arXiv subjects

Jarosław Grytczuk

Publications and source records attributed to Jarosław Grytczuk.

50 records · Page 3Linked to original sources

arXiv2019

Hat chromatic number of graphs

We study the hat chromatic number of a graph defined in the following way: there is one player at each vertex of a loopless graph $G$, an adversary places a hat of one of $K$ colors on the head of each player, two players can see each other's hats if and only if they are at adjacent vertices. All players simultaneously try to guess the color of their hat. The players cannot communicate but collectively determine a strategy before the hats are placed. The hat chromatic number, $μ(G)$, is the largest number $K$ of colors such that the players are able to fix a strategy that will ensure that for every possible placement of hats at least one of the guesses correctly. We compute $μ(G)$ for several classes of graphs, for others we establish some bounds. We establish connections between the hat chromatic number, the chromatic number and the coloring number. We also introduce several variants of the game: with multiple guesses, restrictions on allowed strategies or restrictions on colorings. We show examples how the modified games can be used to obtain interesting results for the original game.

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arXiv2018

Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $ζ^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $ζ^* (T)\leq 2$ for every tree $T$ and give an upper bound on $ζ^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $ζ^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $ζ^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $ζ^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn.

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arXiv2018

The Alon-Tarsi number of a planar graph minus a matching

This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable.

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arXiv2017

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $ζ(G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $ζ(G)$. On a positive side, we prove that $ζ(G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $ζ(G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane.

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arXiv2016

Majority choosability of digraphs

A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of lists of colors of size $k$ to the vertices there is a majority coloring of $D$ from these lists. We prove that every digraph is majority $4$-choosable. This gives a positive answer to a question posed recently by Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain this result as a consequence of a more general theorem, in which majority condition is profitably extended. For instance, the theorem implies also that every digraph has a coloring from arbitrary lists of size three, in which at most $2/3$ of the out-neighbors of any vertex share its color. This solves another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable.

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arXiv2015

Fractional and $j$-fold colouring of the plane

We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance $1$ having the same colour. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval $[a,b]$. Exoo conjectured $χ(G_{[a,b]}) = 7$ for sufficiently small but positive difference between $a$ and $b$. We partially answer this conjecture by proving that $χ(G_{[a,b]}) \geq 5$ for $b > a$. A $j$-fold colouring of graph $G = (V,E)$ is an assignment of $j$-elemental sets of colours to the vertices of $G$, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number $χ_f(G)$ is the infimum of fractions $k/j$ for $j$-fold colouring of $G$ using $k$ colours. We generalize a method by Hochberg and O'Donnel (who proved that $G_{[1,1]} \leq 4.36$) for fractional colouring of graphs $G_{[a,b]}$, obtaining a bound dependant on $\frac{a}{b}$. We also present few specific and two general methods for $j$-fold coloring of $G_{[a,b]}$ for small $j$, in particular for $G_{[1,1]}$ and $G_{[1,2]}$. The $j$-fold colouring for small $j$ has strong practical motivation especially in scheduling theory, while graph $G_{[1,2]}$ is often used to model hidden conflicts in radio networks.

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arXiv2014

Splitting necklaces and measurable colorings of the real line

A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$ cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every $k\geq 1$ there is a measurable $(k+3)$-coloring of the real line such that no interval can be fairly split using at most $k$ cuts. In particular, there is a measurable $4$-coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative in 1991 by Keränen. Curiously, in the discrete case the desired coloring also uses four colors.

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arXiv2013

Strong chromatic index of sparse graphs

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $χ_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $χ_{s}^{\prime}(G)\leq (4k-1)Δ(G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $% χ_{s}^{\prime}(G)\leq 4Δ-3$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs.

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arXiv2012

Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erdős on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljević. The question how many axes aligned hyperplanes are needed for a fair splitting of a d-dimensional k-colored cube remains open.

