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 -colouring of a graph is a function that assigns one of colours to each vertex of . A path of even order in is repetitively coloured by if for . A colouring of is nonrepetitive if no path of is repetitively coloured by . Observe that a nonrepetitive colouring is proper, in the sense that adjacent vertices are coloured differently. The nonrepetitive chromatic number is the minimum integer such that admits a nonrepetitive -colouring.
The classical result in this area is by Thue , who proved in 1906 that every path is nonrepetitively 3-colourable. Starting with the seminal work of Alon et al. , nonrepetitive colourings of general graphs have recently been widely studied; see the surveys [27, 28, 11, 26] and other references [23, 32, 12, 33, 27, 28, 17, 42, 16, 2, 7, 5, 20, 10, 30, 9, 13, 40, 36, 21, 41, 6, 25, 3, 31, 38, 4, 37, 8, 22, 29, 49, 51, 35, 14, 1, 43, 44, 47, 34, 26, 50].
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) , trees are nonrepetitively 4-colourable [9, 36], outerplanar graphs are nonrepetitively -colourable [36, 5], and more generally, every graph with treewidth is nonrepetitively -colourable . Graphs with maximum degree are nonrepetitively -colourable [3, 27, 32, 17], and graphs excluding a fixed immersion have bounded nonrepetitive chromatic number .
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. . The best known lower bound is , due to Ochem (see ). The best known upper bound is where is the number of vertices, due to Dujmović, Frati, Joret, and Wood . Note that several works have studied colourings of planar graphs in which only facial paths are required to be nonrepetitively coloured [33, 43, 44, 4, 47, 34, 8].
This paper proves that planar graphs have bounded nonrepetitive chromatic number.
Every planar graph satisfies .
We generalise this result for graphs of bounded Euler genus, for graphs excluding any fixed minor, and for graphs excluding any fixed topological minor.111The Euler genus of the orientable surface with handles is . The Euler genus of the non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum integer such that embeds in a surface of Euler genus . Of course, a graph is planar if and only if it has Euler genus 0; see  for more about graph embeddings in surfaces. A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. A graph is a topological minor of a graph if a subdivision of is a subgraph of . If contains as a topological minor, then contains as a minor. If contains no minor, then is -minor-free. If contains no topological minor, then is -topological-minor-free.
Every graph with Euler genus satisfies .
For every graph , there is an integer such that every -minor-free graph satisfies .
For every graph , there is a constant such that every -topological-minor-free graph satisfies .
The proofs of Theorems 2 and 1 are given in Section 3, and the proofs of Theorems 4 and 3 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.  that show that planar graphs and other graph classes are subgraphs of certain strong products.
Undefined terms and notation can be found in .
2.1 Strongly Nonrepetitive Colourings
A key to all our proofs is to consider a strengthening of nonrepetitive colouring defined below.
For a graph , a lazy walk in is a sequence of vertices such that for each , either is an edge of , or . A lazy walk can be thought of as a walk in the graph obtained from by adding a loop at each vertex. For a colouring of , a lazy walk is -repetitive if for each .
A colouring is strongly nonrepetitive if for every -repetitive lazy walk , there exists such that . Let be the minimum number of colours in a strongly nonrepetitive colouring of . Since a path has no repeated vertices, every strongly nonrepetitive colouring is nonrepetitive, and thus for every graph .
A layering of a graph is a partition of such that for every edge , if and , then . If is a vertex in a connected graph and is the set of vertices at distance exactly from in for all , then the layering is called a BFS layering of .
Consider a layering of a graph . Let be a connected component of , for some . The shadow of is the set of vertices in adjacent to some vertex in . The layering is shadow-complete if every shadow is a clique. This concept was introduced by Kündgen and Pelsmajer  and implicitly by Dujmović et al. .
We will need the following result.
A tree-decomposition of a graph consists of a collection of subsets of , called bags, indexed by the vertices of a tree , and with the following properties:
for every vertex of , the set induces a non-empty (connected) subtree of , and
for every edge of , there is a vertex for which .
The width of such a tree-decomposition is . The treewidth of a graph is the minimum width of a tree-decomposition of . Tree-decompositions were introduced by Robertson and Seymour . Treewidth measures how similar a given graph is to a tree, and is particularly important in structural and algorithmic graph theory.
Barát and Varjú  and Kündgen and Pelsmajer  independently proved that graphs of bounded treewidth have bounded nonrepetitive chromatic number. Specifically, Kündgen and Pelsmajer  proved that every graph with treewidth is nonrepetitively -colourable, which is the best known bound. Theorem 7 below strengthens this result. The proof is almost identical to that of Kündgen and Pelsmajer  and depends on the following lemma. A lazy walk is boring if for each .
Lemma 6 ().
Every path has a -colouring such that every -repetitive lazy walk is boring.
For every graph of treewidth at most , we have .
The proof proceeds by induction on . If , then 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 . By adding edges if necessary, we may assume that is connected and chordal, with clique-number at most .
