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Andrzej Ruciński

Publications and source records attributed to Andrzej Ruciński.

At least 19 recordsLinked to original sources
arXiv2026

Rainbow Turán numbers for paths of length four

Given a set $V$ of $n$ vertices and an integer $k\ge1$, our goal is to maximize the number of edges in graphs $G_1, G_2, \ldots, G_k$, defined on $V$, under the constraint that the union of all graphs, thought of as a multi-graph, does not contain a rainbow copy of the path $P_5$ on $5$ vertices, that is, a copy of $P_5$ with each of its four edges belonging to a different $G_i$. We consider two versions of the problem, in which, respectively, $\sum_i e(G_i)$ and $\min_i e(G_i)$ is maximized. In the former case, we determine the maximum precisely for all $k\le n-1$ (and also for $P_4$). In the latter, we obtain an asymptotic value for $k\in\{5,6,9\}$ and formulate a very plausible conjecture for all other values of $k$. We also solve the problem for $k=4$, but under an additional assumption of completeness.

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arXiv2026

Homogeneous substructures in random ordered hyper-matchings

An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges order-isomorphic to a member of $\mathcal{P}$. In this paper we are interested in the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching selected uniformly from all such matchings on a fixed vertex set $[rn]$. We determine this size (up to multiplicative constants) for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$, the set $\mathcal{R}^{(r)}$ of all $r$-partite patterns, as well as sets $\mathcal{P}$ enjoying a Boolean-like, symmetric structure.

math.CO↗
arXiv2026

Homogeneous substructures in random ordered uniform matchings

An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges belonging to $\mathcal{P}$. In this paper we determine the order of magnitude of the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$ and the set $\mathcal{R}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns.

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arXiv2025

Powers of Hamiltonian cycles in randomly augmented Pósa-Seymour graphs

We study the question of the least number of random edges that need to be added to a Pósa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle, $m>k$. It turns out that, depending on $k$ and $m$, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large $m$. Indeed, for each $k\ge2$ and $m>m_0(k)$, we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of $m$ the two bounds coincide. In addition, we also determine the thresholds for some small values of $k$ and $m$.

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arXiv2025

Shuffle squares and ordered nest-free graphs

A shuffle square is a word consisting of two shuffled copies of the same word. For instance, the Turkish word $\mathtt{\color{red}{ik}\color{blue}{i}\color{red}{li}\color{blue}{kli}}$ (binary in English) is a shuffle square, as it can be split into two copies of the word $\mathtt{ikli}$. We explore a representation of shuffle squares in terms of \emph{ordered nest-free graphs} and demonstrate the usefulness of this approach by applying it to several families of binary words. Among others, we characterize shuffle squares with four and five runs, as well as shuffle squares with all $\mathtt1$-runs of length one (and with the $\mathtt1$'s alternating between the two copies). In our main result we provide quite general sufficient conditions for a binary word not to be a shuffle square. In particular, it follows that binary words of the type $(\mathtt{1001})^n$, $n$ odd, are not shuffle squares. We complement it by showing that all other words whose every $\mathtt{1}$-run has length one or two, while every $\mathtt{0}$-run has length two, are shuffle squares. We also provide a counterexample to a believable stipulation that binary words of the form $\mathtt1^{m}\mathtt0^{m-2}\mathtt1^{m-4}\cdots$, $m$ odd, are far from being shuffle squares (the distance measured by the minimum number of letters one has to delete in order to turn a word into a shuffle square).

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arXiv2024

Erdős-Szekeres type Theorems for ordered uniform matchings

For $r,n\ge2$, an ordered $r$-uniform matching of size $n$ is an $r$-uniform hypergraph on a linearly ordered vertex set $V$, with $|V|=rn$, consisting of $n$ pairwise disjoint edges. There are $\tfrac12\binom{2r}r$ different ways two edges may intertwine, called here patterns. Among them we identify $3^{r-1}$ collectable patterns $P$, which have the potential of appearing in arbitrarily large quantities called $P$-cliques. We prove an Erdős-Szekeres type result guaranteeing in every ordered $r$-uniform matching the presence of a $P$-clique of a prescribed size, for some collectable pattern $P$. In particular, in the diagonal case, one of the $P$-cliques must be of size $Ω\left( n^{3^{1-r}}\right)$. In addition, for each collectable pattern $P$ we show that the largest size of a $P$-clique in a random ordered $r$-uniform matching of size $n$ is, with high probability, $Θ\left(n^{1/r}\right)$.

