On low rank-width colorings

TL;DR

This paper introduces low rank-width colorings, a generalization of low tree-depth colorings, and demonstrates their applicability to various graph classes, including powers of bounded expansion graphs, unit interval graphs, and bipartite permutation graphs.

Contribution

It defines low rank-width colorings and proves their existence for several important graph classes, extending the scope of graph coloring techniques beyond bounded rank-width classes.

Findings

Graph classes of bounded expansion and their powers admit low rank-width colorings.

Unit interval graphs and bipartite permutation graphs admit low rank-width colorings.

Interval graphs and permutation graphs do not admit low rank-width colorings.

Abstract

We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Ne\v{s}et\v{r}il and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in \mathbb{N}$, every graph $G\in \mathcal{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q(i)$. Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class $\mathcal{C}$ of bounded expansion and every positive integer $r$, the class $\{G^r\colon G\in \mathcal{C}\}$ of $r$th powers of graphs from $\mathcal{C}$, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every graph class admitting low rank-width colorings has the Erd\H{o}s-Hajnal property and is $\chi$-bounded.

Figures

Equations

$$\mathrm{td}(G)=\{\begin{array}{ll}1& \text{if ∣V(G)∣=1}\\ 1+{\min }_{v\in V(G)}\mathrm{td}(G-v)& \text{if ∣V(G)∣>1 and s=1}\\ {\max }_{1\le i\le s}\mathrm{td}(G_i)& \text{otherwise}.\end{array}$$

$${\mathrm{wcol}}_r(G)$$

$$a_1\le (b_1-1)m+c_1<a_2\le (b_2-1)m+c_2<\cdots <a_k\le (b_k-1)m+c_k.$$

$$a_1\le (c_1-1)m+b_1<a_2\le (c_2-1)m+b_2<\cdots <a_k\le (c_k-1)m+b_k.$$

$$a_1\le (b_1-1)m+c_1<a_2\le (b_2-1)m+c_2<\cdots <a_k\le (b_k-1)m+c_k,$$

$$\epsilon :=\min \left(\frac{\delta (R(1))}{2},\frac{1}{2{\log }_{2}N(1)}\right),$$

$${\left|V(H)\right|}^{\delta (R(1))}\ge {\left(\frac{n}{N(1)}\right)}^{\delta (R(1))}\ge n^{\frac{\delta (R(1))}{2}}\ge n^{\epsilon }.$$

Full text

Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class $\mathscr{C}$ of bounded expansion and every positive integer $r$, the class $\{G^{r}\colon G\in\mathscr{C}\}$ of $r$th powers of graphs from $\mathscr{C}$ admits low rank-width colorings. On the negative side, we show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every hereditary graph class admitting low rank-width colorings has the Erdős-Hajnal property and is $\chi$-bounded.

††journal: European Journal of Combinatorics

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5(11.13, 13.45)

1 Introduction and main results

A proper vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that no two adjacent vertices are assigned the same color, or in other words, so that every single color class induces an independent set. In this simplest form, a vertex coloring does not have to satisfy any constraints on the interaction between two or more color classes. By adding such constraints we may be able to derive a lot more structural information about a graph than from a simple proper vertex coloring. For example, DeVos et al. [15] introduced colorings, later called $p$-tree-width colorings, that, for a fixed integer parameter $p$, have the property that the union of any $i\leq p$ color classes induces a subgraph of tree-width at most $i-1$. In such a coloring, every single color class induces a graph of tree-width [math], that is, it is an independent set. Hence low tree-width colorings are in particular proper vertex colorings. The union of any two color classes induces a graph of tree-width $1$, that is, a forest, the union of any three color classes induces a graph of tree-width $2$, and so on. Using the structure theorem of Robertson and Seymour [43] for graphs excluding a fixed graph as a minor, De Vos et al. proved that for every graph $H$ and every integer $p\geq 1$, there is an integer $N=N(H,p)$, such that every $H$-minor-free graph $G$ admits a $p$-tree-width coloring with $N$ colors.

Despite the graph theoretic interest in such colorings, we also find nice algorithmic applications. Consider e.g. the subgraph isomorphism problem. Here, we are given two graphs $G$ and $H$ as input, and we are asked to determine whether $G$ contains a subgraph isomorphic to $H$. In many natural settings the pattern graph $H$ we are looking for is small and in such case a low-tree-width coloring as described above is very useful. Assume $H$ has $p$ vertices and assume we can efficiently compute a $p$-tree-width coloring of $G$ with $N$ colors. We will find a copy of $H$ in $G$, if it exists, in at most $p$ color classes. We can hence iterate through all combinations of at most $p$ colors and, by the properties of the coloring, look for $H$ in a subgraph of $G$ of tree-width at most $p-1$. From an algorithmic point of view, graphs of bounded tree-width are very well behaved graphs. Many NP-complete problems, in fact, all problems that can be formulated in monadic second order logic, are solvable in linear time on graphs of bounded tree-width [8, 9]. In particular, the subgraph isomorphism problem for every fixed pattern graph $H$ is solvable in linear time on any graph of bounded tree-width, in our case, in time $f(p)\cdot n$ for some computable function $f$. Hence, if we have already computed a $p$-tree-width coloring of $G$ with $N$ colors, we need only an additional running time of $\mathcal{O}(N^{p}\cdot f(p)\cdot n)$ to solve the subgraph isomorphism problem.

Tree-depth is another important and useful graph invariant. It was introduced under this name in [35], but equivalent notions were known before, including the notion of rank [38], vertex ranking number and minimum height of an elimination tree [1, 14, 44], etc. In [35], Nešetřil and Ossona de Mendez introduced the notion of $p$-tree-depth colorings as vertex colorings of a graph such that the union of any $i\leq p$ color classes induces a subgraph of tree-depth at most $i$. Note that the tree-depth of a graph is always larger (at least by $1$) than its tree-width, hence a low tree-depth coloring is a stronger requirement than a low tree-width coloring. Also based on the structure theorem, Nešetřil and Ossona de Mendez [35] proved that proper minor-closed classes admit even low tree-depth colorings.

Not much later, Nešetřil and Ossona de Mendez [36] proved that proper minor-closed classes are unnecessarily restrictive for the existence of low tree-depth colorings. They introduced the notion of bounded expansion classes of graphs, a concept that generalizes the concept of classes with excluded minors and with excluded topological minors. While the original definition of bounded expansion is in terms of density of shallow minors, it turns out that low tree-depth colorings give an alternative characterisation: a class $\mathscr{C}$ of graphs has bounded expansion if and only if for all $p\in\mathbb{N}$ there exists an integer $N=N(\mathscr{C},p)$ such that every graph $G\in\mathscr{C}$ admits a $p$-tree-depth coloring with $N$ colors [36]. For the even more general notion of nowhere dense classes of graphs [37], it turns out that a class $\mathscr{C}$ of graphs closed under taking subgraphs is nowhere dense if and only if for every $p\in\mathbb{N}$ and every $\varepsilon>0$ there exists $n_{0}$ such that every $n$-vertex graph $G\in\mathscr{C}$ with $n\geq n_{0}$ admits a $p$-tree-depth coloring with $n^{\varepsilon}$ colors.

Furthermore, there is a simple algorithm to compute such a coloring in time $\mathcal{O}(n)$ in case $\mathscr{C}$ has bounded expansion and in time $\mathcal{O}(n^{1+\varepsilon})$ for any $\varepsilon>0$ in case $\mathscr{C}$ is nowhere dense. As a result, the subgraph isomorphism problem for every fixed pattern $H$ can be solved in linear time on any class of bounded expansion and in almost linear time on any nowhere dense class. More generally, it was shown in [18, 24] that every fixed first order property can be tested in linear time on graphs of bounded expansion, implicitly using the notion of low tree-depth colorings, and in almost linear time on nowhere dense classes [25].

Note that bounded expansion and nowhere dense classes of graphs are uniformly sparse graphs. In fact, bounded expansion classes of graphs can have at most a linear number of edges and nowhere dense classes can have no more than $\mathcal{O}(n^{1+\varepsilon})$ many edges, for any fixed $\varepsilon>0$. This motivates our new definition of low rank-width colorings which extends the coloring technique to dense classes of graphs which are closed under taking induced subgraphs.

