k$-uniform hypergraph with maximum degree $O(k)$ has an equitable coloring with $\frac{k}{\ln k}(1-o_k(1))$ parts. The result is asymptotically tight. The proof uses a double application of the non-symmetric version of the Lovász Local Lemma.

math.CO↗

arXiv2002

Edge coloring complete uniform hypergraphs with many components

Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. Let $f_k(H)$ be the maximum possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. If $H$ is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. Several results concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform hypergraph on $n$ vertices, are also established.

math.CO↗

arXiv2001

The domatic number of regular and almost regular graphs

The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices has $dom(G) \geq 2$. For every $k$, it is known that there are graphs with minimum degree at least $k$ and with $dom(G)=2$. In this paper we prove that this is not the case if $G$ is $k$-regular or {\em almost} $k$-regular (by ``almost'' we mean that the minimum degree is $k$ and the maximum degree is at most $Ck$ for some fixed real number $C \geq 1$). In this case we prove that $dom(G) \geq (1+o_k(1))k/(2\ln k)$. We also prove that the order of magnitude $k/\ln k$ cannot be improved. One cannot replace the constant 2 with a constant smaller than 1. The proof uses the so called {\em semi-random method} which means that combinatorial objects are generated via repeated applications of the probabilistic method; in our case iterative applications of the Lovász Local Lemma.

math.CO↗