arXiv subjects

Jarosław Grytczuk

Publications and source records attributed to Jarosław Grytczuk.

At least 19 recordsLinked to original sources

arXiv2026

A rainbow partition theorem for trees and connected maximin share allocations of chores

Xiao, Qiu, and Huang (AAMAS 2023) and independently Lonc (personal communication) asked whether indivisible chores located at the vertices of a tree can always be allocated to $n$ agents in connected bundles so that the cost of every agent is at most its connected maximin share; for goods, this is a theorem of Bouveret, Cechlárová, Elkind, Igarashi, and Peters. We answer the question affirmatively, even for monotone costs. The answer follows from a combinatorial theorem: if $\mathcal P_1,\ldots,\mathcal P_k$ are partitions of the vertex set of a finite tree, each into at most $k$ connected parts, then the vertex set can be split into disjoint sets $B_1,\ldots,B_k$, some possibly empty, such that each nonempty $B_i$ is connected and contained in a part of $\mathcal P_i$. Equivalently, if each of $k$ colours occurs on at most $k-1$ edges of a tree, then the vertices can be partitioned into connected sets labelled by distinct colours, none containing an edge of its own colour; in particular, one can choose for every colour a component of the forest obtained by deleting that colour so that the chosen components cover the tree. The bound is already best possible for paths, and for additive costs, the theorem is equivalent to the fair-division statement. The proof reduces the problem to inward partitions of oriented trees, which we obtain from the colourful KKM theorem on a simplex of edge weights, using a leaf-elimination labelling that remains compatible when weights vanish. We also give an algorithm running in time $k^{O(k)}$ plus polynomial time.

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arXiv2026

Finite and infinite barrycades

An $n$-barrycade of height $h$ is a set of $h$ permutations of $[n]$ such that all the prefix sums are different. This notion was described in 2020 by Richard Guy, together with the central problem to determine for which values of $n$ there exists a break-free $n$-barrycade, that is, one in which every possible prefix sum from $1$ to $\frac{n(n+1)}{2}-1$ is covered. The height of such barrycade would be $\frac{n+2}{2}$. We prove that barrycades of linear height exist for infinitely many sizes: there is a constant $c>0$ such that for infinitely many positive integers $n$ there exists an $n$-barrycade of height at least $cn$. We also explore an infinite variant of the problem and propose three constructions -- greedy, grasshopper and precise grasshopper -- that give increasingly more satisfying results. Along the way we encounter new integer sequences and formulate conjectures concerning omitted elements, missing partial sums, and word representations of infinite barrycades, which are firmly supported by computational data.

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arXiv2026

Epistemic fair division of independence structures

We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

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arXiv2026

Packing arithmetic progressions

Let $\mathcal{F}=\{A_1,A_2,\ldots,A_k\}$ be a collection of finite arithmetic progressions, where each $A_d$ is an initial segment of the set $D_d=\{d,2d,3d,\ldots\}$ of consecutive multiples of a positive integer $d$. Let $m(\mathcal{F})$ denote the minimum length of an interval containing pairwise disjoint \emph{shifted} copies of all members of the family $\mathcal{F}$. We study this parameter in the following two cases: for a fixed positive integer $n$, (1) each progression in $\mathcal{F}$ has the form $A_d=D_d\cap\{1,2,\ldots,n\}$, and (2) all progressions $A_d$ of $\mathcal{F}$ have the same size $n$, that is, $A_d=D_d\cap \{1,2,\ldots, nd\}$. We in particular derive the following asymptotic estimates. In case (1), when $k=n$, we get $m(\mathcal{F})=Θ(n^{3/2}/\ln n)$. In case (2), when $k=n$, we get $m(\mathcal{F})=Θ(n^3/\ln n)$, while if $k>k_0(n)$, then $m(\mathcal{F}) < 3kn$. In both cases we additionally determine $m(\mathcal{F})$ asymptotically or settle its order of magnitude for all $k<n$.

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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.

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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

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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arXiv2025

Neighborly boxes and strings with jokers; constructions and asymptotics

We study families of axis-aligned boxes in a $d$-dimensional Euclidean space $\mathbb{R}^d$ whose placement is restricted by bounds on the dimension of their pairwise intersections. More specifically, two such boxes in $\mathbb{R}^d$ are said to be \emph{$k$-neighborly} if their intersection has dimension at least $d-k$ and at most $d-1$. The maximum number of pairwise $k$-neighborly boxes in $\mathbb{R}^d$ is denoted by $n(k,d)$. It is known that $n(k,d)=Θ(d^k)$, for fixed $1\leqslant k\leqslant d$, however, exact formulas are known only in three cases: $k=1$, $k=d-1$, and $k=d$. In particular, the equality $n(1,d)=d+1$ is equivalent to the famous theorem of Graham and Pollak concerning partitions of complete graphs into complete bipartite graphs. In our main result we give a new construction of families of $k$-neighborly boxes which improves the lower bound for $n(k,d)$ when $k$ is close to $d$. Together with some recent upper bounds on $n(k,d)$, it gives the asymptotic equality $n(d-s,d)\thicksim\frac{2^s+1}{2^{s+1}}\cdot2^d$, for every fixed $s\leqslant d/2$. In our constructions we use a familiar interpretation of the problem in the language of Hamming cubes represented by binary strings with a special blank symbol, called \emph{joker}.

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arXiv2025

Mrs. Correct and Majority Colorings

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an on-line analog of the list coloring.