Rank-width was introduced by Oum and Seymour [42] and aims to extend tree-width by allowing well behaved dense graphs to have small rank-width. Also for graphs of bounded rank-width there are many efficient algorithms based on dynamic programming. Here, we have the important meta-theorem of Courcelle, Makowsky, and Rotics [12], stating that for every monadic second-order formula (with set quantifiers ranging over sets of vertices) and every positive integer $k$, there is an $\mathcal{O}(n^{3})$-time algorithm to determine whether an input graph of rank-width at most $k$ satisfies the formula. There are several parameters which are equivalent to rank-width in the sense that one is bounded if and only if the other is bounded. These include clique-width [10], NLC-width [47], and Boolean-width [4].

Low rank-width colorings

We now introduce our main object of study.

Definition 1**.**

A class $\mathscr{C}$ of graphs admits low rank-width colorings if there exist functions $N:\mathbb{N}\rightarrow\mathbb{N}$ and $Q:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in\mathbb{N}$, every graph $G\in\mathscr{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q(i)$.

As proved by Oum [39], every graph $G$ with tree-width $k$ has rank-width at most $k+1$, hence every graph class which admits low tree-width colorings (that is, every class of bounded expansion) also admits low rank-width colorings. On the other hand, graphs admitting a low rank-width coloring can be very dense. We also remark that graph classes admitting low rank-width colorings can be closed under taking induced subgraphs without spoiling this property, as rank-width does not increase by removing vertices.

Let us remark that due to the model-checking algorithm of Courcelle et al. [12], the (induced) subgraph isomorphism problem is solvable in cubic time for every fixed pattern $H$ whenever the input graph is given together with a low rank-width coloring for $p=|V(H)|$, using $N(p)$ colors. Again, it suffices to iterate through all $p$-tuples of color classes and look for the pattern $H$ in the subgraph induced by these color classes; this can be done efficiently since this subgraph has rank-width at most $Q(p)$. The caveat is that the graph has to be supplied with an appropriate coloring. In this work we do not investigate the algorithmic aspects of low rank-width colorings, and rather concentrate on the combinatorial question of which classes admit such colorings, and which do not.

Related concepts

Let us discuss the relation of our newly introduced notion of low rank-width colorings with other related concepts studied in the literature. The concept of bounded local clique-width was introduced by Grohe and Turán in [26]. A class $\mathscr{C}$ of graphs has bounded local clique-width if there is a function $f\colon\mathbb{N}\times\mathbb{N}$ such that for every $r\geq 1$, $G\in\mathscr{C}$ and $v\in V(G)$, the clique-width of the subgraph of $G$ induced by the $r$-neighborhood of $v$, $G[N_{r}(v)]$, is bounded by $f(r)$. A more restricted concept, called nice local clique-width decompositions, was introduced by Courcelle, Gavoille, and Kanté [11]. They defined a nice $(r,\ell,g)$-cover of a graph $G$ as a family $\mathcal{F}$ of subsets of $V(G)$ which covers $G$ such that (1) the $r$-neighborhood of every vertex is contained in some set in $\mathcal{F}$, (2) each set of $\mathcal{F}$ intersects at most $\ell$ other sets in $\mathcal{F}$, and (3) any union of $q$ sets in $\mathcal{F}$ induces a subgraph of clique-width at most $g(q)$. A class $\mathscr{C}$ of graphs is nicely locally clique-width decomposable if for each $r$, every graph in $\mathscr{C}$ admits a nice $(r,\ell,g)$-cover for some $\ell$ and $g$. It is implicit in [11] that every nicely locally clique-width decomposable class admits low rank-width colorings (this terminology is not defined in [11] and we will provide an explicit proof below). Also every nicely locally clique-width decomposable class has bounded local clique-width, while there are classes of bounded local clique-width that do not admit low rank-width decompositions. Also not every class that admits low rank-width decompositions admits nice local clique-width decompositions. In particular, both the concept of nice local clique-width decompositions and the concept of bounded local clique-width are not robust under small non-local changes, such as adding a universal vertex, that is, a vertex connected to all other vertices.

On the other hand, our newly introduced notion of low rank-width colorings is robust under this operation, as we can simply assign a new unique color to the newly added vertex and adding one vertex can increase the rank-width of any graph by at most one. In particular, our notion of low rank-width colorable graphs generalizes the notion of low tree-depth colorable graphs, and hence the notion of bounded expansion, while the notion of Coucelle, Gavoille, and Kanté does not even generalize the notion of $H$-minor-free graphs.

Our contribution

We study for several graph classes whether they admit low rank-width covers or not. In section 3, we make explicit the fact that graph classes that admit nice local clique-width decompositions admit low rank-width colorings. Courcelle, Gavoille, and Kanté also proved in [11] that the class of unit interval graphs and the class of bipartite permutation graphs are nicely locally clique-width decomposable. We give a short direct proof for the existence of low rank-width colorings for these classes.

In section 4 we present our main technical contribution and prove that for every class $\mathscr{C}$ of bounded expansion and every integer $r\geq 2$, the class $\{G^{r}\colon G\in\mathscr{C}\}$ of $r$th powers of graphs from $\mathscr{C}$ admits low rank-width colorings. It is easy to see that there are classes of bounded expansion such that $\{G^{r}\colon G\in\mathscr{C}\}$ has both unbounded rank-width and does not admit low tree-depth colorings, hence our notion generalizes both of these concepts. Our results inspired further research, and in a follow-up of this work [22] we proved that in fact every first-order interpretation of a bounded expansion class (the $r$th power of a graph is a simple first-order interpretation) has low rank-width colorings (and in fact low shrub-depth colorings). We comment on these new results in the Conclusion, section 7.

On the negative side, in section 5, we show that the classes of interval graphs and of permutation graphs do not admit low rank-width colorings. To show it, we give a general construction called twisted chain graphs, and show that if a graph class contains a twisted chain graph of order $n$ for every large $n$, then it does not admit low rank-width colorings.

Lastly, in section 6, we show that every hereditary graph class admitting low rank-width colorings has the Erdős-Hajnal property [20] and is $\chi$-bounded [28]. Erdős and Hajnal [20] conjectured that for every graph $H$, there is $\varepsilon=\varepsilon(H)>0$ such that every $n$-vertex graph having no $H$ induced subgraph has either an independent set or a clique of size $n^{\varepsilon}$. A graph class with this property is said to have the Erdős-Hajnal property. We refer to [32, 7, 21, 6, 2] for background and recent progress on the Erdős-Hajnal conjecture. Gyárfás [28] introduced the notion of $\chi$-boundedness as a relaxation of the concept of perfect graphs. For background we refer to a survey of $\chi$-bounded classes by Scott and Seymour [45].

2 Preliminaries

All graphs in this paper are finite, undirected and simple, that is, they do not have loops or parallel edges. Our notation is standard, we refer to [17] for more background on graph theory. We write $V(G)$ for the vertex set of a graph $G$ and $E(G)$ for its edge set. A vertex coloring of a graph $G$ with colors from $S$ is a mapping $c\colon V(G)\rightarrow S$. For each $v\in V(G)$, we call $c(v)$ the color of $v$. The distance between vertices $u$ and $v$ in $G$, denoted $\mathrm{dist}_{G}(u,v)$, is the length of a shortest path between $u$ and $v$ in $G$, or $\infty$ if no such path exists. The $r$th power of a graph $G$ is the graph $G^{r}$ with vertex set $V(G)$, where there is an edge between two vertices $u$ and $v$ in $G^{r}$ if and only if their distance in $G$ is at most $r$.

Rank-width was introduced by Oum and Seymour [42]. We refer to the surveys [29, 40] for more background. For a graph $G$, we denote the adjacency matrix of $G$ by $A_{G}$, where for $x,y\in V(G)$, $A_{G}[x,y]=1$ if and only if $x$ is adjacent to $y$. Let $G$ be a graph. We define the cut-rank function $\mathrm{cutrk}_{G}\colon 2^{V}\rightarrow\mathbb{N}$ such that $\mathrm{cutrk}_{G}(X)$ is the rank of the matrix $A_{G}[X,V(G)\setminus X]$ over the field $GF(2)$ (if $X=\emptyset$ or $X=V(G)$, then we let $\mathrm{cutrk}_{G}(X)=0$).

