Jarosław Grytczuk

Nonrepetitive colorings of graphs

A sequence a = a1a2. . . . an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance, the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thues sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and Δ(G). We show, by an application of the Lovasz Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ cΔ(G)2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n - 3, and π(T) ≤ 4(Δ(T - 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.

Thue type problems for graphs, points, and numbers

A sequence S=s1s2...sn is said to be nonrepetitive if no two adjacent blocks of S are the same. A celebrated 1906 theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set {0,1,2}. This result is the starting point of Combinatorics on Words-a wide area with many deep results, sophisticated methods, important applications and intriguing open problems. The main purpose of this survey is to present a range of new directions relating Thue sequences more closely to Graph Theory, Combinatorial Geometry, and Number Theory. For instance, one may consider graph colorings avoiding repetitions on paths, or colorings of points in the plane avoiding repetitions on straight lines. Besides presenting a variety of new challenges we also recall some older problems of this area.

The Map-Coloring Game

1. INTRODUCTION. Suppose that Alice wants to color a planar map using four colors in a proper way, that is, so that any two adjacent regions get different colors. Despite the fact that she knows for certain that it is eventually possible, she may fail in her first attempts. Indeed, there are usually many proper partial colorings not extend-able to proper colorings of the whole map. Thus, if she is unlucky, she may accidentally create such a bad partial coloring. Now suppose that Alice asks Bob to help her in this task. They color the regions of a map alternately, with Alice going first. Bob agrees to cooperate by respecting the rule of a proper coloring. However, for some reason he does not want the job to be completed—his secret aim is to achieve a bad partial coloring. (For instance, he may wish to start the coloring procedure over and over again just to stay longer in Alices company.) Is it possible for Alice to complete the coloring somehow, in spite of Bobs insidious plan? If not, then how many additional colors are needed to guarantee that the map can be successfully colored, no matter how clever Bob is? This map-coloring game was invented about twenty-five years ago by Steven J. Brams with the hope of finding a game-theoretic proof of the Four Color Theorem, avoiding perhaps the use of computers. Though this approach has not been successful, at least we are left with a new, intriguing map-coloring problem: What is the fewest number of colors allowing a guaranteed win for Alice in the map-coloring game in the plane? Bramss game was published by Martin Gardner in his Mathematical Games column in Scientific American in 1981. Surprisingly, it remained unnoticed by the graph-theoretic community until ten years later, when it was reinvented by Hans L. Bodlaen-der [1] in the wider context of general graphs. In this version Alice and Bob play as before by coloring properly the vertices of a graph G. The game chromatic number χ g (G) of G is the smallest number of colors for which Alice has a winning strategy. As every map is representable by a graph whose edges correspond to adjacent regions of the map, Bramss question is equivalent to determining the game chromatic number of planar graphs. Since then the problem has been analyzed in serious combinatorial journals and gained the …

Nonrepetitive colorings of graphs -- a survey.

A vertex coloring f of a graph G is nonrepetitive if there are no integer r≥1 and a simple path v1,…,v2r in G such that f(vi)=f(vr

Nonrepetitive colorings of trees

A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by @p(G). A famous theorem of Thue asserts that @p(P)=3 for any path P with at least four vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that @p(T) is bounded by 4 in this class we aim to describe the 4-chromatic trees. In particular, we study the 4-critical trees which are minimal with respect to this property. Though there are many trees T with @p(T)=4 we show that any of them has a sufficiently large subdivision H such that @p(H)=3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edge-colored by at most @D+1 colors without repetitions on paths.

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 three symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of three-element sets L1,,Ln there exists a nonrepetitive sequence s1,,sn with si∈Li. We propose a new non-constructive way to build long nonrepetitive sequences and provide an elementary proof that sets of size 4 suffice confirming the best known bound. The simple double counting in the heart of the argument is inspired by the recent algorithmic proof of the Lovasz local lemma due to Moser and Tardos. Furthermore we apply this approach and present game-theoretic type results on nonrepetitive sequences. 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 greater than 1. Our techniques allow to reduce 37-6. Another game we consider is the 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.

Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u) S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,...,k} is the lucky number of G, denoted by @h(G). Using algebraic methods we prove that @h(G)=

Nonrepetitive list colourings of paths

A vertex colouring c of a graph G is called nonrepetitive if for every integer r ≥ 1 and every path P = (v1,v2,…,v2r) in G, the first half of P is coloured differently from the second half of P, that is, c(vj)≠c(vr+j) for some j = 1,2,…,r. This notion was inspired by a striking result of Thue asserting that the path Pn on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by πch(G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. Using the Lefthanded Local Lemma we prove that πch(Pn) ≤ 4 for every n.

Pattern avoidance on graphs

In this paper we consider colorings of graphs avoiding certain patterns on paths. Let X be a set of variables and let p=x1x2...xr,x[emailxa0protected]?X, be a pattern, that is, any sequence of variables. A finite sequence s is said to match a pattern p if s may be divided into non-empty blocks s=B1B2...Br, such that xi=xj implies Bi=Bj, for all i,j=1,2,...,r. A coloring of vertices (or edges) of a graph G is said to be p-free if no path in G matches a pattern p. The pattern chromatic [emailxa0protected]p(G) is the minimum number of colors used in a p-free coloring of G. Extending the result of Alon et al. [Non-repetitive colorings of graphs, Random Struct. Alg. 21 (2002) 336-346] we prove that if each variable occurs in a pattern p at least m>=2 times then @pp(G)=

Nonrepetitive Graph Coloring

A coloring of the vertices of a graph G is nonrepetitive if no simple path in G looks like a 1 a 2 ... a n a 1 a 2 ... a n. The minimum number of colors needed for a graph G is denoted by π(G). For instance, by the famous 1906 theorem of Thue, π(G) = 3 if G is a simple path with at least 4 vertices. This implies that π(G) ≤ 4 if Δ(G) ≤ 2. But how large can π(G) be for cubic graphs, κ-trees, or planar graphs? This paper is a small survey of problems and results of the above type.