Planar graphs have bounded nonrepetitive chromatic number

TL;DR

This paper proves that planar graphs and certain other graph classes can be nonrepetitively coloured with a bounded number of colours, confirming a longstanding conjecture and extending to broader graph families.

Contribution

It establishes that planar graphs and related classes have bounded nonrepetitive chromatic number, solving a conjecture and generalising to graphs of bounded genus and excluded minors.

Findings

Planar graphs have bounded nonrepetitive chromatic number.

The result extends to graphs of bounded Euler genus.

Applicable to graphs excluding fixed minors or topological minors.

Abstract

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Haluszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.

Equations

$$\pi (G)⩽{\pi }^{\ast }(G)⩽{\pi }^{\ast }(H⊠P⊠K_{\_}3)⩽3{\pi }^{\ast }(H⊠P)⩽3\cdot 4{\pi }^{\ast }(H)⩽3\cdot 4\cdot 4^3=768,$$

$$\pi (G)⩽{\pi }^{\ast }(G)⩽{\pi }^{\ast }(H⊠P⊠K_{\_}\max \{2g,3\})⩽\max \{2g,3\}\cdot {\pi }^{\ast }(H⊠P)$$

$$⩽\max \{2g,3\}\cdot 4^4$$

$$=256\max \{2g,3\},$$

$$\pi (G)⩽{\pi }^{\ast }(G)⩽{\pi }^{\ast }(K_{\_}k+(H⊠P⊠K_{\_}\max \{6k,1\}))⩽k+6k{\pi }^{\ast }(H⊠P)⩽k+6k\cdot 4{\pi }^{\ast }(H)⩽k+6k\cdot 4^{11(k+1)}.$$

$$\pi (G)⩽{\pi }^{\ast }(G)⩽(k+6k\cdot 4^{11(k+1)})4^r,$$

Full text

\aicAUTHORdetails

title = Planar Graphs Have Bounded Nonrepetitive Chromatic Number, author = Vida Dujmović, Louis Esperet, Gwenaël Joret, Bartosz Walczak, David R. Wood, plaintextauthor = Vida Dujmovic, Louis Esperet, Gwenael Joret, Bartosz Walczak, David R. Wood, \aicEDITORdetailsyear=2020, number=5, received=11 April 2019, published=6 March 2020, updated=21 January 2022,

[classification=text]

1 Introduction

A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. More precisely, a $k$-colouring of a graph $G$ is a function $\phi$ that assigns one of $k$ colours to each vertex of $G$. A path $(v_{\_}1,v_{\_}2,\dots,v_{\_}{2t})$ of even order in $G$ is repetitively coloured by $\phi$ if $\phi(v_{\_}i)=\phi(v_{\_}{t+i})$ for $i\in\{1,\dots,t\}$. A colouring $\phi$ of $G$ is nonrepetitive if no path of $G$ is repetitively coloured by $\phi$. Observe that a nonrepetitive colouring is proper, in the sense that adjacent vertices are coloured differently. The nonrepetitive chromatic number $\pi(G)$ is the minimum integer $k$ such that $G$ admits a nonrepetitive $k$-colouring.

The classical result in this area is by Thue [52], who proved in 1906 that every path is nonrepetitively 3-colourable. Starting with the seminal work of Alon et al. [3], nonrepetitive colourings of general graphs have recently been widely studied; see the surveys [28, 29, 11, 27, 51] and other references [24, 33, 12, 34, 29, 18, 44, 17, 2, 7, 5, 21, 10, 31, 9, 13, 41, 37, 22, 42, 6, 26, 3, 32, 39, 4, 38, 8, 52, 23, 30, 53, 55, 36, 14, 1, 45, 46, 49, 35, 54, 50, 43, 28, 11, 27, 51].

Several graph classes are known to have bounded nonrepetitive chromatic number. In particular, cycles are nonrepetitively 3-colourable (except for a finite number of exceptions) [12], trees are nonrepetitively 4-colourable [9, 37], outerplanar graphs are nonrepetitively $12$-colourable [37, 5], and more generally, every graph with treewidth $k$ is nonrepetitively $4^{k}$-colourable [37]. Graphs with maximum degree $\Delta$ are nonrepetitively $O(\Delta^{2})$-colourable [3, 28, 33, 18], and graphs excluding a fixed immersion have bounded nonrepetitive chromatic number [54].