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arXiv2024

Largest bipartite sub-matchings of a random ordered matching or a problem with socks

Let $M$ be an ordered matching of size $n$, that is, a partition of the set $[2n]$ into 2-element subsets. The sock number of $M$ is the maximum size of a sub-matching of $M$ in which all left-ends of the edges precede all the right-ends (such matchings are also called bipartite). The name of this parameter comes from an amusing "real-life" problem posed by Bosek, concerning an on-line pairing of randomly picked socks from a drying machine. Answering one of Bosek's questions we prove that the sock number of a random matching of size $n$ is asymptotically equal to $n/2$. Moreover, we prove that the expected average number of socks waiting for their match during the whole process is equal to $\frac{2n+1}{6}$. Analogous results are obtained if socks come not in pairs, but in sets of size $r\geq 2$, which corresponds to a similar problem for random ordered $r$-matchings. We also attempt to enumerate matchings with a given sock number.

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arXiv2024

Ordered unavoidable sub-structures in matchings and random matchings

An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment $\{1,2\},\{3,4\}$, a nesting $\{1,4\},\{2,3\}$, and a crossing $\{1,3\},\{2,4\}$. Accordingly, there are three basic homogeneous types of ordered matchings (with all pairs of edges arranged in the same way) which we call, respectively, lines, stacks, and waves. We prove an Erdős-Szekeres type result guaranteeing in every ordered matching of size $n$ the presence of one of the three basic sub-structures of a given size. In particular, one of them must be of size at least $n^{1/3}$. We also investigate the size of each of the three sub-structures in a random ordered matching. Additionally, the former result is generalized to $3$-uniform ordered matchings. Another type of unavoidable patterns we study are twins, that is, pairs of order-isomorphic, disjoint sub-matchings. By relating to a similar problem for permutations, we prove that the maximum size of twins that occur in every ordered matching of size $n$ is $O\left(n^{2/3}\right)$ and $Ω\left(n^{3/5}\right)$. We conjecture that the upper bound is the correct order of magnitude and confirm it for almost all matchings. In fact, our results for twins are proved more generally for $r$-multiple twins, $r\ge2$.

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arXiv2023

Twins in ordered hyper-matchings

An ordered $r$-matching of size $n$ is an $r$-uniform hypergraph on a linearly ordered set of vertices, consisting of $n$ pairwise disjoint edges. Two ordered $r$-matchings are isomorphic if there is an order-preserving isomorphism between them. A pair of twins in an ordered $r$-matching is formed by two vertex disjoint isomorphic sub-matchings. Let $t^{(r)}(n)$ denote the maximum size of twins one may find in every ordered $r$-matching of size $n$. By relating the problem to that of largest twins in permutations and applying some recent Erdős-Szekeres-type results for ordered matchings, we show that $t^{(r)}(n)=Ω\left(n^{\frac{3}{5\cdot(2^{r-1}-1)}}\right)$ for every fixed $r\geqslant 2$. On the other hand, $t^{(r)}(n)=O\left(n^{\frac{2}{r+1}}\right)$, by a simple probabilistic argument. As our main result, we prove that, for almost all ordered $r$-matchings of size $n$, the size of the largest twins achieves this bound.

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arXiv2023

Long twins in random words

Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a $k$-letter alphabet. The obtained lower bounds for small values of $k$ significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length $n$ contains twins of length at least $0.34n$. Our main result states that in a random ternary word of length $n$, with high probability, one can find twins of length at least $0.41n$. In the general case of alphabets of size $k\geq 3$ we obtain analogous lower bounds of the form $\frac{1.64}{k+1}n$ which are better than the known deterministic bounds for $k\leq 354$. In addition, we present similar results for multiple twins in random words.