A rank-decomposition of $G$ is a pair $(T,L)$, where $T$ is a subcubic tree (i.e. a tree where every node has degree $1$ or $3$) with at least $2$ nodes and $L$ is a bijection from $V(G)$ to the set of leaves of $T$. For every edge $e\in E(T)$, $T-e$ has exactly two components $T_{1},T_{2}$, and hence $e$ induces a vertex bipartition $(A^{e}_{1},A^{e}_{2})$ of $G$, where $A^{e}_{i}=L^{-1}(T_{i})$. The width of $e$ is defined as $\mathrm{cutrk}_{G}(A^{e}_{1})$ (which is equal to $\mathrm{cutrk}_{G}(A^{e}_{2})$). The width of $(T,L)$ is the maximum width over all edges in $T$, and the rank-width of $G$, denoted by $\mathrm{rw}(G)$, is the minimum width over all rank-decompositions of $G$. If $\left\lvert V(G)\right\rvert\leq 1$, then $G$ has no rank-decompositions, and the rank-width of $G$ is defined to be [math].

A tree-decomposition of a graph $G$ is a pair $(T,\mathcal{B})$ consisting of a tree $T$ and a family $\mathcal{B}=\{B_{t}\}_{t\in V(T)}$ of sets $B_{t}\subseteq V(G)$, satisfying the following three conditions:

1. (T1)

$V(G)=\bigcup_{t\in V(T)}B_{t}$; 2. (T2)

for every $uv\in E(G)$, there exists a node $t$ of $T$ such that $\{u,v\}\subseteq B_{t}$; 3. (T3)

for $t_{1},t_{2},t_{3}\in V(T)$, $B_{t_{1}}\cap B_{t_{3}}\subseteq B_{t_{2}}$ whenever $t_{2}$ is on the path from $t_{1}$ to $t_{3}$ in $T$.

The width of a tree-decomposition $(T,\mathcal{B})$ is $\max\{\left\lvert B_{t}\right\rvert-1:t\in V(T)\}$. The tree-width of $G$ is the minimum width over all tree-decompositions of $G$.

Let $G$ be a graph and let $G_{1},\ldots,G_{s}$ be its connected components. Then the tree-depth of $G$ is recursively defined as

[TABLE]

A graph is an interval graph if it is the intersection graph of a family $\mathcal{I}$ of intervals on the real line. An interval graph is a unit interval graph if all intervals in $\mathcal{I}$ have the same length. A graph is a permutation graph if it is the intersection graph of line segments whose endpoints lie on two parallel lines.

3 Nice local clique-width decompositions

In this section we first study the connections between the newly introduced notion of low rank-width colorings and the notion of nice local clique-width decompositions that was introduced by Courcelle, Gavoille, and Kanté [11]. We show that graph classes that admit nice local clique-width decompositions admit low rank-width colorings. We then prove that unit interval graphs and bipartite permutation graphs admit low rank-width colorings. As discussed in the introduction, this can be obtained by the result that these classes are nicely locally clique-width decomposable, as shown by Courcelle, Gavoille, and Kanté [11]. On the other hand, it is a simple application of Lozin’s characterisations of those two classes; so we would like to present direct proofs.

Recall that for integers $r,\ell\geq 1$ and a function $g\colon\mathbb{N}\rightarrow\mathbb{N}$, a family $\mathcal{F}$ of vertex subsets of $G$ is called a nice $(r,\ell,g)$-cover if

$\bigcup_{S\in\mathcal{F}}S=V(G)$,

• 2.

for every vertex $v$ of $G$, the $r$-neighborhood of $v$ is contained in some set of $\mathcal{F}$,

• 3.

each set of $\mathcal{F}$ intersects at most $\ell$ other sets in $\mathcal{F}$,

• 4.

any union of $q$ sets in $\mathcal{F}$ induces a subgraph of clique-width at most $g(q)$.

A class $\mathcal{C}$ of graphs is nicely locally clique-width decomposable if for each $r$, every graph in $\mathcal{C}$ admits a nice $(r,\ell,g)$-cover for some $\ell$ and $g$.

Theorem 1** (Implicit in [11]).**

*Every nicely locally clique-width decomposable class admits low rank-width colorings. *

Proof.

Let $\mathscr{C}$ be a nicely locally clique-width decomposable class. Consider any $G\in\mathscr{C}$ and let $p\geq 1$ be an integer. We want to show that $G$ admits a low rank-width coloring with $N=N(p)$ colors so that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q=Q(i)$ for constants $N$ and $Q$ depending only on $p$ and $i$. By assumption, $G$ admits a nice $(1,\ell,g)$-cover $\mathcal{F}$ for a constant $\ell$ and function $g$. Without loss of generality assume that $\ell\geq 3$.

As the neighborhood of each $v\in V(G)$ is contained in some set of $\mathcal{F}$, we can assign a mapping $f\colon V(G)\rightarrow\mathcal{F}$ such that the neighborhood of $v$ is fully contained in $f(v)$.

Now, we define an auxiliary graph $H$ on vertex set $\mathcal{F}$ such that for $P,Q\in\mathcal{F}$, $P$ and $Q$ are adjacent in $H$ if and only if $P\cap Q\neq\emptyset$. As $\mathcal{F}$ is a nice $(1,\ell,g)$-cover, $H$ has maximum degree at most $\ell$. We conclude that the $p$-th power $H^{p}$ of $H$ has maximum degree bounded by $\sum_{i=0}^{p}\ell^{i}<1+(p+1)\cdot\ell^{p}\eqqcolon N(p)$. Thus, $H^{p}$ admits a proper coloring $\lambda$ with $N$ colors. We define a coloring $\eta$ of $G$ by assigning $v\in V(G)$ the color $\lambda(f(v))$.

We let $Q(i)\coloneqq g(i)$ and show that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q(i)$; as $G$ and $p$ were chosen arbitrarily, this means that $\mathscr{C}$ admits low rank-width colorings. For this, it suffices to prove the following claim: if a vertex subset $X\subseteq V(G)$ receives at most $i$ different colors in the coloring $\eta$, then $G[X]$ has rank-width at most $Q(i)$. Since the rank-width of a disjoint union of graphs of rank-width at most $Q(i)$ also has rank-width at most $Q(i)$, we may focus on the case when $G[X]$ is connected.

Let $\mathcal{X}=\{f(v)\colon v\in X\}\subseteq\mathcal{F}$. By the definition of $\eta$ we have that $\mathcal{X}$ receives at most $i$ different colors under $\lambda$. Note that whenever $uv$ is an edge in $G$, $f(u)$ and $f(v)$ are either equal or adjacent in $H$. Therefore, we conclude that since $G[X]$ is connected, $H[\mathcal{X}]$ is connected as well.

Suppose for a moment that $|\mathcal{X}|>i$. Then we would be able to find a subset $\mathcal{X}^{\prime}\subseteq\mathcal{X}$ with $|\mathcal{X}^{\prime}|=i+1$ such that $H[\mathcal{X}^{\prime}]$ is connected a well. Observe that then the vertices of $\mathcal{X}^{\prime}$ would be pairwise at distance at most $p$ in $H$, implying, by the construction of $\lambda$, that they would receive pairwise different colors in $\lambda$. This is a contradiction with the fact that $\mathcal{X}$ receives at most $i$ colors under $\lambda$, hence we conclude that $|\mathcal{X}|\leq i$.

Since $\mathcal{F}$ is a $(1,\ell,g)$-cover, we conclude that the graph $G[\bigcup\mathcal{X}]$ has clique-width bounded by $g(i)$, hence also its rank-width is bounded by $g(i)$. As $G[X]$ is an induced subgraph of $G[\bigcup\mathcal{X}]$, we are done. $\square$

Courcelle, Gavoille, and Kanté [11] showed that the class of unit interval graphs is nicely locally clique-width decomposable. So, theorem 1 implies that the class of unit interval graphs admits low rank-width colorings. We nevertheless give an explicit proof of this observation.