It is widely recognised that the most important open problem in the field of nonrepetitive graph colouring is whether planar graphs have bounded nonrepetitive chromatic number. It was first asked by Alon et al. [3]. The best known lower bound is $11$, due to Ochem (see [17]). The best known upper bound is $O(\log n)$ where $n$ is the number of vertices, due to Dujmović, Frati, Joret, and Wood [17]. Note that several works have studied colourings of planar graphs in which only facial paths are required to be nonrepetitively coloured [34, 45, 46, 4, 49, 35, 8].

This paper proves that planar graphs have bounded nonrepetitive chromatic number.

Theorem 1**.**

Every planar graph $G$ satisfies $\pi(G)\leqslant 768$.

We generalise this result for graphs of bounded Euler genus, for graphs excluding any fixed minor, and for graphs excluding any fixed topological minor.222The Euler genus of the orientable surface with $h$ handles is $2h$. The Euler genus of the non-orientable surface with $c$ cross-caps is $c$. The Euler genus of a graph $G$ is the minimum integer $k$ such that $G$ embeds in a surface of Euler genus $k$. Of course, a graph is planar if and only if it has Euler genus 0; see [40] for more about graph embeddings in surfaces. A graph $X$ is a minor of a graph $G$ if a graph isomorphic to $X$ can be obtained from a subgraph of $G$ by contracting edges. A graph $X$ is a topological minor of a graph $G$ if a subdivision of $X$ is a subgraph of $G$. If $G$ contains $X$ as a topological minor, then $G$ contains $X$ as a minor. If $G$ contains no $X$ minor, then $G$ is $X$-minor-free. If $G$ contains no $X$ topological minor, then $G$ is $X$-topological-minor-free. The result for graphs excluding a fixed minor confirms a conjecture of Grytczuk [28, 29].

Theorem 2**.**

Every graph $G$ with Euler genus $g$ satisfies $\pi(G)\leqslant 256\max\{2g,3\}$.

Theorem 3**.**

For every graph $X$, there is an integer $k$ such that every $X$-minor-free graph $G$ satisfies $\pi(G)\leqslant k$.

Theorem 4**.**

For every graph $X$, there is an integer $k$ such that every $X$-topological-minor-free graph $G$ satisfies $\pi(G)\leqslant k$.

The proofs of Theorems 1 and 2 are given in Section 3, and the proofs of Theorems 3 and 4 are given in Section 4. Before that in Section 2 we introduce the tools used in our proofs, namely so-called strongly nonrepetitive colourings, tree-decompositions and treewidth, and strong products. With these tools in hand, the above theorems quickly follow from recent results of Dujmović et al. [19] that show that planar graphs and other graph classes are subgraphs of certain strong products.

2 Tools

Undefined terms and notation can be found in [15].

2.1 Strongly Nonrepetitive Colourings

A key to all our proofs is to consider a strengthening of nonrepetitive colouring defined below.

For a graph $G$, a lazy walk in $G$ is a sequence of vertices $v_{\_}1,\ldots,v_{\_}k$ such that for each $i\in\{1,\dots,k\}$, either $v_{\_}iv_{\_}{i+1}$ is an edge of $G$, or $v_{\_}i=v_{\_}{i+1}$. A lazy walk can be thought of as a walk in the graph obtained from $G$ by adding a loop at each vertex. For a colouring $\phi$ of $G$, a lazy walk $v_{\_}1,\ldots,v_{\_}{2k}$ is $\phi$-repetitive if $\phi(v_{\_}i)=\phi(v_{\_}{i+k})$ for each $i\in\{1,\dots,k\}$.

