Nonrepetitive colourings of graphs excluding a fixed immersion or topological minor

TL;DR

This paper proves that certain classes of graphs, specifically those excluding a fixed immersion or a particular planar graph as a topological minor, have bounded nonrepetitive chromatic number, expanding understanding of graph coloring constraints.

Contribution

It establishes the boundedness of nonrepetitive chromatic number for graphs excluding a fixed immersion or a specific planar graph as a topological minor, a new class with this property.

Findings

Graphs excluding a fixed immersion have bounded nonrepetitive chromatic number.

Graphs excluding a certain planar graph as a topological minor also have bounded nonrepetitive chromatic number.

This class of graphs is the largest known to have this property.

Abstract

We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ 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 $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.

Figures

Full text

**Nonrepetitive Colourings of Graphs Excluding

a Fixed Immersion or Topological Minor**

††14th March 2024

Paul Wollan 222Department of Computer Science, University of Rome, “La Sapienza”, Rome, Italy (wollan@di.uniroma1.it). Supported by the European Research Council under the European Union s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement no. 279558.  David R. Wood 333School of Mathematical Sciences, Monash University, Melbourne, Australia (david.wood@monash.edu).

Research supported by the Australian Research Council.

Abstract. We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ 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 $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.

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 $\psi$ 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 $\psi$ if $\psi(v_{i})=\psi(v_{t+i})$ for $i\in\{1,\dots,t\}$. A colouring $\psi$ of $G$ is nonrepetitive if no path of $G$ of even order is repetitively coloured by $\psi$. 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. We only consider simple graphs with no loops or parallel edges.

The seminal result in this area is by Thue [41], who in 1906 proved that every path is nonrepetitively 3-colourable. Thue expressed his result in terms of strings over an alphabet of three characters—Alon et al. [3] introduced the generalisation to graphs in 2002. Nonrepetitive graph colourings have since been widely studied [26, 27, 10, 30, 11, 31, 39, 2, 4, 7, 5, 9, 28, 8, 12, 37, 33, 21, 38, 6, 25, 3, 29, 32, 35]. The principle result of Alon et al. [3] was that graphs with maximum degree $\Delta$ are nonrepetitively $\mathcal{O}(\Delta^{2})$-colourable. Several subsequent papers improved the constant [26, 30, 16]. The best known bound is due to Dujmović et al. [16].

Theorem 1** ([16]).**

Every graph with maximum degree $\Delta$ is nonrepetitively $(1+o(1))\Delta^{2}$-colourable.

A number of other graph classes are known to have bounded nonrepetitive chromatic number. In particular, trees are nonrepetitively 4-colourable [8, 33], outerplanar graphs are nonrepetitively $12$-colourable [33, 5], and graphs with bounded treewidth have bounded nonrepetitive chromatic number [33, 5]. (See Section 2 for the definition of treewidth.) The best known bound is due to Kündgen and Pelsmajer [33].

Theorem 2** ([33]).**

Every graph with treewidth $k$ is nonrepetitively $4^{k}$-colourable.

The primary contribution of this paper is to provide a qualitative generalisations of Theorems 1 and 2 via the notion of graph immersions and excluded topological minors.

A graph $G$ contains a graph $H$ as an immersion if the vertices of $H$ can be mapped to distinct vertices of $G$, and the edges of $H$ can be mapped to pairwise edge-disjoint paths in $G$, such that each edge $vw$ of $H$ is mapped to a path in $G$ whose endpoints are the images of $v$ and $w$. The image in $G$ of each vertex in $H$ is called a branch vertex. Structural and colouring properties of graphs excluding a fixed immersion have been widely studied [24, 22, 42, 20, 18, 13, 34, 1, 36, 19, 14, 23, 40]. We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number.

Theorem 3**.**

For every graph $H$ with $t$ vertices, every graph that does not contain $H$ as an immersion is nonrepetitively $4^{t^{4}+O(t^{2})}$-colourable.

Since a graph with maximum degree $\Delta$ contains no star with $\Delta+1$ leaves as an immersion, Theorem 3 implies that graphs with bounded degree have bounded nonrepetitive chromatic number (as in Theorem 1).

We strengthen Theorem 3 as follows (although without explicit bounds). A graph $G$ contains a graph $H$ as a strong immersion if $G$ contains $H$ as an immersion, such that for each edge $vw$ of $H$, no internal vertex of the path in $G$ corresponding to $vw$ is a branch vertex.

