Nonrepetitive Colorings Research Articles

Graph product structure for non-minor-closed classes

Dujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, p-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is k-planar graphs (those with a drawing in the plane in which each edge is involved in at most k crossings). We prove that every k-planar graph is isomorphic to a subgraph of the strong product of a graph of treewidth O(k5) and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that k-planar graphs have non-repetitive chromatic number upper-bounded by a function of k. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a more general setting based on so-called shortcut systems, which are of independent interest. This leads to analogous results for certain types of map graphs, string graphs, graph powers, and nearest neighbour graphs.

On nonrepetitive colorings of cycles

We say that a sequence a1 . a2t of integers is repetitive if ai = ai+t for every i ϵ {1,...,t}. A walk in a graph G is a sequence v1 . vr of vertices of G in which vivi+1 ϵ E(G) for every i ϵ {1,..., r - 1}. Given a k-coloring c: V(G) → {1,..., k} of V(G), we say that c is walk-nonrepetitive if for every t ϵ N, for every walk v1 . v2t in G the sequence c(V1) . c(v2t) is not repetitive unless vi = vi+t for every i ϵ {1,..., t}, and the walk-nonrepetitive chromatic number σ(G) of G is the minimum k for which G has a walk-nonrepetitive k-coloring. Let Cn denote the cycle with n vertices. In this paper we show that σ(Cn) = 4 whenever n ≥ 4 and n ∉ {5,7}, which answers a question posed by Barát and Wood in 2008.

Fractional meanings of nonrepetitiveness

A sequence S is called r-nonrepetitive if no r sequentially adjacent blocks in S are identical. By the classic results of Thue from the beginning of the 20th century, we know that there exist arbitrarily long binary 3-nonrepetitive sequences and ternary 2-nonrepetitive sequences. This discovery stimulated over the years intensive research leading to various generalizations and many exciting problems and results in combinatorics on words.In this paper, we study two fractional versions of nonrepetitive sequences. In the first one, we demand that all subsequences of a sequence S, with gaps bounded by a fixed integer j≥1, are r-nonrepetitive. (This variant emerged from studying nonrepetitive colorings of the Euclidean plane.) Let πj(r) denote the least size of an alphabet guaranteeing existence of arbitrarily long such sequences. We prove that ⌈jr−1⌉+1≤πj(r)≤2⌈jr−1⌉+1, for all r≥3 and j≥1. We also consider a more general situation with the gap bound j being a real number, and apply this to nonrepetitive coloring of the plane.The second variant allows for using a “fractional” alphabet, analogously as for the fractional coloring of graphs. More specifically, we look for sequences of b-element subsets B1,B2,… of an a-element alphabet, with the ratio a/b as small as possible, such that every member of the Cartesian product B1×B2×⋯ is r-nonrepetitive. By using the entropy compression argument, we prove that the corresponding parameter πf(r)=inf⁡ab can be arbitrarily close to 1 for sufficiently large r.

Nonrepetitive List Colorings of the Integers

A coloring of the integers is nonrepetitive if no two adjacent intervals have the same color sequence. A beautiful theorem of Thue asserts that there exists a nonrepetitive coloring of {mathbb {N}} using only three colors. We obtain some generalizations of this result in which the adjacency of intervals is specified by more general graphs. We focus on the list variant of the problem, in which every integer gets a color from its own set of colors. For instance, we prove that there exists a coloring of {mathbb {N}} from arbitrary lists of size 8, such that the following property holds for every nge 1: among any 2^n consecutively adjacent intervals, each of length n, no two have the same color sequence. Another result is related to the possible extension of the famous Dejean’s conjecture to the list setting. It asserts that for every kge 1, there is a coloring of {mathbb {N}} from lists of size k+2sqrt{k}, such that no two among any k consecutively adjacent intervals have the same color sequence.

Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

We propose a new proof technique that applies to the same problems as the Lovász Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach.
 In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive chromatic number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive chromatic number, and a $ \Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})$ upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound.

Nonrepetitive colorings of graphs excluding a fixed immersion or topological minor

AbstractWe prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.

A note on the Thue chromatic number of lexicographic produts of graphs

A sequence is called non-repetitive if none of its subsequences forms a repetition (a sequence r1r2 · · · r2n such that ri = rn+i for all 1 ≤ i ≤ n). Let G be a graph whose vertices are coloured. A colouring ϕ of the graph G is non-repetitive if the sequence of colours on every path in G is non-repetitive. The Thue chromatic number, denoted by π(G), is the minimum number of colours of a non-repetitive colouring of G.

Anagram-free colorings of graphs

A sequence S is called anagram-free if it contains no segment r1r2…rkrk+1…r2k such that rk+1…r2k is a permutation of the block r1r2…rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [Alon, N., J. Grytczuk, M. Hałuszczak and O. Riordan, Non-repetitive colorings of graphs, Random Struct. Algor. 21 (2002), 336–346], we consider a natural generalisation of anagram-free sequences for graph colorings. A coloring of the vertices of a given graph G is called anagram-free if the sequence of colors on any path in G is anagram-free. We call the minimal number of colors needed for such a coloring the anagram-chromatic number of G.We have studied the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we basically give each vertex a separate color.