A colouring $\phi$ is strongly nonrepetitive if for every $\phi$-repetitive lazy walk $v_{\_}1,\ldots,v_{\_}{2k}$, there exists $i\in\{1,\dots,k\}$ such that $v_{\_}i=v_{\_}{i+k}$. Let $\pi^{*}(G)$ be the minimum number of colours in a strongly nonrepetitive colouring of $G$. Since a path has no repeated vertices, every strongly nonrepetitive colouring is nonrepetitive, and thus $\pi(G)\leqslant\pi^{*}(G)$ for every graph $G$.

2.2 Layerings

A layering of a graph $G$ is a partition $(V_{\_}0,V_{\_}1,\dots)$ of $V(G)$ such that for every edge $vw\in E(G)$, if $v\in V_{\_}i$ and $w\in V_{\_}j$, then $|i-j|\leqslant 1$. If $r$ is a vertex in a connected graph $G$ and $V_{\_}i$ is the set of vertices at distance exactly $i$ from $r$ in $G$ for all $i\geqslant 0$, then the layering $(V_{\_}0,V_{\_}1,\dots)$ is called a BFS layering of $G$.

Consider a layering $(V_{\_}0,V_{\_}1,\dots)$ of a graph $G$. Let $H$ be a connected component of $G[V_{\_}i\cup V_{\_}{i+1}\cup\cdots]$, for some $i\geqslant 1$. The shadow of $H$ is the set of vertices in $V_{\_}{i-1}$ adjacent to some vertex in $H$. The layering is shadow-complete if every shadow is a clique. This concept was introduced by Kündgen and Pelsmajer [37] and implicitly by Dujmović et al. [20].

We will need the following result.

Lemma 5** ([37, 20]).**

Every BFS-layering of a connected chordal graph is shadow-complete.

2.3 Treewidth

A tree-decomposition of a graph $G$ consists of a collection $\{B_{\_}x\subseteq V(G):x\in V(T)\}$ of subsets of $V(G)$, called bags, indexed by the vertices of a tree $T$, and with the following properties:

• •

for every vertex $v$ of $G$, the set $\{x\in V(T):v\in B_{\_}x\}$ induces a non-empty (connected) subtree of $T$, and

• •

for every edge $vw$ of $G$, there is a vertex $x\in V(T)$ for which $v,w\in B_{\_}x$.

The width of such a tree-decomposition is $\max\{|B_{\_}x|:x\in V(T)\}-1$. The treewidth of a graph $G$ is the minimum width of a tree-decomposition of $G$. Tree-decompositions were introduced by Robertson and Seymour [47]. Treewidth measures how similar a given graph is to a tree, and is particularly important in structural and algorithmic graph theory.

Barát and Varjú [5] and Kündgen and Pelsmajer [37] independently proved that graphs of bounded treewidth have bounded nonrepetitive chromatic number. Specifically, Kündgen and Pelsmajer [37] proved that every graph with treewidth $k$ is nonrepetitively $4^{k}$-colourable, which is the best known bound. Theorem 7 below strengthens this result. The proof is almost identical to that of Kündgen and Pelsmajer [37] and depends on the following lemma. A lazy walk $v_{\_}1,\ldots,v_{\_}{2k}$ is boring if $v_{\_}i=v_{\_}{i+k}$ for each $i\in\{1,\dots,k\}$.

Lemma 6** ([37]).**

Every path $P$ has a $4$-colouring $\phi$ such that every $\phi$-repetitive lazy walk is boring.

Theorem 7**.**

For every graph $G$ of treewidth at most $k\geqslant 0$, we have $\pi^{*}(G)\leqslant 4^{k}$.

Proof.

The proof proceeds by induction on $k$. If $k=0$, then $G$ has no edges, so assigning the same colour to all the vertices gives a strongly nonrepetitive colouring. For the rest of the proof, assume that $k\geqslant 1$. Consider a tree-decomposition of $G$ of width at most $k$. By adding edges if necessary, we may assume that every bag of the tree-decomposition is a clique. Thus, $G$ is connected and chordal, with clique-number at most $k+1$.