Theorem 2**.**

*The class of unit interval graphs and the class of bipartite permutation graphs admit low rank-width colorings. *

We recall the characterizations of these classes obtained by Lozin [33]. Let $n,m\geq 1$. We denote by $H_{n,m}$ the graph with $n\cdot m$ vertices which can be partitioned into $n$ independents sets $V_{1}=\{v_{1,1},\ldots,v_{1,m}\},\ldots,V_{n}=\{v_{n,1},\ldots,v_{n,m}\}$ so that for each $i\in\{1,\ldots,n-1\}$ and for each $j,j^{\prime}\in\{1,\ldots,m\}$, vertex $v_{i,j}$ is adjacent to $v_{i+1,j^{\prime}}$ if and only if $j^{\prime}\in\{1,\ldots,j\}$, and there are no edges between $V_{i}$ and $V_{j}$ if $\left\lvert i-j\right\rvert\geq 2$. The graph $\widetilde{H}_{n,m}$ is the graph obtained from $H_{n,m}$ by replacing each independent set $V_{i}$ by a clique.

Lemma 3**.**

The following statements hold:

(Lozin **[33]**) The rank-width of $H_{n,m}$ and of $\widetilde{H}_{n,m}$ is at most $3n$. 2. 2.

(Lozin **[33]**) Every bipartite permutation graph on $n$ vertices is isomorphic to an induced subgraph of $H_{n,n}$. 3. 3.

(Lozin **[34]**) Every unit interval graph on $n$ vertices is isomorphic to an induced subgraph of $\widetilde{H}_{n,n}$.

Hence, in order to prove theorem 2, it suffices to prove that the graphs $H_{n,m}$ and $\widetilde{H}_{n,m}$ admit low rank-width colorings.

Proof (of theorem 2).

For every positive integer $p$, let $N(p):=p+1$ and $Q(i):=3i$ for each $i\in\{1,\ldots,p\}$. We prove that for all $n,m\geq 1$, the graphs $H_{n,m}$ and $\widetilde{H}_{n,m}$ can be vertex colored using $N(p)$ colors so that each of the connected components of the subgraph induced by any $i\leq p$ color classes has rank-width at most $Q(i)$. As rank-width and rank-width colorings are monotone under taking induced subgraphs, the statement of the theorem follows from lemma 3.

Assume that the vertices of $H_{n,m}$ (and $\widetilde{H}_{n,m}$, respectively) are, as in the definition, named $v_{1,1},\ldots,v_{1,m},\ldots,v_{n,1},\ldots,v_{n,m}$. We color the vertices in the $i$th row, $v_{i,1},\ldots,v_{i,m}$, with color $j+1$ where $j\in\{0,1,\ldots,p\}$ and $i\equiv j{\pmod{p+1}}$. Then any connected component $H$ of a subgraph induced by $i\leq p$ colors is isomorphic to $H_{i^{\prime},m}$ ($\widetilde{H}_{i^{\prime},m}$, respectively) for some $i^{\prime}\leq i$. Hence, according to lemma 3, $H$ has rank-width at most $3i=Q(i)$, as claimed. $\square$

4 Powers of sparse graphs

In this section we show that the class of $r$th powers of graphs from a bounded expansion class admit low rank-width colorings. The original definition of bounded expansion classes by Nešetřil and Ossona de Mendez [36] is in terms of bounds on the density of bounded depth minors. We will work with the characterisation by the existence of low tree-depth colorings as well as by a characterisation in terms of bounds on generalized coloring numbers.

Theorem 4** (Nešetřil and Ossona de

Mendez [36]).**

*A class $\mathscr{C}$ of graphs has bounded expansion if and only if for all integers $p\geq 0$ there exists an integer $N=N(\mathscr{C},p)$ such that every graph $G\in\mathscr{C}$ admits a $p$-tree-depth coloring with $N$ colors. *

Our main result in this section is the following.

Theorem 5**.**

*Let $\mathscr{C}$ be a class of bounded expansion and $r\geq 2$ be an integer. Then the class $\{G^{r}\colon G\in\mathscr{C}\}$ of $r$th powers of graphs from $\mathscr{C}$ admits low rank-width colorings. *

For a graph $G$, we denote by $\Pi(G)$ the set of all linear orders of $V(G)$. For $u,v\in V(G)$ and an integer $r\geq 0$, we say that $u$ is weakly $r$-reachable from $v$ with respect to a linear order $L$, if there is a path $P$ of length at most $r$ between $u$ and $v$ such that $u$ is the smallest among the vertices of $P$ with respect to $L$. We denote by $\mathrm{WReach}_{r}[G,L,v]$ the set of vertices that are weakly $r$-reachable from $v$ with respect to $L$. The weak $r$-coloring number $\mathrm{wcol}_{r}(G)$ of $G$ is defined as

[TABLE]

The weak coloring number was introduced by Kierstead and Yang [30] in the context of coloring and marking games on graphs. As shown by Zhu [48], classes of bounded expansion can be characterised by the weak coloring numbers.

Theorem 6** (Zhu [48]).**

*A class $\mathscr{C}$ has bounded expansion if and only if for all integers $r\geq 0$ there is an integer $f(r)$ such that for all $G\in\mathscr{C}$ we have $\mathrm{wcol}_{r}(G)\leq f(r)$. *

In order to prove theorem 5, we will first compute a low tree-depth coloring. We would like to apply the following theorem, relating the tree-width (and hence in particular the tree-depth) of a graph and the rank-width of its $r$th power.

Theorem 7** (Gurski and Wanke [27]).**

*Let $p\geq 0,r\geq 2$ be integers. If a graph $H$ has tree-width at most $p$, then $H^{r}$ has rank-width at most $2(r+1)^{p+1}-2$. *

We remark that Gurski and Wanke [27] proved this bound for clique-width instead of rank-width, but clique-width is never smaller than the rank-width [42].

The natural idea would be just to combine the bound of theorem 7 with low tree-depth coloring given by theorem 4. Note however, that when we consider any subgraph $H$ induced by $i\leq p$ color classes, the graph $H^{r}$ may be completely different from the graph $G^{r}[V(H)]$, due to paths that are present in $G$ but disappear in $H$. Hence we cannot directly apply theorem 7. Instead, we will prove the existence of a refined coloring of $G$ such that for any subgraph $H$ induced by $i\leq p$ color classes, in the refined coloring there is a subgraph $H^{\prime}$ such that $G^{r}[V(H)]\subseteq H^{\prime r}$ and such that $H^{\prime}$ gets only $g(i)$ colors in the original coloring, for some fixed function $g$. We can then apply theorem 7 to $H^{\prime}$ and use the fact that rank-width is monotone under taking induced subgraphs.

In the following, we will say that a vertex subset $X$ receives a color $i$ under a coloring $c$ if $i\in c^{-1}(X)$. We first need the following definitions.

Definition 2**.**

Let $G$ be a graph, $X\subseteq V(G)$ and $r\geq 2$. A superset $X^{\prime}\supseteq X$ is called an $r$-shortest path hitter for $X$ if for all $u,v\in X$ with $1<\mathrm{dist}_{G}(u,v)\leq r$, $X^{\prime}$ contains an internal vertex of some shortest path between $u$ and $v$.

Definition 3**.**

Let $G$ be a graph, let $c$ be a coloring of $G$, and let $p\geq 1,r\geq 2$ and $d\geq 1$ be integers. A coloring $c^{\prime}$ is a $(d,r)$-good refinement of $c$ if for every vertex set $X\subseteq V(G)$, there exists an $r$-shortest path hitter $X^{\prime}$ of $X$ such that if $X$ receives at most $p$ colors under $c^{\prime}$, then $X^{\prime}$ receives at most $d\cdot p$ colors under $c$.

We use the weak coloring numbers to prove the existence of a good refinement.

Lemma 8**.**

*Let $G$ be a graph and $k\geq 1,r\geq 2$ be integers. Then every coloring $c$ of $G$ using $k$ colors has a $(2\mathrm{wcol}_{r}(G),r)$-good refinement coloring using $k^{2\mathrm{wcol}_{r}(G)}$ colors. *

Proof.