New Bounds for Facial Nonrepetitive Colouring

We prove that the facial nonrepetitive chromatic number of any outerplanar graph is at most 11 and of any planar graph is at most 22.

Facially-constrained colorings of plane graphs: A survey

In this survey the following types of colorings of plane graphs are discussed, both in their vertex and edge versions: facially proper coloring, rainbow coloring, antirainbow coloring, loose coloring, polychromatic coloring, ℓ-facial coloring, nonrepetitive coloring, odd coloring, unique-maximum coloring, WORM coloring, ranking coloring and packing coloring.In the last section of this paper we show that using the language of words these different types of colorings can be formulated in a more general unified setting.

Nonrepetitive and pattern-free colorings of the plane

A classic result of Thue states that using 3 symbols one can construct an infinite nonrepetitive sequence, that is, a sequence without two identical adjacent blocks. In this paper, we present Thue-type results concerning coloring of the Euclidean plane, related to the famous Hadwiger–Nelson problem. This old problem asks for the chromatic number of the unit distance graph of the plane, the infinite graph with the set of vertices of R2 and edges between points at distance 1.We solve a problem posed by Grytczuk (2008) by showing that any coloring of the unit distance graph of R2 in which the sequence of colors of every simple path is nonrepetitive needs more than countable number of colors. Grytczuk, Kosiński and Zmarz (2016) introduced a relaxation of this problem, by restricting the nonrepetitiveness condition to a certain class of paths. Let a path in the unit distance graph of R2 be called line path if it consists of collinear points. A coloring of R2 is line nonrepetitive if the sequence of colors on every line path is nonrepetitive. We present a line nonrepetitive 18-coloring of R2, which improves the result of Grytczuk, Kosiński and Zmarz with 36 colors. Furthermore, we construct a line nonrepetitive coloring, which additionally avoids palindromes and uses 32 colors.We also consider a natural generalization of this problem to other patterns. We say that a patternq (finite sequence over a set of variables E) occurs in a sequence w (over an alphabet A) if there exists a substitution f from E to the set of nonempty sequences over A such that f(q) is a block in w. A sequence wavoids a pattern q if q does not occur in w. For a pattern q, a coloring of R2 is called linep-free if the sequence of colors on every line path avoids q. We present a 9-coloring of R2 which is line ααα-free and line αβαβ-free at once. Moreover, we give a non-constructive result for a class of patterns. Namely, let q be a pattern with no variable occurring exactly once or satisfying ℓ⩾2d, where ℓ is the length of q and d is the number of variables in q. Then there exists a line q-free coloring of R2 using a finite number of colors. In fact, this result holds even if (instead of taking into account only line paths) we consider all sequences of points in R2 with consecutive distances in some fixed interval [a,b] and without any pair of vertices at distance smaller than a fixed ε>0.

$$(2+\epsilon )$$ ( 2 + ϵ ) -Nonrepetitive List Colouring of Paths

A sequence $$a_1a_2\ldots a_p$$a1a2?ap is an r-repetition (for a real number $$r >1 $$r>1) if $$p=\lceil rq \rceil $$p=?rq? for some positive integer q, and $$a_j=a_{j+q}$$aj=aj+q for $$j=1,2,\ldots , p-q$$j=1,2,?,p-q. In other words, the sequence can be divided into $$\lceil r \rceil $$?r? blocks where all the blocks are the same, say, all the blocks equal to $$a_1a_2\ldots a_q$$a1a2?aq for some $$q \ge 1$$q?1, except that when r is not an integer, the last block is the prefix of $$a_1...a_q$$a1...aq of length $$ \lceil (r - \lfloor r \rfloor )q \rceil $$?(r-?r?)q?. A colouring of the vertices of a graph G is r-nonrepetitive if there is no path in G for which the colour sequence of its vertices forms an r-repetition. The r-nonrepetitive chromatic number $$\pi _r(G)$$?r(G) of G is the minimum number of colours needed in an r-nonrepetitive colouring of G. 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 r-nonrepetitive choice number $$\pi _{rch}(G)$$?rch(G) of G is the least integer k such that for every k-list assignment L, there is an r-nonrepetitive colouring c of G satisfying $$c(v)\in L(v)$$c(v)?L(v) for every vertex v of G. A classical result of Thue asserts that $$\pi _2(P_n)\le 3$$?2(Pn)≤3 for all n. It is known that $$ \pi _{2ch}(P_n) \le 4$$?2ch(Pn)≤4 for all n. However, it remains an open problem whether $$\pi _{2ch}(P_n) \le 3$$?2ch(Pn)≤3 for all n. This paper proves that for any $$\epsilon > 0$$∈>0, $$\pi _{(2+\epsilon )ch}(P_n) \le 3$$?(2+∈)ch(Pn)≤3 for all n.