Let $(V_{\_}0,V_{\_}1,\ldots)$ be a BFS-layering of $G$. We refer to $V_{\_}i$ as the set of vertices at depth $i$. Note that the subgraph $G[V_{\_}i]$ of $G$ induced by each layer $V_{\_}i$ has treewidth at most $k-1$.333This is clear for $i=0$ (since $k\geqslant 1$), and for $i\geqslant 1$ this follows from the fact that the graph $G[V_{\_}i]$ plus a universal vertex is a minor of $G$ (contract $V_{\_}0\cup\cdots\cup V_{\_}{i-1}$ into a single vertex and remove $V_{\_}{i+1},V_{\_}{i+2},\dots$), and thus has treewidth at most $k$. Since removing a universal vertex decreases the treewidth by exactly one, it follows that $G[V_{\_}i]$ has treewidth at most $k-1$. Thus the spanning subgraph $H$ of $G$ induced by all edges whose endpoints have the same depth also has treewidth at most $k-1$. By the induction hypothesis, $H$ has a strongly nonrepetitive colouring $\phi_{\_}1$ with $4^{k-1}$ colours. The graph $P$ obtained from $G$ by contracting each set $V_{\_}i$ (which might not induce a connected graph) into a single vertex $x_{\_}i$ is a path, and thus, by Lemma 6, $P$ has a $4$-colouring $\phi_{\_}2$ such that every $\phi_{\_}2$-repetitive walk is boring. For each $i\geqslant 0$ and each vertex $u\in V_{\_}i$, set $\phi(u):=(\phi_{\_}1(u),\phi_{\_}2(x_{\_}i))$. The colouring $\phi$ of $G$ uses at most $4\cdot 4^{k-1}=4^{k}$ colours.

We now prove that $\phi$ is strongly nonrepetitive. Let $W$ be a $\phi$-repetitive lazy walk $v_{\_}1,\ldots,v_{\_}{2k}$. Our goal is to prove that $v_{\_}j=v_{\_}{j+k}$ for some $j\in\{1,\ldots,k\}$. Let $d$ be the minimum depth of a vertex in $W$.

Let $W^{\prime}$ be the sequence of vertices obtained from $W$ by removing all vertices at depth greater than $d$. We claim that $W^{\prime}$ is a lazy walk. To see this, consider vertices $v_{\_}i,v_{\_}{i+1},\ldots,v_{\_}{i+t}$ of $W$ such that $v_{\_}i$ and $v_{\_}{i+t}$ have depth $d$ but $v_{\_}{i+1},\ldots,v_{\_}{i+t-1}$ all have depth greater than $d$; thus, $v_{\_}{i+1},\ldots,v_{\_}{i+t-1}$ were removed when constructing $W^{\prime}$. Then, the vertices $v_{\_}{i+1},\ldots,v_{\_}{i+t-1}$ lie in a connected component of the graph induced by the vertices of depth greater than $d$, thus it follows that $v_{\_}i$ and $v_{\_}{i+t}$ are adjacent or equal by Lemma 5. This shows that $W^{\prime}$ is a lazy walk in $G[V_{\_}d]$.

The projection of $W$ on $P$ is a $\phi_{\_}2$-repetitive lazy walk in $P$, and is thus boring by Lemma 6. It follows that the vertices $v_{\_}j$ and $v_{\_}{j+k}$ of $W$ have the same depth for every $j\in\{1,\dots,k\}$. In particular, $v_{\_}j$ was removed from $W^{\prime}$ if and only if $v_{\_}{j+k}$ was. Hence, there are indices $1\leqslant i_{\_}1<i_{\_}2<\cdots<i_{\_}{\ell}\leqslant k$ such that $W^{\prime}=v_{\_}{i_{\_}1},v_{\_}{i_{\_}2},\dots,v_{\_}{i_{\_}{\ell}},v_{\_}{i_{\_}1+k},v_{\_}{i_{\_}2+k},\dots,v_{\_}{i_{\_}{\ell}+k}$. Since $W$ is $(\phi_{\_}1,\phi_{\_}2)$-repetitive, it follows that $W^{\prime}$ is also $(\phi_{\_}1,\phi_{\_}2)$-repetitive and in particular $W^{\prime}$ is $\phi_{\_}1$-repetitive. By the definition of $\phi_{\_}1$, there is an index $i_{\_}r$ such that $v_{\_}{i_{\_}r}=v_{\_}{i_{\_}r+k}$, which completes the proof. ∎