Let $\Gamma$ be the set of colors used by $c$, and let $d:=2\mathrm{wcol}_{r}(G)$. The $(d,r)$-good refinement $c^{\prime}$ that we are going to construct will use subsets of $\Gamma$ of size at most $d$ as the color set; the number of such subsets is at most $k^{2\mathrm{wcol}_{r}(G)}$. Let $L$ be a linear order of $V(G)$ with \max_{v\in V(G)}\>\bigl{|}\mathrm{WReach}_{r}[G,L,v]\bigr{|}=\mathrm{wcol}_{r}(G). We construct a new coloring $c^{\prime}$ as follows:

1. (1)

Start by setting $c^{\prime}(v):=\emptyset$ for each $v\in V(G)$. 2. (2)

For each pair of vertices $u$ and $v$ such that $u\in\mathrm{WReach}_{r}[G,L,v]$, we add the color $c(u)$ to $c^{\prime}(v)$. 3. (3)

For each pair $u,v$ of non-adjacent vertices such that $u<_{L}v$ and $u\in\mathrm{WReach}_{r}[G,L,v]$, we do the following. Check whether there is a path $P$ of length at most $r$ connecting $u$ and $v$ such that all the internal vertices of $P$ are larger than both $u$ and $v$ in $L$. If there is no such path, we do nothing for the pair $u,v$. Otherwise, fix one such path $P$, chosen to be the shortest possible, and let $z$ be the vertex traversed by $P$ that is the largest in $L$. Then add the color $c(z)$ to $c^{\prime}(v)$.

Thus, every vertex $v$ receives in total at most $2\mathrm{wcol}_{r}(G)$ colors of $\Gamma$ to its final color $c^{\prime}(v)$: at most $\mathrm{wcol}_{r}(G)$ in step (2), and at most $\mathrm{wcol}_{r}(G)$ in step (3), because we add at most one color per each $u\in\mathrm{WReach}_{r}[G,L,v]$. It follows that each final color $c^{\prime}(v)$ is a subset of $\Gamma$ of size at most $2\mathrm{wcol}_{r}(G)=d$.

We claim that $c^{\prime}$ is a $(d,r)$-good refinement of $c$. Let $X\subseteq V(G)$ be a set that receives at most $p$ colors under $c^{\prime}$, say colors $A_{1},\ldots,A_{p}\subseteq\Gamma$. Let $X^{\prime}$ be the set of vertices of $G$ that are colored by colors in $A_{1}\cup\cdots\cup A_{p}$ under $c$. Since $|A_{i}|\leq d$ for each $i\in\{1,\ldots,p\}$, we have that $X^{\prime}$ receives at most $d\cdot p$ colors under $c$.

To show that $X^{\prime}$ is an $r$-shortest path hitter of $X$, let us choose any two vertices $u$ and $v$ in $X$ with $u<_{L}v$ and $1<\mathrm{dist}_{G}(u,v)\leq r$. If there is a shortest path from $u$ to $v$ whose all internal vertices are larger than $u$ and $v$ in $L$, by step (3), $X^{\prime}$ contains a vertex that is contained in one such path. Otherwise, a shortest path from $u$ to $v$ contains a vertex $z$ with $L(z)<L(v)$ other than $u$ and $v$. This implies that there exists $z^{\prime}\in\mathrm{WReach}_{r}[G,L,v]\setminus\{u\}$ on the path such that $c(z^{\prime})\in c^{\prime}(v)$, and hence $z^{\prime}\in X^{\prime}$ by step (2). Therefore, $X^{\prime}$ is an $r$-shortest path hitter of $X$, as required. $\square$

Definition 4**.**

Let $G$ be a graph, let $X\subseteq V(G)$, and let $r\geq 1$ be an integer. A superset $X^{\prime}\supseteq X$ is called an $r$-shortest path closure of $X$ if for each $u,v\in X$ with $\mathrm{dist}_{G}(u,v)=\ell\leq r$, $G[X^{\prime}]$ contains a path of length $\ell$ between $u$ and $v$.

Definition 5**.**

Let $G$ be a graph, let $c$ be a coloring of $G$, and let $r\geq 2$ and $d\geq 1$ be integers. A coloring $c^{\prime}$ is a $(d,r)$-excellent refinement of $c$ if for every vertex set $X\subseteq V(G)$ there exists an $r$-shortest path closure $X^{\prime}$ of $X$ such that if $X$ receives at most $p$ colors in $c^{\prime}$, then $X^{\prime}$ receives at most $d\cdot p$ colors in $c$.

We inductively define excellent refinements from good refinements.

Lemma 9**.**

*Let $G$ be a graph, let $k\geq 1,r\geq 2$ be integers, and let $d_{r}\coloneqq\prod_{2\leq\ell\leq r}2\mathrm{wcol}_{\ell}(G)$. Then every coloring $c$ of $G$ using at most $k$ colors has a $(d_{r},r)$-excellent refinement coloring using at most $k^{d_{r}}$ colors. *

Proof.

The proof is by induction on $r$. For $r=2$, an $r$-shortest path hitter of a set $X$ is an $r$-shortest path closure, and vice versa. Hence, the statement immediately follows from lemma 8. Now assume $r\geq 3$. By induction hypothesis, there is a $(d_{r-1},r-1)$-excellent refinement $c_{1}$ of $c$ with at most $k^{d_{r-1}}$ colors. By applying lemma 8 to $c_{1}$, we obtain a $(2\mathrm{wcol}_{r}(G),r)$-good refinement $c^{\prime}$ of $c_{1}$ with at most $(k^{d_{r-1}})^{2\mathrm{wcol}_{r}(G)}=k^{d_{r}}$ colors. We claim that $c^{\prime}$ is a $(d_{r},r)$-excellent refinement of $c$. Any set $X$ which gets at most $p$ colors from $c^{\prime}$ can be first extended to an $r$-shortest path hitter $X^{\prime}$ for $X$ which receives at most $2\mathrm{wcol}_{r}(G)\cdot p$ colors. Then $X^{\prime}$ can be extended by induction hypothesis to an $(r-1)$-shortest path closure $X^{\prime\prime}$ of $X^{\prime}$ that receives at most $d_{r-1}\cdot 2\mathrm{wcol}_{r}(G)\cdot p=d_{r}\cdot p$ colors.

It remains to show that $X^{\prime\prime}$ is an $r$-shortest path closure of $X$. Take any $u,v\in X$ with $\mathrm{dist}_{G}(u,v)=\ell\leq r$. If $\ell\leq 1$, then $u,v$ are already adjacent in $G[X]$. Otherwise, since $X^{\prime}$ is an $r$-shortest path hitter for $X$, there is a vertex $z\in X^{\prime}$ that lies on some shortest path connecting $u$ and $v$ in $G$. In particular, $\mathrm{dist}_{G}(u,z)=\ell_{1}$ and $\mathrm{dist}_{G}(z,v)=\ell_{2}$ for $\ell_{1},\ell_{2}$ satisfying $\ell_{1},\ell_{2}<\ell$ and $\ell_{1}+\ell_{2}=\ell$. Since $X^{\prime\prime}$ is an $(r-1)$-shortest path closure of $X^{\prime}$, we infer that $\mathrm{dist}_{G[X^{\prime\prime}]}(u,z)=\ell_{1}$ and $\mathrm{dist}_{G[X^{\prime\prime}]}(z,v)=\ell_{2}$. Hence $\mathrm{dist}_{G[X^{\prime\prime}]}(u,v)=\ell$ by the triangle inequality. $\square$

Proof (of theorem 5).