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arXiv2012

Online version of the theorem of Thue

A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3 symbols. We consider the online variant of this result in which a nonrepetitive sequence is constructed during a play between two players: Bob is choosing a position in a sequence and Alice is inserting a symbol on that position taken from a fixed set A. The goal of Bob is to force Alice to create a repetition, and if he succeeds, then the game stops. The goal of Alice is naturally to avoid that and thereby to construct a nonrepetitive sequence of any given length. We prove that Alice has a strategy to play arbitrarily long provided the size of the set A is at least 12. This is the online version of the Theorem of Thue. The proof is based on nonrepetitive colorings of outerplanar graphs. On the other hand, one can prove that even over 4 symbols Alice has no chance to play for too long. The minimum size of the set of symbols needed for the online version of Thue's theorem remains unknown.

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arXiv2012

Additive colorings of planar graphs

An \emph{additive coloring} of a graph $G$ is an assignment of positive integers $\{1,2,...,k\}$ to the vertices of $G$ such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number $k$ for which there exists an additive coloring of $G$ is denoted by $η(G)$. We prove that $η(G)\leqslant 468$ for every planar graph $G$. This improves a previous bound $η(G)\leqslant 5544$ due to Norin. The proof uses Combinatorial Nullstellensatz and coloring number of planar hypergrahs. We also demonstrate that $η(G)\leqslant 36$ for 3-colorable planar graphs, and $η(G)\leqslant 4$ for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each $r\geqslant 2$ there is an $r$-chromatic graph $G_{r}$ with no additive coloring by elements of any Abelian group of order $r$.

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arXiv2011

Nonrepetitive games

(Note. The results of this manuscript has been merged and published with another paper of the same authors: A new approach to nonrepetitve sequences.) A repetition of size $h$ ($h\geqslant1$) in a given sequence is a subsequence of consecutive terms of the form: $xx=x_1... x_hx_1... x_h$. A sequence is nonrepetitive if it does not contain a repetition of any size. The remarkable construction of Thue asserts that 3 different symbols are enough to build an arbitrarily long nonrepetitive sequence. We consider game-theoretic versions of results on nonrepetitive sequences. A nonrepetitive game is played by two players who pick, one by one, consecutive terms of a sequence over a given set of symbols. The first player tries to avoid repetitions, while the second player, in contrast, wants to create them. Of course, by simple imitation, the second player can force lots of repetitions of size 1. However, as proved by Pegden, there is a strategy for the first player to build an arbitrarily long sequence over 37 symbols with no repetitions of size $>1$. Our techniques allow to reduce 37 to 6. Another game we consider is an erase-repetition game. Here, whenever a repetition occurs, the repeated block is immediately erased and the next player to move continues the play. We prove that there is a strategy for the first player to build an arbitrarily long nonrepetitive sequence over 8 symbols. Our approach is inspired by a new algorithmic proof of the Lovász Local Lemma due to Moser and Tardos and previous work of Moser (his so called entropy compression argument).

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arXiv2011

A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that 3 symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of 3-element sets $L_1,..., L_n$ there exists a nonrepetitive sequence $s_1, ..., s_n$ with $s_i\in L_i$. Applying the probabilistic method one can prove that this is true for sufficiently large sets $L_i$. We present an elementary proof that sets of size 4 suffice (confirming the best known bound). The argument is a simple counting with Catalan numbers involved. Our approach is inspired by a new algorithmic proof of the Lovász Local Lemma due to Moser and Tardos and its interpretations by Fortnow and Tao. The presented method has further applications to nonrepetitive games and nonrepetitive colorings of graphs.

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arXiv2011

Nonrepetitive sequences on arithmetic progressions

A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every $k\geqslant 1$ and every $c\geqslant 1$ there exist arbitrarily long sequences over at most $(1+\frac{1}{c})k+18k^{c/c+1}$ symbols whose subsequences indexed by arithmetic progressions with common differences from the set $\{1,2,...,k\}$ are nonrepetitive. This improves a previous bound obtained in \cite{Grytczuk Rainbow}. Our approach is based on a technique introduced recently in \cite{GrytczukKozikMicek}, which was originally inspired by a constructive proof of the Lovász Local Lemma due to Moser and Tardos \cite{MoserTardos}. We also discuss some related problems that can be successfully attacked by this method.

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