2.4 Strong Products

The strong product of graphs $A$ and $B$, denoted by $A\boxtimes B$, is the graph with vertex set $V(A)\times V(B)$, where distinct vertices $(v,x),(w,y)\in V(A)\times V(B)$ are adjacent if (1) $v=w$ and $xy\in E(B)$, or (2) $x=y$ and $vw\in E(A)$, or (3) $vw\in E(A)$ and $xy\in E(B)$. Nonrepetitive colourings of graph products have been studied in [37, 36, 7, 43]. Indeed, Kündgen and Pelsmajer [37] note that their method shows that the strong product of $k$ paths is nonrepetitively $4^{k}$-colourable, which is a precursor to the following results.

Lemma 8**.**

Let $H$ be a graph with an $\ell$-colouring $\phi_{\_}2$ such that every $\phi_{\_}2$-repetitive lazy walk is boring. For every graph $G$, we have $\pi^{*}(G\boxtimes H)\leqslant\ell\,\pi^{*}(G)$.

Proof.

Consider a strongly nonrepetitive colouring $\phi_{\_}1$ of $G$ with $\pi^{*}(G)$ colours. For any two vertices $u\in V(G)$ and $v\in V(H)$, we define the colour $\phi(u,v)$ of the vertex $(u,v)\in V(G\boxtimes H)$ by $\phi(u,v):=(\phi_{\_}1(u),\phi_{\_}2(v))$. We claim that this is a strongly nonrepetitive colouring of $G\boxtimes H$. To see this, consider a $\phi$-repetitive lazy walk $W=(u_{\_}1,v_{\_}1),\ldots,(u_{\_}{2k},v_{\_}{2k})$ in $G\boxtimes H$. By the definition of the strong product and the definition of $\phi$, the projection $W_{\_}G=u_{\_}1,u_{\_}2,\ldots,u_{\_}{2k}$ of $W$ on $G$ is a $\phi_{\_}1$-repetitive lazy walk in $G$ and the projection $W_{\_}H=v_{\_}1,v_{\_}2,\ldots,v_{\_}{2k}$ of $W$ on $H$ is a $\phi_{\_}2$-repetitive lazy walk in $H$. By the definition of $\phi_{\_}1$, there is an index $i$ such that $u_{\_}i=u_{\_}{i+k}$. By the definition of $\phi_{\_}2$, we have $v_{\_}j=v_{\_}{j+k}$ for every $j\in\{1,\dots,k\}$. In particular, $v_{\_}i=v_{\_}{i+k}$ and $(u_{\_}i,v_{\_}i)=(u_{\_}{i+k},v_{\_}{i+k})$, which completes the proof. ∎

Applying Lemma 6, we obtain the following immediate corollary.

Corollary 9**.**

For every graph $G$ and every path $P$, we have $\pi^{*}(G\boxtimes P)\leqslant 4\pi^{*}(G)$.

By taking $H=K_{\_}\ell$ and a proper $\ell$-colouring $\phi_{\_}2$ of $K_{\_}\ell$, we obtain the following corollary of Lemma 8.

Corollary 10**.**

For every graph $G$ and every integer $\ell\geqslant 1$, we have $\pi^{*}(G\boxtimes K_{\_}{\ell})\leqslant\ell\,\pi^{*}(G)$.

3 Planar Graphs and Graphs of Bounded Genus

The following recent result by Dujmović et al. [19] is a key theorem.

Theorem 11** ([19]).**

Every planar graph is a subgraph of $H\boxtimes P\boxtimes K_{\_}{3}$ for some graph $H$ with treewidth at most $3$ and some path $P$.

Corollaries 9, 7 and 11 imply that for every planar graph $G$,

[TABLE]

which proves Theorem 1.

For graphs of bounded Euler genus, Distel et al. [16] proved the following strengthening of Theorem 11.