Nonrepetitive colorings of line arrangements

A sequence S is nonrepetitive if no two adjacent segments of S are identical. A famous result of Thue from 1906 asserts that there are arbitrarily long nonrepetitive sequences over 3 symbols. We study the following geometric variant of this problem. Given a set P of points in the plane and a set L of lines, what is the least number of colors needed to color P so that every line in L is nonrepetitive? If P consists of all intersection points of a prescribed set of lines L, then we prove that there is such coloring using at most 405 colors. The proof is based on a theorem of Thue and on a result of Alon and Marshall concerning homomorphisms of edge colored planar graphs. We also consider nonrepetitive colorings involving other geometric structures. For instance, a nonrepetitive analog of the famous Hadwiger–Nelson problem is formulated as follows: what is the least number of colors needed to color the plane so that every path of the unit distance graph whose vertices are colinear is nonrepetitive? Using a theorem of Thue we prove that this number is at most 36.

Measurable patterns, necklaces and sets indiscernible by measure

In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem', see Alon et al. (Proc. Amer. Math. Soc., 2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and Laso\'{n} (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of $\mathbb{R}^d$ such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Laso\'{n}, we show that for every collection $\mu_1,\ldots,\mu_{2d-1}$ of $2d-1$ continuous, signed locally finite measures on $\mathbb{R}^d$, there exist two nontrivial axis-aligned $d$-dimensional cuboids (rectangular parallelepipeds) $C_1$ and $C_2$ such that $\mu_i(C_1)=\mu_i(C_2)$ for each $i\in\{1,\ldots,2d-1\}$. We also show by examples that the bound $2d-1$ cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.

Nonrepetitive colorings of lexicographic product of graphs

Special issue PRIMA 2013 A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2\textellipsisv2l for which c(vi)=c(vl+i) for all 1<=i<=l. Given graphs G and H with |V(H)|=k, the lexicographic product G[H] is the graph obtained by substituting every vertex of G by a copy of H, and every edge of G by a copy of Kk,k. We prove that for a sufficiently long path P, a nonrepetitive coloring of P[Kk] needs at least 3k+⌊k/2⌋ colors. If k>2 then we need exactly 2k+1 colors to nonrepetitively color P[Ek], where Ek is the empty graph on k vertices. If we further require that every copy of Ek be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P[Ek] is at least 3k+1 and at most 3k+⌈k/2⌉. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.

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 Thueʼs theorem remains unknown.

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 three 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 $\pi_l(G)$, is the minimum $k$ such that for any list assignment $\{{L_v}\}$ of $G$ with each $|{L_v}|\geqslant k$ there is a nonrepetitive coloring of $G$ chosen from $\{{L_v}\}$. Alon et al. proved in 2002 that $\pi_l(G)=O(\Delta^2)$ for every graph $G$ with maximum degree at most $\Delta$. We propose an almost linear bound in $\Delta$ for trees, namely, for any $\varepsilon>0$ there is a...

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let G → be a directed graph. A transitive fraternal augmentation of G → is a directed graph H → with the same vertex set, including all the arcs of G → and such that for any vertices x , y , z , 1. if ( x , y ) ∈ E ( G → ) and ( x , z ) ∈ E ( G → ) then ( y , z ) ∈ E ( H → ) or ( z , y ) ∈ E ( H → ) (fraternity); 2. if ( x , y ) ∈ E ( G → ) and ( y , z ) ∈ E ( G → ) then ( x , z ) ∈ E ( H → ) (transitivity). In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col 2 ( G ) ≤ O ( ∇ 1 ( G ) ∇ 0 ( G ) 2 ) , where ∇ k ( G ) ( k ≥ 0 ) denotes the greatest reduced average density with depth k of a graph G ; we give a constructive proof that ∇ k ( G ) bounds the distance ( k + 1 ) -coloring number col k + 1 ( G ) with a function f ( ∇ k ( G ) ) . On the other hand, ∇ k ( G ) ≤ ( col 2 k + 1 ( G ) ) 2 k + 1 . We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.

The complexity of nonrepetitive coloring

A coloring of a graph is nonrepetitive if the graph contains no path that has a color pattern of the form x x (where x is a sequence of colors). We show that determining whether a particular coloring of a graph is nonrepetitive is coNP-hard, even if the number of colors is limited to four. The problem becomes fixed-parameter tractable, if we only exclude colorings x x up to a fixed length k of x .

Nonrepetitive colorings of graphs of bounded tree-width

A sequence of the form s 1 s 2 … s m s 1 s 2 … s m is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [Thue type problems for graphs, points and numbers, manuscript] by proving that every outerplanar graph has a nonrepetitive 12-coloring. We also show that graphs of tree-width t have nonrepetitive 4 t -colorings.