Let $G\in\mathscr{C}$ and let $d_{r}:=\prod_{2\leq\ell\leq r}2\mathrm{wcol}_{\ell}(G)$. Since $\mathscr{C}$ has bounded expansion, by theorem 6, for each $r$, $\mathrm{wcol}_{r}(G)$ is bounded by a constant only depending on $\mathscr{C}$. We start by taking $c$ to be a $(d_{r}\cdot p)$-tree-depth coloring with $N(d_{r}\cdot p)$ colors, where $N$ is the function from theorem 4. Then its $(d_{r},r)$-excellent refinement $c^{\prime}$ has the property that $c^{\prime}$ uses at most $N(d_{r}\cdot p)^{d_{r}}$ colors, and every subset $X$ which receives at most $p$ colors in $c^{\prime}$ has an $r$-shortest path closure $X^{\prime}$ that receives at most $d_{r}\cdot p$ colors in $c$. Thus, the graph induced by $X$ in the $r$th power $G^{r}$ is the same as the graph induced by $X$ in the $r$th power $G[X^{\prime}]^{r}$. Since $G[X^{\prime}]$ has tree-depth at most $d_{r}\cdot p$, by theorem 7, $G[X^{\prime}]^{r}$ has rank-width at most $2(r+1)^{d_{r}\cdot p+1}-2$. Therefore, $G^{r}[X]$ has rank-width at most $2(r+1)^{d_{r}\cdot p+1}-2$ as well. $\square$

We now give two example applications of theorem 5. A map graph is a graph that can be obtained from a plane graph by making a vertex for each face, and adding an edge between two vertices, if the corresponding faces share a vertex. We use the following characterization of map graphs as half-squares of planar graphs, due to Chen et al. [5].

Lemma 10** ([5]).**

*Every map graph is an induced subgraph of the second power of a planar graph. *

As the class of planar graphs is a bounded expansion class, we conclude that map graphs admit low rank-width colorings. A similar reasoning can be performed for line graphs of graphs from any bounded expansion graph class. Thus, both map graphs and line graphs of graphs from any fixed bounded expansion graph class admit low rank-width colorings.

Lemma 11**.**

*If $\mathscr{C}$ is a graph class of bounded expansion, then there is a graph class of bounded expansion $\mathscr{C}_{1}$ such that all line graphs of graphs from $\mathscr{C}$ are induced subgraphs of graphs from $\mathscr{C}_{1}^{2}$. *

Proof.

Observe that for any graph $G$, if by $G_{1}$ we denote $G$ with every edge subdivided once, then the line graph of $G$ is a subgraph of $G_{1}^{2}$ induced by the subdividing vertices. It follows that we may take $\mathscr{C}_{1}$ to be the class of all $1$-subdivisions of graphs from $\mathscr{C}$. This class also has bounded expansion. $\square$

5 Negative results

In contrast to the result in section 3, we prove that interval graphs and permutation graphs do not admit low rank-width colorings. For this, we introduce twisted chain graphs. Briefly, a twisted chain graph $G$ consists of three vertex sets $A,B,C$ where each of $G[A\cup C]$ and $G[B\cup C]$ is a chain graph, but the ordering of $C$ with respect to the chain graphs $G[A\cup C]$ and $G[B\cup C]$ are distinct. See Figure 1 for an illustration.

Definition 6**.**

For $n\in\mathbb{N}$, a graph on the set of $3n^{2}$ vertices $A\cup B\cup C$, where $A=\{v_{1},\ldots,v_{n^{2}}\}$, $B=\{w_{1},\ldots,w_{n^{2}}\}$, and $C=\{z_{(i,j)}\colon 1\leq i,j\leq n\}$, is called a twisted chain graph of order $n$ if

for integers $x,y,i,j\in\{1,\ldots,n\}$ and $k=n(x-1)+y$, $v_{k}$ is adjacent to $z_{(i,j)}$ if and only if either ($x<i$) or ($x=i$ and $y\leq j$);

• 2.

for integers $x,y,i,j\in\{1,\ldots,n\}$ and $k=n(x-1)+y$, $w_{k}$ is adjacent to $z_{(i,j)}$ if and only if either ($x<j$) or ($x=j$ and $y\leq i$);

• 3.

the edge relation within $A\cup B$ and within $C$ is arbitrary.

We first show that a large twisted chain graph has large rank-width. We remark that a similar construction based on merging two chain graphs in a mixed order can be found in Brandstädt et al [3]. Also, a slightly more general construction was given by Dabrowski and Paulusma [13]. Obtaining any lower bound seems to follow from a careful examination and modification of the constructions given in [3] or [13]; however, we prefer to give our own direct proof for the sake of completeness. Also, in those papers, the authors provided a lower bound of clique-width, and its direct application to rank-width does not provide a linear lower bound.

Lemma 12**.**

*For every integer $n>0$, every twisted chain graph of order $12n$ has rank-width at least $n$. *

Before we proceed to the proof of lemma 12, we need to introduce some basic tools. A vertex bipartition $(X,Y)$ of a graph $G$ is balanced with respect to a set $C\subseteq V(G)$ if $\frac{|C|}{3}\leq|X\cap C|,|Y\cap C|$. We will need the following standard fact.

Lemma 13**.**

*If $G$ is a graph of rank-width at most $w$ and $C\subseteq V(G)$ with $|C|\geq 3$, then $G$ admits a vertex bipartition $(X,Y)$ with $\mathrm{cutrk}_{G}(X)\leq w$ that is balanced with respect to $C$. *

Proof.

Let $(T,L)$ be a rank-decomposition of $G$ of width at most $w$. We subdivide an edge of $T$, and regard the new vertex as a root node. For each node $t\in V(T)$, let $\mu(t)$ be the number of leaves of $T$ that are descendants of $t$ and correspond to vertices of $C$. Now, we choose a node $t$ that is farthest from the root node and such that $\mu(t)\geq\frac{|C|}{3}$. By the choice of $t$, either $t$ is a leaf or for each child $t^{\prime}$ of $t$ we have $\mu(t^{\prime})<\frac{|C|}{3}$. Therefore, since $|C|\geq 3$, in any case $\frac{|C|}{3}\leq\mu(t)<\frac{2|C|}{3}$. Let $e$ be the edge connecting the node $t$ and its parent. By the construction, the vertex bipartition associated with $e$ satisfies the required property. $\square$

We now proceed to the proof of lemma 12.

Proof (of lemma 12).

Let $m:=12n$ and let $G$ be a twisted chain graph of order $m$. Adopt the notation from Definition 6 for $G$. Suppose for the sake of contradiction that the rank-width of $G$ is smaller than $n$. By lemma 13, there exists a vertex bipartition $(S,T)$ of $G$ with $\mathrm{cutrk}_{G}(S)<n$ such that $|C\cap S|\geq\frac{|C|}{3}=m^{2}/3$ and similarly $|C\cap T|\geq m^{2}/3$.

Suppose we have vertices $v_{a_{1}},\ldots,v_{a_{k}}\in A\cap S$ and $z_{(b_{1},c_{1})},\ldots,z_{(b_{k},c_{k})}\in C\cap T$ with the following property satisfied:

[TABLE]

Such a structure will be called an $A$-ordered $(S,T)$-matching of order $k$. By the definition of adjacency in $G$ it follows that the submatrix of $A_{G}[S,T]$ induced by rows corresponding to vertices $v_{a_{i}}$ and columns corresponding to vertices $z_{(b_{i},c_{i})}$ has ones in the upper triangle and on the diagonal, and zeroes in the lower triangle. The rank of this submatrix is $k$, so since $\mathrm{cutrk}_{G}(A)<n$, there is no $A$-ordered $(S,T)$-matching of order $n$. We similarly define the $A$-ordered $(T,S)$-matching of rank $n$, where the vertices $v_{a_{i}}$ belong to $T$ and vertices $z_{(b_{i},c_{i})}$ belong to $S$. Likewise, there is no $A$-ordered $(T,S)$-matching of order $n$.

Suppose now we have vertices $w_{a_{1}},\ldots,w_{a_{k}}\in B\cap S$ and $z_{(b_{1},c_{1})},\ldots,z_{(b_{k},c_{k})}\in C\cap T$ with the following property satisfied:

[TABLE]

Such a structure will be called a $B$-ordered $(S,T)$-matching of order $k$, and a $B$-ordered $(T,S)$-matching of order $k$ is defined analogously. Again, the same reasoning as above shows that there is no $B$-ordered $(S,T)$-matching of order $n$, and no $B$-ordered $(T,S)$-matching of order $n$.

Let $\preceq_{1}$ and $\preceq_{2}$ be lexicographic orders on $\{1,\ldots,m\}\times\{1,\ldots,m\}$, with the leading coordinate being the first one for $\preceq_{1}$ and the second for $\preceq_{2}$.