Theorem 4**.**

For every fixed graph $H$, there exists a constant $k$, such that every graph $G$ that does not contain $H$ as a strong immersion is nonrepetitively $k$-colourable.

Note that planar graphs with $n$ vertices are nonrepetitively $\mathcal{O}(\log n)$-colourable [15], and the same is true for graphs excluding a fixed graph as a minor or topological minor [17]. It is unknown whether any of these classes have bounded nonrepetitive chromatic number. Our final result shows that excluding a special type of topological minor gives bounded nonrepetitive chromatic number.

Theorem 5**.**

Let $H$ be a fixed planar graph that has a planar embedding with all the vertices of $H$ with degree at least 4 on a single face. Then there exists a constant $k$, such that every graph $G$ that does not contain $H$ as a topological minor is nonrepetitively $k$-colourable.

Graphs with bounded treewidth exclude fixed walls as topological minors. Since walls are planar graphs with maximum degree 3, Theorem 5 implies that graphs of bounded treewidth have bounded nonrepetitive chromatic number (as in Theorem 2). Similarly, for every graph $H$ with $t$ vertices, the ‘fat star’ graph (which is the 1-subdivision of the $t$-leaf star with edge multiplicity $t$) contains $H$ as a strong immersion. Since fat stars embed in the plane with all vertices of degree at least 4 on a single face, Theorem 5 implies that graphs excluding a fixed graph as a strong immersion have bounded nonrepetitive chromatic number (as in Theorem 4). In this sense, Theorem 5 generalises all of Theorems 1, 2, 3 and 4.

The results of this paper, in relation to the best known bounds on the nonrepetitive chromatic number, are summarised in Figure 1.

2 Tree Decompositions

For a graph $G$ and tree $T$, a tree decomposition or $T$-decomposition of $G$ consists of a collection $(T_{x}\subseteq V(G):x\in V(T))$ of sets of vertices of $G$, called bags, indexed by the nodes of $T$, such that for each vertex $v\in V(G)$ the set $\{x\in V(T):v\in T_{x}\}$ induces a connected subtree of $T$, and for each edge $vw$ of $G$ there is a node $x\in V(T)$ such that $v,w\in T_{x}$. The width of a $T$-decomposition is the maximum, taken over the nodes $x\in V(T)$, of $|T_{x}|-1$. The treewidth of a graph $G$ is the minimum width of a tree decomposition of $G$. The adhesion of a tree decomposition $(T_{x}:x\in V(T))$ is $\max\{|T_{x}\cap T_{y}|:xy\in E(T)\}$. The torso of each node $x\in V(T)$ is the graph obtained from $G[T_{x}]$ by adding a clique on $T_{x}\cap T_{y}$ for each edge $xy\in E(T)$ incident to $x$. Dujmović et al. [17] generalised Theorem 2 as follows:

Lemma 6** ([17]).**

If a graph $G$ has a tree decomposition with adhesion $k$ such that every torso is nonrepetitively $c$-colourable, then $G$ is nonrepetitively $c\,4^{k}$-colourable.

For integers $c,d\geqslant 0$ a graph $G$ has $(c,d)$-bounded degree if $G$ contains at most $c$ vertices with degree greater than $d$.

Lemma 7**.**

Every graph with $(c,d)$-bounded degree is nonrepetitively $c+(1+o(1))d^{2}$-colourable.

Proof.

Assign a distinct colour to each vertex of degree at least $d$, and colour the remaining graph by Theorem 1. For each vertex $v$ of degree at least $d$, no other vertex is assigned the same colour as $v$. Thus $v$ is in no repetitively coloured path. The result then follows from Theorem 1. ∎

Dvořák [18] proved the following structure theorem for graphs excluding a strong immersion.

Theorem 8** ([18]).**

For every fixed graph $H$, there exists a constant $k$, such that every graph $G$ that does not contain $H$ as a strong immersion has a tree decomposition such that each torso is $(k,k)$-bounded degree.

Lemmas 7, 6 and 8 imply Theorem 4.

3 Weak Immersions

The proof of Theorem 4 gives no explicit bound on the constant $k$. In this section we prove an explicit bound on the nonrepetitive chromatic number of graphs excluding a weak immersion. Theorem 3 follows from Lemma 6 and the following structure theorem of independent interest.

Theorem 9**.**

For every graph $H$ with $t$ vertices, every graph that does not contain $H$ as a weak immersion has a tree decomposition with adhesion at most $t^{2}$ such that every torso has $(t,t^{4}+2t^{2})$-bounded degree.