Let be a BFS-layering of . We refer to as the set of vertices at depth . Note that the subgraph of induced by each layer has treewidth at most , and thus the spanning subgraph of induced by all edges whose endpoints have the same depth also has treewidth at most . By the induction hypothesis, has a strongly nonrepetitive colouring with colours. The graph obtained from by contracting each set (which might not induce a connected graph) into a single vertex is a path, and thus, by Lemma 6, has a -colouring such that every -repetitive walk is boring. For each and each vertex , set . The colouring of uses at most colours.
We now prove that is strongly nonrepetitive. Let be a -repetitive lazy walk . Our goal is to prove that for some . Let be the minimum depth of a vertex in .
Let be the sequence of vertices obtained from by removing all vertices at depth greater than . We claim that is a lazy walk. To see this, consider vertices of such that and have depth but all have depth greater than ; thus, were removed when constructing . Then, the vertices lie in a connected component of the graph induced by the vertices of depth greater than , thus it follows that and are adjacent or equal by Lemma 5. This shows that is a lazy walk in .
The projection of on is a -repetitive lazy walk in , and is thus boring by Lemma 6. It follows that the vertices and of have the same depth for every . In particular, was removed from if and only if was. Hence, there are indices such that . Since is -repetitive, it follows that is also -repetitive and in particular is -repetitive. By the definition of , there is an index such that , which completes the proof. ∎
2.4 Strong Products
The strong product of graphs and , denoted by , is the graph with vertex set , where distinct vertices are adjacent if (1) and , or (2) and , or (3) and . Nonrepetitive colourings of graph products have been studied in [36, 35, 7]. Indeed, Kündgen and Pelsmajer  note that their method shows that the strong product of paths is nonrepetitively -colourable, which is a precursor to the following results.
Let be a graph with an -colouring such that every -repetitive lazy walk is boring. For every graph , we have .
Consider a strongly nonrepetitive colouring of with colours. For any two vertices and , we define the colour of the vertex by . We claim that this is a strongly nonrepetitive colouring of . To see this, consider a -repetitive lazy walk in . By the definition of the strong product and the definition of , the projection of on is a -repetitive lazy walk in and the projection of on is a -repetitive lazy walk in . By the definition of , there is an index such that . By the definition of , we have for every . In particular, and , which completes the proof. ∎
Applying Lemma 6, we obtain the following immediate corollary.
For every graph and every path , we have .
By taking and a proper -colouring of , we also obtain the following direct corollary to Lemma 8.
For every graph and every integer , we have .
3 Planar Graphs and Graphs of Bounded Genus
The following recent result by Dujmović et al.  is a key theorem.
Theorem 11 ().
Every planar graph is a subgraph of for some graph with treewidth at most and some path .
Theorem 12 ().
Every graph of Euler genus is a subgraph of for some graph with treewidth at most and some path .
4 Excluded Minors
Our results for graphs excluding a minor depend on the following graph minor structure theorem of Robertson and Seymour .
Theorem 13 ().
For every graph , there is an integer such that every -minor-free graph has a tree-decomposition in which each torso is -almost-embeddable.
We omit the definition of -almost embeddable from this paper, since we do not need it. All we need to know is the following theorem of Dujmović et al. .
Theorem 14 ().
Every -almost embeddable graph is a subgraph of for some graph with treewidth at most .
Dujmović et al.  proved the following lemma, which generalises a result of Kündgen and Pelsmajer . A tree-decomposition of a graph is -rich if is a clique in on at most vertices, for each edge .
Lemma 15 ().
Let be a graph that has an -rich tree decomposition such that the subgraph induced by each bag is nonrepetitively -colourable. Then .
Rich tree-decompositions are implicit in the graph minor structure theorem, as demonstrated by the following lemma, which is little more than a restatement of the graph minor structure theorem.
Lemma 16 ().
For every graph , there are constants and such that every -minor-free graph is a spanning subgraph of a graph that has an -rich tree-decomposition such that each bag induces an -almost-embeddable subgraph of .
which implies Theorem 3 since and depend only on .
To obtain our result for graphs excluding a fixed topological minor, we use the following structure theorem due to Grohe and Marx .
Theorem 17 ().
For every graph , there is a constant such that every graph excluding as a topological minor has a tree decomposition such that each torso is -almost-embeddable or has at most vertices with degree greater than .
Theorem 18 ().
Every graph with maximum degree is nonrepetitively -colourable.
Theorem 17 leads to the next lemma.
Lemma 19 ().
For every graph , there are integers and such that every -topological-minor-free graph is a spanning subgraph of a graph that has an -rich tree decomposition such that the subgraph induced by each bag is -almost-embeddable or has at most vertices with degree greater than .
Theorem 18 implies that if a graph has at most vertices with degree greater than , then it is nonrepetitively -colourable. Lemmas 15 and 19 and (1) with imply that for every graph , every -topological-minor-free graph satisfies , which implies Theorem 4, since and depend only on .
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.
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