Claim 1**.**

*If there is a sequence $(x_{1},y_{1})\prec_{1}(x_{2},y_{2})\prec_{1}\ldots\prec_{1}(x_{4k},y_{4k})$ such that $z_{(x_{j},y_{j})}\in S$ for odd $j$ and $z_{(x_{j},y_{j})}\in T$ for even $j$, then there is either an $A$-ordered $(S,T)$-matching of order $k$, or an $A$-ordered $(T,S)$-matching of order $k$. *

Proof.

Denote $r_{j}=(x_{j}-1)m+y_{j}$ for $j=1,2,\ldots,4k$. For $i=1,2,\ldots,2k$, we define $a_{i}$ and $(b_{i},c_{i})$ as follows:

if $v_{r_{2i-1}}\in S$, then $a_{i}=r_{2i-1}$ and $(b_{i},c_{i})=(x_{2i},y_{2i})$, and

• 2.

if $v_{r_{2i-1}}\in T$, then $a_{i}=r_{2i-1}$ and $(b_{i},c_{i})=(x_{2i-1},y_{2i-1})$.

It follows that vertices $v_{a_{1}},\ldots,v_{a_{2k}}$ and $z_{(b_{1},c_{1})},\ldots,z_{(b_{2k},c_{2k})}$ satisfy

[TABLE]

and for each $i$ we have that $v_{a_{i}}$ and $z_{(b_{i},c_{i})}$ belong to different sides of the bipartition $(S,T)$. Now, if for at least $k$ indices $i$ we have $v_{a_{i}}\in S$ and $z_{(b_{i},c_{i})}\in T$, then the subsequence induced by elements with these indices gives an $A$-ordered $(S,T)$-matching of order $k$. Otherwise, there are at least $k$ indices $i$ with $v_{a_{i}}\in T$ and $z_{(b_{i},c_{i})}\in S$, and the subsequence induced by them gives an $A$-ordered $(T,S)$-matching of order $k$. $\lrcorner$

A symmetric proof gives the following.

Claim 2**.**

*If there is a sequence $(x_{1},y_{1})\prec_{2}(x_{2},y_{2})\prec_{2}\ldots\prec_{2}(x_{4k},y_{4k})$ such that $z_{(x_{j},y_{j})}\in S$ for odd $j$ and $z_{(x_{j},y_{j})}\in T$ for even $j$, then there is either a $B$-ordered $(S,T)$-matching of order $k$, or a $B$-ordered $(T,S)$-matching of order $k$. *

From 1 and 2 it follows that the largest possible length of sequences as in the statements is smaller than $4n$. For $i,j\in\{1,\ldots,m\}$, call the sets $\{z_{(i,y)}\colon y\in\{1,\ldots,m\}\}$ and $\{z_{(x,j)}\colon x\in\{1,\ldots,m\}\}$ the $i$th row and the $j$th column, respectively. A row or a column is mixed if it contains both elements of $S$ and elements of $T$. Observe that if there were at least $4n$ mixed rows, then by choosing vertices from $S$ and $T$ alternately from these rows we would obtain a sequence of length $4n$ as in the statement of 1. Then 1 gives us an $A$-ordered $(S,T)$-matching of order $n$ or an $A$-ordered $(T,S)$-matching of order $n$, a contradiction. Hence, there are less than $4n$ mixed rows, and symmetrically there are less than $4n$ mixed columns.

Observe that if there were two non-mixed rows such that the first one was contained in $S$ while the second was contained in $T$, then all the $12n$ columns would be mixed, a contradiction. Hence, either all the non-mixed rows belong to $S$, or all of them belong to $T$. However, there are more than $12n-4n=8n=2m/3$ non-mixed rows, so either $S$ or $T$ contains less than a third of vertices of $C$. This is a contradiction with the assumption that $(S,T)$ is balanced with respect to $C$. $\square$

We now observe that if a graph class contains arbitrarily large twisted chain graphs, then it does not admit low rank-width colorings.

Theorem 14**.**

*Let $\mathscr{C}$ be a hereditary graph class, and suppose for each positive integer $n$ some twisted chain graph of order $n$ belongs to $\mathscr{C}$. Then $\mathscr{C}$ does not admit low rank-width colorings. *

Proof.

We show that for every pair of integers $m\geq 3$ and $n\geq 1$, there is a graph $G\in\mathscr{C}$ such that for every coloring of $G$ with $m$ colors, there is an induced subgraph $H$ that receives at most $3$ colors and has rank-width at least $n$. This implies that $\mathscr{C}$ does not admit low rank-width colorings. We will need the following simple Ramsey-type argument. 3 follows, e.g., from [46, Theorem 11.5], but we give a simple proof for the sake of completeness.

Claim 3**.**

*For all positive integers $k,d$, there exists an integer $M=M(k,d)$ such that for all sets $X,Y$ with $|X|=|Y|=M$ and all functions $f\colon X\times Y\to\{1,\ldots,d\}$, there exist subsets $X^{\prime}\subseteq X$ and $Y^{\prime}\subseteq Y$ with $|X^{\prime}|=|Y^{\prime}|=k$ such that $f$ sends all elements of $X^{\prime}\times Y^{\prime}$ to the same value. *

Proof.

We prove the claim for $M(k,d)=k\cdot d^{dk}$. Let $X_{0}$ be an arbitrary subset $X$ of size $dk$. For each $y\in Y$, define the type of $y$ as the function $g_{y}\colon X_{0}\to\{1,\ldots,d\}$ defined as $g_{y}(x)=f(x,y)$. There are at most $d^{dk}$ different types possible, so there is a subset $Y^{\prime}\subseteq Y$ of size $k$ such that each element of $Y^{\prime}$ has the same type $g$. Since $|X_{0}|=dk$, there is some $i\in\{1,\ldots,d\}$ such that $g$ yields value $i$ for at least $k$ elements of $X_{0}$. Then if we take $X^{\prime}$ to be an arbitrary set of $k$ elements mapped to $i$ by $g$, then $f(x,y)=i$ for each $(x,y)\in X^{\prime}\times Y^{\prime}$, as required. $\lrcorner$

Let $M_{1}:=M(12n,m)$, $M_{2}:=M(M_{1},m)$, and $M_{3}:=M(M_{2},m)$. Let $G\in\cal C$ be a twisted chain graph of order $M_{3}$; adopt the notation from Definition 6 for $G$. Suppose $G$ is colored by $m$ colors by a coloring $c$. By 3, there exist $X_{1},Y_{1}\subseteq\{1,\ldots,M_{3}\}$ with $|X_{1}|=|Y_{1}|=M_{2}$ such that $\{z_{(x,y)}\colon(x,y)\in X_{1}\times Y_{1}\}$ is monochromatic under $c$.

Now, for an index $k\in\{1,\ldots,M_{3}^{2}\}$, let $(i_{1}(k),j_{1}(k))\in\{1,\ldots,m\}\times\{1,\ldots,m\}$ be the unique pair such that $k=(i_{1}(k)-1)M_{3}+j_{1}(k)$, and let $(i_{2}(k),j_{2}(k))\in\{1,\ldots,m\}\times\{1,\ldots,m\}$ be the unique pair such that $k=(j_{2}(k)-1)M_{3}+i_{2}(k)$. By reindexing vertices $A$ and $C$ using pairs $(i_{1}(k),j_{1}(k))$ and $(i_{2}(k),j_{2}(k))$, we may view coloring $c$ on $A$ and $C$ as a coloring on $\{1,\ldots,M_{3}\}\times\{1,\ldots,M_{3}\}$. By applying 3 to the vertices from $A$ indexed by $X_{1}\times Y_{1}$, we obtain subsets $X_{2}\subseteq X_{1}$ and $Y_{2}\subseteq Y_{1}$ such that $|X_{2}|=|Y_{2}|=M_{1}$ and the set $\{v_{(x-1)M_{3}+y}\colon x\in X_{2},y\in Y_{2}\}$ is monochromatic. Finally, by applying 3 to the vertices from $B$ indexed by $X_{2}\times Y_{2}$, we obtain subsets $X_{3}\subseteq X_{2}$ and $Y_{3}\subseteq Y_{2}$ such that $|X_{3}|=|Y_{3}|=12n$ and the set $\{w_{(y-1)M_{3}+x}\colon(x,y)\in X_{3}\times Y_{3}\}$ is monochromatic. Now observe that the subgraph $G[\{v_{(x-1)M_{3}+y},w_{(y-1)M_{3}+x},z_{(x,y)}\colon(x,y)\in X_{3}\times Y_{3}\}]$ receives at most $3$ colors, and is a twisted chain graph of order $12n$. By lemma 12 it has rank-width at least $n$, so this proves the claim. $\square$

We now observe that a twisted chain graph of order $n$ is an interval graph, provided each of $A$, $B$, and $C$ is a clique, and there are no edges between $A$ and $B$. Similarly, for each $n$ there is a twisted chain graph of order $n$ that is a permutation graph. See Figures 2 and 3 for examples of intersection models.