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arXiv2023

Powers of Hamiltonian cycles in randomly augmented Dirac graphs -- the complete collection

We study the powers of Hamiltonian cycles in randomly augmented Dirac graphs, that is, $n$-vertex graphs $G$ with minimum degree at least $(1/2+\varepsilon)n$ to which some random edges are added. For any Dirac graph and every integer $m\ge2$, we accurately estimate the threshold probability $p=p(n)$ for the event that the random augmentation $G\cup G(n,p)$ contains the $m$-th power of a Hamiltonian cycle.

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arXiv2023

Constructing sparsest $\ell$-hamiltonian saturated $k$-uniform hypergraphs for a wide range of $\ell$

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.

math.CO↗
arXiv2021

Sandwiching biregular random graphs

Let $G(n,n,m)$ be a uniformly random $m$-edge subgraph of the complete bipartite graph $K_{n,n}$ with bipartition $(V_1, V_2)$, where $n_i = |V_i|$. Given a real number $p \in [0,1]$ such that $d_1 := pn_2$ and $d_2 := pn_1$ are integers, let $R(n,n,p)$ be a random subgraph of $K_{n,n}$ such that every $v \in V_i$ has degree $d_i$, for $i = 1, 2$. In this paper we determine sufficient conditions on $n_1,n_2,p$, and $m$ under which one can embed $G(n,n,m)$ into $R(n,n,p)$ and vice versa with probability tending to $1$. In particular, in the balanced case $n_1 = n_2$, we show that if $p \gg \log n/n$ and $1 - p \gg \left(\log n/n \right)^{1/4}$, then for some $m \sim pn^2$, asymptotically almost surely one can embed $G(n,n,m)$ into $R(n,n,p)$, while for $p \gg \left(\log^{3} n/n\right)^{1/4}$ and $1-p \gg \log n/n$ we have the opposite embedding. As an extension, we confirm the Kim--Vu Sandwich Conjecture for degrees growing faster than $(n \log n)^{3/4}$.

math.CO↗
arXiv2021

Tight multiple twins in permutations

Two permutations are similar if they have the same length and the same relative order. A collection of $r\ge2$ disjoint, similar subsequences of a permutation $π$ form $r$-twins in $π$. We study the longest guaranteed length of $r$-twins which are tight in the sense that either each twin alone forms a block or their union does. We address the same question with respect to a random permutation.

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arXiv2021

High powers of Hamiltonian cycles in randomly augmented graphs

We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and for any $α>\frac{k}{k+1}$ we show that adding $O(n^{2-2/\ell})$ random edges to an $n$-vertex graph $G$ with minimum degree at least $αn$ yields, with probability close to one, the existence of the $(k\ell+r)$-th power of a Hamiltonian cycle. In particular, for $r=1$ and $\ell=2$ this implies that adding $O(n)$ random edges to such a graph $G$ already ensures the $(2k+1)$-st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of $k$, $\ell$, and $r$ we can show that our result is asymptotically optimal.

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arXiv2020

On weak twins and up-and-down sub-permutations

Two permutations $(x_1,\dots,x_w)$ and $(y_1,\dots,y_w)$ are weakly similar if $x_i π(i_2)<π(i_3)>...$ or $π(i_1)<π(i_2)>π(i_3)<...$. Let $Π_n$ be a random permutation selected uniformly from all $n!$ permutations of $[n]$. It is known that the length of a longest alternating permutation in $Π_n$ is asymptotically almost surely (a.a.s.) close to $2n/3$. We study the maximum length $α(n)$ of a pair of disjoint alternating sub-permutations in $Π_n$ and show that there are two constants $1/3<c_1<c_2<1/2$ such that a.a.s. $c_1n\le α(n)\le c_2n$. In addition, we show that the alternating shape is the most popular among all permutations of a given length.

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arXiv2020

Variations on twins in permutations

Let $π$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $π$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $Ω(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length $Ω(n^{2/3}/\log n^{1/3})$. Recently, Bukh and Rudenko have tweaked our proof and removed the log-factor. For completeness, we also present our version of their proof (see Remark 1.2 below on the interrelation between the two proofs). In addition, we study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation $τ$, then the corresponding extremal function equals $Θ(\sqrt{n})$, provided that $τ$ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is $(1+o(1))\frac{\log n}{\log\log n}$, while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every $n$ there are permutations avoiding them on all segments of length greater than $24$.

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