The starting point for the proof of Theorem 9 is the following structure theorem of Wollan [42]. For a tree $T$ and graph $G$, a $T$-partition of $G$ is a partition $(T_{x}\subseteq V(G):x\in V(T))$ of $V(G)$ indexed by the nodes of $T$. Each set $T_{x}$ is called a bag. Note that a bag may be empty. For each edge $xy$ of a tree $T$, let $T(xy)$ and $T(yx)$ be the components of $T-xy$ where $x$ is in $T(xy)$ and $y$ is in $T(yx)$. For each edge $xy\in E(T)$, let $G(T,xy):=\bigcup\{T_{z}:z\in V(T(xy))\}$ and $G(T,yx):=\bigcup\{T_{z}:z\in V(T(yx))\}$. Let $E(T,xy)$ be the set of edges in $G$ between $G(T,xy)$ and $G(T,yx)$. The adhesion of a $T$-partition $(T_{x}:x\in V(T))$ is the maximum, taken over all edges $xy$ of $T$, of $|E(T,xy)|$. For each node $x$ of $T$, the torso of $x$ (with respect to a $T$-partition) is the graph obtained from $G$ by identifying $G(T,yx)$ into a single vertex for each edge $xy$ incident to $x$ (deleting resulting parallel edges and loops).

Theorem 10** ([42]).**

For every graph $H$ with $t$ vertices, for every graph $G$ that does not contain $H$ as a weak immersion, there is a $T$-partition of $G$ with adhesion at most $t^{2}$ such that each torso has $(t,t^{2})$-bounded degree.

Proof of Theorem 9.

Let $G$ be a graph that does not contain $H$ as a weak immersion. Consider the $T$-partition $(T_{x}:x\in V(T))$ of $G$ from Theorem 10.

Let $T^{\prime}$ be obtained from $T$ by orienting each edge towards some root vertex. We now define a tree decomposition $(T^{*}_{x}:x\in V(T))$ of $G$. Initialise $T^{*}_{x}:=T_{x}$ for each node $x\in V(T)$. For each edge $vw$ of $G$, if $v\in T_{x}$ and $w\in T_{y}$ and $z$ is the least common ancestor of $x$ and $y$ in $T^{\prime}$, then add $v$ to $T^{*}_{\alpha}$ for each node $\alpha$ on the $\overrightarrow{xz}$ path in $T^{\prime}$, and add $w$ to $T^{*}_{\alpha}$ for each node $\alpha$ on the $\overrightarrow{yz}$ path in $T^{\prime}$. Thus each vertex $v\in T_{x}$ is in a sequence of bags that correspond to a directed path from $x$ to some ancestor of $x$ in $T^{\prime}$. By construction, the endpoints of each edge are in a common bag. Thus $(T^{*}_{x}:x\in V(T))$ is a tree decomposition of $G$.

Consider a vertex $v\in T^{*}_{x}\cap T^{*}_{y}$ for some edge $\overrightarrow{xy}$ of $T^{\prime}$. Then $v$ has a neighbour $w$ in $G(T,yx)$, and $vw\in E(T,xy)$. Thus $|T^{*}_{x}\cap T^{*}_{y}|\leqslant|E(T,xy)|\leqslant t^{2}$. That is, the tree decomposition $(T^{*}_{x}:x\in V(T))$ has adhesion at most $t^{2}$.

Let $G^{+}_{x}$ be the torso of each node $x\in V(T)$ with respect to the tree decomposition $(T^{*}_{x}:x\in V(T))$. That is, $G^{+}$ is obtained from $G[T^{*}_{x}]$ by adding a clique on $T^{*}_{x}\cap T^{*}_{y}$ for each edge $xy$ of $T$. Our goal is to prove that $G^{+}_{x}$ has $(t,t^{4}+2t^{2})$-bounded degree.

Consider a vertex $v$ of $G^{+}_{x}$. Then $v$ is in at most one child bag $y$ of $x$, as otherwise $v$ would belong to a set of bags that do not correspond to a directed path in $T^{\prime}$. Since $(T^{*}_{x}:x\in V(T))$ has adhesion at most $t^{2}$, $v$ has at most $t^{2}$ neighbours in $T^{*}_{x}\cap T^{*}_{p}$, where $p$ is the parent of $x$ and $v$ has at most $t^{2}$ neighbours in $T^{*}_{x}\cap T^{*}_{y}$. Thus the degree of $v$ in $G^{+}_{x}$ is at most the degree of $v$ in $G[T^{*}_{x}]$ plus $2t^{2}$. Call this property ($\star$).

