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Andrzej Dudek

Publications and source records attributed to Andrzej Dudek.

At least 55 records · Page 3Linked to original sources
arXiv2017

Approximate Counting of Matchings in $(3,3)$-Hypergraphs

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite $(3,3)$-hypergraphs. The proof technique of this paper uses the general correlation decay technique and a new combinatorial analysis of the underlying structures of the intersection graphs. The proof method could be also of independent interest.

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arXiv2017

Square of a Hamilton cycle in a random graph

We show that the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle is $p=\frac{1}{\sqrt{n}}$. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić. In addition we consider how many random edges need to be added to a graph of order $n$ with minimum degree $αn$ in order that it contains the square of a Hamilton cycle w.h.p.

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arXiv2017

On offset Hamilton cycles in random hypergraphs

An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) satisfies $|E_{i-1}\cap E_i|=\ell$ and every pair of consecutive edges $E_{i},E_{i+1}$ in $C$ satisfies $|E_{i}\cap E_{i+1}|=k-\ell$. We show that in general $\sqrt{e^{k}\ell!(k-\ell)!/n^k}$ is the sharp threshold for the existence of the $\ell$-offset Hamilton cycle in the random $k$-uniform hypergraph $H_{n,p}^{(k)}$. We also examine this structure's natural connection to the 1-2-3 Conjecture.

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arXiv2016

Embedding the Erdős-Rényi Hypergraph into the Random Regular Hypergraph and Hamiltonicity

We establish an inclusion relation between two uniform models of random $k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$, the random $d$-regular $k$-graph. We show that if $n\log n\ll m\ll n^k$ we can choose $d = d(n) \sim {km}/n$ and couple $\mathbb G^{(k)}(n,m)$ and $\mathbb R^{(k)}(n,d)$ so that the latter contains the former with probability tending to one as $n\to\infty$. This extends an earlier result of Kim and Vu about "sandwiching random graphs". In view of known threshold theorems on the existence of different types of Hamilton cycles in $\mathbb G^{(k)}(n,m)$, our result allows us to find conditions under which $\mathbb R^{(k)}(n,d)$ is Hamiltonian. In particular, for $k\ge 3$ we conclude that if $n^{k-2} \ll d \ll n^{k-1}$, then a.a.s. $\mathbb R^{(k)}(n,d)$ contains a tight Hamilton cycle.

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arXiv2016

Square of Hamilton cycle in a random graph

We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić.

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arXiv2016

Size-Ramsey numbers of cycles versus a path

The size-Ramsey number $\hat{R}(\mathcal{F},H)$ of a family of graphs $\mathcal{F}$ and a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours, say, red and blue, yields a red copy of a graph from $\mathcal{F}$ or a blue copy of $H$. In this paper we first focus on $\mathcal{F} = \mathcal{C}_{\le cn}$, where $\mathcal{C}_{\le cn}$ is the family of cycles of length at most $cn$, and $H = P_n$. In particular, we show that $2.00365 n \le \hat{R}(\mathcal{C}_{\le n},P_n) \le 31n$. Using similar techniques, we also managed to analyze $\hat{R}(C_n,P_n)$, which was investigated before but only using the regularity method.

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arXiv2016

On the number of alternating paths in bipartite complete graphs

Let $C \subseteq [r]^m$ be a code such that any two words of $C$ have Hamming distance at least $t$. It is not difficult to see that determining a code $C$ with the maximum number of words is equivalent to finding the largest $n$ such that there is an $r$-edge-coloring of $K_{m, n}$ with the property that any pair of vertices in the class of size $n$ has at least $t$ alternating paths (with adjacent edges having different colors) of length $2$. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum $t$ such that there is an $r$-edge-coloring of $K_{m,n}$ such that any pair of vertices in class of size $n$ is connected by $t$ internally disjoint and alternating paths of length $2k$. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs.

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arXiv2016

On some multicolour Ramsey properties of random graphs

The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, $\hat{R}(P_n) \ge (1+\sqrt{2})n-O(1)$, due to Bollobás, and the upper bound, $\hat{R}(P_n) \le 91n$, due to Letzter. Next we study long monochromatic paths in edge-coloured random graph $G(n,p)$ with $pn \to \infty$. Let $α> 0$ be an arbitrarily small constant. Recently, Letzter showed that a.a.s.\ any $2$-edge colouring of $G(n,p)$ yields a monochromatic path of length $(2/3-α)n$, which is optimal. Extending this result, we show that a.a.s.\ any $3$-edge colouring of $G(n,p)$ yields a monochromatic path of length $(1/2-α)n$, which is also optimal. In general, we prove that for $r\ge 4$ a.a.s.\ any $r$-edge colouring of $G(n,p)$ yields a monochromatic path of length $(1/r-α)n$. We also consider a related problem and show that for any $r \ge 2$, a.a.s.\ any $r$-edge colouring of $G(n,p)$ yields a monochromatic connected subgraph on $(1/(r-1)-α)n$ vertices, which is also tight.

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arXiv2015

Weak and strong versions of the 1-2-3 conjecture for uniform hypergraphs

Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function $ω:E\to\{1,\dots,w\}$, a coloring of vertices of $H$, induced by $ω$, is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that $H$ is strongly $w$-weighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that $H$ is weakly $w$-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all $5$-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for $r\ge 6$ we show that almost all $r$-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3-uniform hypergraphs are weakly 2-weighted but not 1-weighted and for $r\ge 4$ almost all $r$-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are $r$-uniform hypergraphs which are not strongly $(r^2-r)$-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete.

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arXiv2015

On the Ramsey-Turán number with small $s$-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Turán number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $α_s(G) < f$, where $α_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^δ)$ for fixed $δ<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<δ< 1$, $RT_s(n,K_{s+1}, n^δ) = Ω(n^{1+δ-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^δ) = O(n^{1+δ})$. Furthermore, the range of $δ$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^δ)$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.

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arXiv2015

The Total Acquisition Number of Random Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ can be moved to a neighbouring vertex $v$, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process. LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = Θ(n)$, confirming a conjecture of West.

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arXiv2015

On the size-Ramsey number of hypergraphs

The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for $k$-uniform hypergraphs. Analogous to the graph case, we consider the size-Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size-Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.

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arXiv2015

The set chromatic number of random graphs

In this paper we study the set chromatic number of a random graph $G(n,p)$ for a wide range of $p=p(n)$. We show that the set chromatic number, as a function of $p$, forms an intriguing zigzag shape.

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arXiv2014

Rainbow Connection of Random Regular Graphs

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove that with probability tending to one as $n$ goes to infinity the rainbow connection of $G$ satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.

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arXiv2014

An alternative proof of the linearity of the size-Ramsey number of paths

The size Ramsey number $\hat{r}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In 1983, Beck provided a beautiful argument that shows that $\hat{r}(P_n)$ is linear, solving a problem of Erdős. In this short note, we provide an alternative but elementary proof of this fact that actually gives a better bound, namely, $\hat{r}(P_n) < 137n$ for $n$ sufficiently large.

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arXiv2014

Acquaintance time of random graphs near connectivity threshold

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and the second author made a major step towards this conjecture by showing that asymptotically almost surely $AC(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs.

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arXiv2013

Maker-Breaker games on random geometric graphs

In a Maker-Breaker game on a graph $G$, Breaker and Maker alternately claim edges of $G$. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between $n$ points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as $n\to\infty$, the graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the $H$-game as soon as there is a subgraph from a finite list of "minimal graphs". These results also allow us to give precise expressions for the limiting probability that $G(n,r)$ is Maker-win in each case, where $G(n,r)$ is the graph on $n$ points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most $r$.

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