arXiv subjects

Jakub Kozik

Publications and source records attributed to Jakub Kozik.

23 records · Page 2Linked to original sources

arXiv2012

Towards on-line Ohba's conjecture

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 χ(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 χ(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 χ(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead's result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le χ(G)+\sqrt{χ(G)}$ then its on-line choice number equals chromatic number.

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arXiv2012

Nonrepetitive choice number of trees

A nonrepetitive coloring of a path is a coloring of its vertices such that the sequence of colors along the path does not contain two identical, consecutive blocks. The remarkable construction of Thue asserts that 3 colors are enough to color nonrepetitively paths of any length. A nonrepetitive coloring of a graph is a coloring of its vertices such that all simple paths are nonrepetitively colored. Assume that each vertex $v$ of a graph $G$ has assigned a set (list) of colors $L_v$. A coloring is chosen from $\{L_v\}_{v\in V(G)}$ if the color of each $v$ belongs to $L_v$. The Thue choice number of $G$, denoted by $π_l(G)$, is the minimum $k$ such that for any list assignment $\set{L_v}$ of $G$ with each $|L_v|\geq k$ there is a nonrepetitive coloring of $G$ chosen from $\{L_v\}$. Alon et al. (2002) proved that $π_l(G)=O(Δ^2)$ for every graph $G$ with maximum degree at most $Δ$. We propose an almost linear bound in $Δ$ for trees, namely for any $\epsi>0$ there is a constant $c$ such that $π_l(T)\leq cΔ^{1+\epsi}$ for every tree $T$ with maximum degree $Δ$. The only lower bound for trees is given by a recent result of Fiorenzi et al. (2011) that for any $Δ$ there is a tree $T$ such that $π_l(T)=Ω(\frac{\logΔ}{\log\logΔ})$. We also show that if one allows repetitions in a coloring but still forbid 3 identical consecutive blocks of colors on any simple path, then a constant size of the lists allows to color any tree.

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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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