First consider the case that $v\not\in T_{x}$. Let $z$ be the node of $T$ for which $v\in T_{z}$. Since $v\in T^{*}_{x}$, by construction, $x$ is an ancestor of $z$. Let $y$ be the node immediately before $x$ on the $\overrightarrow{zx}$ path in $T^{\prime}$. We now bound the number of neighbours of $v$ in $T^{*}_{x}$. Say $w\in N_{G}(v)\cap T^{*}_{x}$. If $w$ is in $G(T,xy)$ then let $e_{w}$ be the edge $vw$. Otherwise, $w$ is in $G(T,yx)$ and thus $w$ has a neighbour $u$ in $G(T,xy)$ since $w\in T^{*}_{x}$; let $e_{w}$ be the edge $wu$. Observe that $\{e_{w}:w\in N_{G}(v)\cap T^{*}_{x}\}\subseteq E(T,xy)$, and thus $|\{e_{w}:w\in N_{G}(v)\cap T^{*}_{x}\}|\leqslant t^{2}$. Since $e_{u}\neq e_{w}$ for distinct $u,w\in N_{G}(v)\cap T^{*}_{x}$, we have $|N_{G}(v)\cap T^{*}_{x}|\leqslant t^{2}$. By ($\star$), the degree of $v$ in $G^{+}_{x}$ is at most $3t^{2}$.

Now consider the case that $v\in T_{x}$. Suppose further that $v$ is not one of the at most $t$ vertices of degree greater than $t^{2}$ in the torso $Q$ of $x$ with respect to the given $T$-partition. Suppose that in $Q$, $v$ has $d_{1}$ neighbours in $T_{x}$ and $d_{2}$ neighbours not in $T_{x}$ (the identified vertices). So $d_{1}+d_{2}\leqslant t^{2}$. Consider a neighbour $w$ of $v$ in $G[T^{*}_{x}]$ with $w\not\in T_{x}$. Then $w\in G(T,yx)$ for some child $y$ of $x$. For at most $d_{2}$ children $y$ of $x$, there is a neighbour of $v$ in $G(T,yx)$. Furthermore, for each child $y$ of $x$, $v$ has at most $t^{2}$ neighbours in $G(T,yx)$ since the $T$-partition has adhesion at most $t^{2}$. Thus $v$ has degree at most $d_{1}+d_{2}t^{2}\leqslant t^{4}$ in $G[T^{*}_{x}]$. By ($\star$), $v$ has degree at most $2t^{2}+t^{4}$ in $G^{+}_{x}$.

Since $3t^{2}\leqslant t^{4}+2t^{2}$, the torso $G^{+}_{x}$ has $(t,t^{4}+2t^{2})$-bounded degree. ∎

4 Excluding a Topological Minor

Theorem 5 is an immediate corollary of Lemma 6 and the following structure theorem of Dvořák [18] that extends Theorem 8.

Theorem 11** ([18]).**

Let $H$ be a fixed planar graph that has a planar embedding with all the vertices of $H$ with degree at least 4 on a single face. Then there exists a constant $k$, such that every graph $G$ that does not contain $H$ as a topological minor has a tree decomposition such that each torso has $(k,k)$-bounded degree.

While Theorem 11 is not explcitly stated in [18], we now explain that it is in fact a special case of Theorem 3 in [18]. This result provides a structural description of graphs excluding a given topological minor in terms of the following definition. For a graph $H$ and surface $\Sigma$, let $\operatorname{mf}(H,\Sigma)$ be the minimum, over all possible embeddings of $H$ in $\Sigma$, of the minimum number of faces such that every vertex of degree at least 4 is incident with one of these faces. By assumption, for our graph $H$ and for every surface $\Sigma$, we have $\operatorname{mf}(H,\Sigma)=1$. In this case, Theorem 3 of Dvořák [18] says that for some integer $k=k(H)$, every graph $G$ that does not contain $H$ as a topological minor is a clique sum of $(k,k)$-bounded degree graphs. It immediately follows that $G$ has the desired tree decomposition. See Corollary 1.4 in [34] for a closely related structure theorem.

The following natural open problem arises from this research: Do graphs excluding a fixed planar graph as a topological minor have bounded nonrepetitive chromatic number? And what is the structure of such graphs?

Acknowledgement

This research was initiated at the Workshop on New Trends in Graph Coloring held at the Banff International Research Station in October 2016. Thanks to the organisers. And thanks to Chun-Hung Liu and Zdeněk Dvořák for stimulating conversations.