Theorem 12** ([16]).**

Every graph $G$ of Euler genus $g$ is a subgraph of $H\boxtimes P\boxtimes K_{\_}{\max\{2g,3\}}$ for some graph $H$ with treewidth at most $3$ and some path $P$.

Corollaries 9, 7 and 12 imply that for every graph $G$ with Euler genus $g$,

[TABLE]

which proves Theorem 2.

4 Excluded Minors

Our results for graphs excluding a minor depend on the following version of the graph minor structure theorem of Robertson and Seymour [48]. A tree-decomposition $(B_{\_}x:x\in V(T))$ of a graph $G$ is $r$-rich if $B_{\_}x\cap B_{\_}y$ is a clique in $G$ on at most $r$ vertices, for each edge $xy\in E(T)$.

Theorem 13** ([21]).**

For every graph $X$, there are integers $r\geqslant 1$ and $k\geqslant 1$ such that every $X$-minor-free graph $G_{\_}0$ is a spanning subgraph of a graph $G$ that has an $r$-rich tree-decomposition such that each bag induces a $k$-almost-embeddable subgraph of $G$.

We omit the definition of $k$-almost embeddable from this paper, since we do not need it. All we need to know is the following theorem of Dujmović et al. [19], where $A+B$ is the complete join of graphs $A$ and $B$.

Theorem 14** ([19]).**

Every $k$-almost embeddable graph is a subgraph of $K_{\_}k+(H\boxtimes P\boxtimes K_{\_}{\max\{6k,1\}})$ for some graph $H$ with treewidth at most $11k+10$.

Observe that $\pi^{*}(G+K_{\_}k)=\pi^{*}(G)+k$ for every graph $G$ and integer $k\geqslant 0$. Thus Theorems 14, 9 and 7 imply that for every $k$-almost embeddable graph $G$,

[TABLE]

Dujmović et al. [21] proved the following lemma, which generalises a result of Kündgen and Pelsmajer [37].

Lemma 15** ([21]).**

Let $G$ be a graph that has an $r$-rich tree-decomposition such that the subgraph induced by each bag is nonrepetitively $c$-colourable. Then $\pi(G)\leqslant c\,4^{r}$.

Theorems 13 and 15 and (1) with $c=k+6k\cdot 4^{11(k+1)}$ imply that for every graph $X$ and every $X$-minor-free graph $G$,

[TABLE]

which implies Theorem 3 since $k$ and $r$ depend only on $X$.

To obtain our result for graphs excluding a fixed topological minor, we use the following version of the structure theorem of Grohe and Marx [25].

Theorem 16** ([21]).**

For every graph $X$, there are integers $r\geqslant 1$ and $k\geqslant 1$ such that every $X$-topological-minor-free graph $G_{\_}0$ is a spanning subgraph of a graph $G$ that has an $r$-rich tree-decomposition such that the subgraph induced by each bag is $k$-almost-embeddable or has at most $k$ vertices with degree greater than $k$.

Alon et al. [3] proved that graphs with maximum degree $\Delta$ are nonrepetitively $O(\Delta^{2})$-colourable. The best known bound is due to Dujmović et al. [18].

Theorem 17** ([18]).**

Every graph with maximum degree $\Delta\geqslant 2$ is nonrepetitively $(\Delta^{2}+O(\Delta^{5/3}))$-colourable.

Theorem 17 implies that if a graph has at most $k$ vertices with degree greater than $k$, then it is nonrepetitively $c^{\prime}$-colourable for some constant $c^{\prime}=k^{2}+O(k^{5/3})+k$. Theorems 16 and 15 and (1) with $c=\max\{k+6k\cdot 4^{11(k+1)},c^{\prime}\}$ imply that for every graph $X$, every $X$-topological-minor-free graph $G$ satisfies $\pi(G)\leqslant\pi^{*}(G)\leqslant c\cdot 4^{r}$, which implies Theorem 4, since $c$ and $r$ depend only on $X$.

Acknowledgements

This research was completed at the Workshop on Graph Theory held at Bellairs Research Institute in April 2019. Thanks to the other workshop participants for creating a productive working atmosphere. Thanks to André Kündgen and Jarosław Grytczuk for helpful comments on an early draft.