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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

Words Avoiding Tangrams

A \emph{tangram} is a word in which every letter occurs an even number of times. Such word can be cut into parts that can be arranged into two identical words. The minimum number of cuts needed is called the \emph{cut number} of a tangram. For example, the word $\mathtt{\color{red}{0102}\color{blue}{0102}}$ is a tangram with cut number one, while the word $\mathtt{\color{red}{01}\color{blue}{01023}\color{red}{023}}$ is a tangram with cut number two. Clearly, tangrams with cut number one coincide with the well known family of words, known as \emph{squares}, having the form $UU$ for some nonempty word $U$. A word $W$ \emph{avoids} a word $T$ if it is not possible to write $W=ATB$, for any words $A$ and $B$ (possibly empty). The famous 1906 theorem of Thue asserts that there exist arbitrarily long words avoiding squares over alphabet with just \emph{three} letters. Given a fixed number $k\geqslant 1$, how many letters are needed to avoid tangrams with the cut number at most $k$? Let $t(k)$ denote the minimum size of an alphabet needed for that purpose. By Thue's result we have $t(1)=3$, which easily implies $t(2)=3$. Curiously, these are currently the only known exact values of this function. In our main result we prove that $t(k)=Θ(\log_2k)$. The proof uses \emph{entropy compression} argument and \emph{Zimin words}. By using a different method we prove that $t(k)\leqslant k+1$ for all $k\geqslant 4$, which gives more exact estimates for small values of $k$. The proof makes use of \emph{Dejean words} and a curious property of \emph{Gauss words}, which is perhaps of independent interest.

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arXiv2024

Variations on shuffle squares

We study decompositions of words into subwords that are in some sense similar, which means that one subword may be obtained from the other by a relatively simple transformation. Our main inspiration are shuffle squares, an intriguing class of words arising in various contexts, from purely combinatorial to more applied, like modeling concurrent processes or DNA sequencing. These words can be split into two parts that are just identical. For example, $ACTACATAGG$ is a shuffle square consisting of two copies of the word $ACTAG$. Of course, each letter must appear any even number of times in each shuffle square. We call words with that property even. We mainly discuss new problems concerning generalized shuffle squares. We propose a number of conjectures and provide some initial results towards them. We prove that every binary word is a cyclic shuffle square, meaning that it splits into two subwords, one of which is a~cyclic permutation of the other. The same statement is no longer true over larger alphabets, but it seems plausible that a similar property should hold with slightly less restricted permutation classes. For instance, we conjecture that every even ternary word is a dihedral shuffle square, which means that it splits into two subwords, one of which can be obtained from the other by a permutation corresponding to the~symmetry of a~regular polygon. We propose a general conjecture stating that a linear number of permutations is sufficient to express all even $k$-ary words as generalized shuffle squares. Our discussion is complemented by some enumerative and computational experiments. In particular, we disprove our former conjecture stating that every even binary word can be turned into a shuffle square by a cyclic permutation. The smallest counterexample has length $24$. We call words of this type shuffle anti-squares. We determined all of them up to the length $28$.

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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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arXiv2024

Neighborly boxes and bipartite coverings; constructions and conjectures

Two axis-aligned boxes in $\mathbb{R}^d$ are \emph{$k$-neighborly} if their intersection has dimension at least $d-k$ and at most $d-1$. The maximum number of pairwise $k$-neighborly boxes in $\mathbb{R}^d$ is denoted by $n(k,d)$. It is known that $n(k,d)=Θ(d^k)$, for fixed $1\leqslant k\leqslant d$, but exact formulas are known only in three cases: $k=1$, $k=d-1$, and $k=d$. In particular, the formula $n(1,d)=d+1$ is equivalent to the famous theorem of Graham and Pollak on bipartite partitions of cliques. In this paper we are dealing with the case $k=2$. We give a new construction of $k$-neighborly \emph{codes} giving better lower bounds on $n(2,d)$. The construction is recursive in nature and uses a kind of ``algebra'' on \emph{lists} of ternary strings, which encode neighborly boxes in a familiar way. Moreover, we conjecture that our construction is optimal and gives an explicit formula for $n(2,d)$. This supposition is supported by some numerical experiments and some partial results on related open problems which are recalled.

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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

Strong arboricity of graphs

An edge coloring of a graph $G$ is \emph{woody} if no cycle is monochromatic. The \emph{arboricity} of a graph $G$, denoted by $\arb (G)$, is the least number of colors needed for a woody coloring of $G$. A coloring of $G$ is \emph{strongly woody} if after contraction of any single edge it is still woody. In other words, not only any cycle in $G$ can be monochromatic but also any \emph{broken cycle}, i.e., a simple path arising by deleting a single edge from the cycle. The least number of colors in a strongly woody coloring of $G$ is denoted by $ζ(G)$ and called the \emph{strong arboricity} of $G$. We prove that $ζ(G)\leqslant χ_a(G)$, where $χ_a(G)$ is the \emph{acyclic chromatic number} of $G$ (the least number of colors in a proper vertex coloring without a $2$-colored cycle). In particular, we get that $ζ(G)\leqslant 5$ for planar graphs and $ζ(G)\leqslant 4$ for otuterplanar graphs. We conjecture that $ζ(G)\leqslant 4$ holds for all planar graphs. We also prove that $ζ(G)\leqslant 4(\arb(G))^2$ holds for arbitrary graph $G$. A natural generalziation of strong arboricity to \emph{matroids} is also discussed, with a special focus on cographic matroids.

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