Lemma 15**.**

*Let $G$ be a twisted chain graph of order $n$, for some positive integer $n$. If each of $A$, $B$, and $C$ is a clique, and there are no edges between $A$ and $B$, then $G$ is an interval graph. *

Proof.

Let $M>2n^{2}$. Consider the following interval model:

For each $i\in\{1,\ldots,n^{2}\}$, assign interval $[0,i]$ to $v_{i}$.

• 2.

For each $i\in\{1,\ldots,n^{2}\}$, assign interval $[M-i,M]$ to $w_{i}$.

• 3.

For each $x,y\in\{1,\ldots,n\}\times\{1,\ldots,n\}$, assign interval $[(x-1)n+y,M-(y-1)n-x]$ to $z_{(x,y)}$.

It can be easily seen that this is an interval model of the twisted chain graph of order $n$. $\square$

Lemma 16**.**

*For each positive integer $n$, there is a twisted chain graph of order $n$ that is a permutation graph. *

Proof.

Let $M>10n^{2}$. Consider the following permutation model, spanned between horizontal lines $\ell_{1}$ with $y$-coordinate [math] and $\ell_{2}$ with $y$-coordinate $1$:

For each $i\in\{1,\ldots,n^{2}\}$, assign the segment with endpoints $(i,0)$ and $(i,1)$ to $v_{i}$.

• 2.

For each $i\in\{1,\ldots,n^{2}\}$, assign the segment with endpoints $(M-i,0)$ and $(M-i,1)$ to $w_{i}$.

• 3.

For each $x,y\in\{1,\ldots,n\}\times\{1,\ldots,n\}$, assign the segment with endpoints $((x-1)n+y,0)$ and $(M-(y-1)n-x,1)$ to $z_{(x,y)}$

It can be easily seen that this is a permutation model of some twisted chain graph of order $n$. $\square$

By theorem 14, we obtain the following.

Theorem 17**.**

*The classes of interval graphs and permutation graphs do not admit low rank-width colorings. *

6 Erdős-Hajnal property and $\chi$-boundedness

A hereditary graph class $\mathscr{C}$ has the Erdős-Hajnal property if there is $\varepsilon>0$, depending only on $\mathscr{C}$, such that every $n$-vertex graph in $\mathscr{C}$ has either an independent set or a clique of size $n^{\varepsilon}$. The conjecture of Erdős and Hajnal [20] states that for every fixed graph $H$, the class of graphs not having $H$ as an induced subgraph has the Erdős-Hajnal property; cf. [6]. We prove that every class admitting low rank-width colorings has the Erdős-Hajnal property. The proof is based on the fact that every class of graphs of bounded rank-width has this property, which was shown by Oum and Seymour [41]. Since this claim is not written in any published work, we include the proof for the completeness.

A graph is a cograph if it can be recursively constructed from isolated vertices by means of the following two operations: (1) taking disjoint union of two graphs and (2) joining two graphs, i.e., taking their disjoint union and adding all possible edges with one endpoint in the first graph and the second endpoint in the second. It is well-known that every $n$-vertex cograph contains either an independent set or a clique of size $n^{1/2}$; this follows from the fact that cographs are perfect.

Lemma 18** (Oum and Seymour [41]).**

*For every positive integer $p$, there exists a constant $\delta=\delta(p)$ such that every $n$-vertex graph of rank-width at most $p$ contains either an independent set or a clique of size at least $n^{\delta}$. *

Proof.

Let $\kappa(p):=\frac{1}{\log_{2}3+p}$ and $\delta(p):=\frac{\kappa(p)}{2}$. We first prove by induction on $n$ that every $n$-vertex graph of rank-width at most $p$ contains a cograph of size at least $n^{\kappa(p)}$. Assume $n\geq 3$, for otherwise every graph on at most $2$ vertices is a cograph. Let $G$ be an $n$-vertex graph of rank-width at most $p$. By lemma 13, $G$ has a vertex bipartition $(A,B)$ where $\mathrm{cutrk}_{G}(A)\leq p$ and $\left\lvert A\right\rvert,\left\lvert B\right\rvert\geq\frac{n}{3}$. Since $\mathrm{cutrk}_{G}(A)\leq p$, there exist $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ with $\left\lvert A^{\prime}\right\rvert,\left\lvert B^{\prime}\right\rvert\geq\frac{n}{3\cdot 2^{p}}$, such that either there are no edges between $A^{\prime}$ and $B^{\prime}$, or every vertex in $A^{\prime}$ is adjacent to every vertex in $B^{\prime}$. By induction hypothesis, $G[A^{\prime}]$ contains a cograph $H_{1}$ and $G[B^{\prime}]$ contains a cograph $H_{2}$, both as induced subgraphs, such that $\left\lvert V(H_{1})\right\rvert,\left\lvert V(H_{2})\right\rvert\geq(\frac{n}{3\cdot 2^{p}})^{\kappa(p)}$. Note that $G[V(H_{1})\cup V(H_{2})]$ is a cograph. Since $\kappa(p)=\frac{1}{\log_{2}3+p}$, we have $\left\lvert V(H_{1})\right\rvert+\left\lvert V(H_{2})\right\rvert\geq 2(\frac{n}{3\cdot 2^{p}})^{\kappa(p)}=n^{\kappa(p)}$. We conclude that $G$ contains either an independent set or a clique of size at least $(n^{\kappa(p)})^{1/2}=n^{\delta(p)}$. $\square$

Proposition 19**.**

*Every hereditary class of graphs that admits low rank-width colorings has the Erdős-Hajnal property. *

Proof.

Let $\mathscr{C}$ be the class of graphs in question. Fix any $G\in\mathscr{C}$, say on $n$ vertices. Since $\mathscr{C}$ admits low rank-width colorings, there exist functions $N\colon\mathbb{N}\rightarrow\mathbb{N}$ and $R\colon\mathbb{N}\rightarrow\mathbb{N}$, depending only on $\mathscr{C}$, such that for all $p$, $G$ can be colored using $N(p)$ colors so that each induced subgraph of $G$ that receives $i\leq p$ colors has rank-width at most $R(i)$. Let $c$ be such a coloring for $p=1$; then $c$ uses $N(1)$ colors. Let $\delta(p)$ be the function defined in lemma 18. We define

[TABLE]

and claim that $G$ has an independent set or a clique of size at least $n^{\varepsilon}$.

First, assume that $n\geq N(1)^{2}$. Then there is a color $i$ such that $|c^{-1}(i)|\geq\frac{n}{N(1)}$. Thus, the subgraph $H$ induced by the vertices with color $i$ has at least $\frac{n}{N(1)}$ vertices and rank-width at most $R(1)$. Since $n\geq N(1)^{2}$, by lemma 18, $H$, and thus also $G$, contains either an independent set or a clique of size at least

[TABLE]

Second, assume $n<N(1)^{2}$. Then $n^{\varepsilon}<N(1)^{2\varepsilon}\leq 2$, so any two-vertex induced subgraph of $G$ is either a clique or an independent set of size at least $n^{\varepsilon}$. $\square$

A hereditary class $\mathscr{C}$ of graphs is $\chi$-bounded if there exists a function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ such that for every $G\in\mathscr{C}$ and an induced subgraph $H$ of $G$, we have $\chi(H)\leq f(\omega(H))$, where $\chi(H)$ is the chromatic number of $H$ and $\omega(H)$ is the size of a maximum clique in $H$. It was proved by Dvořák and Král’ [19] that for every $p$, the class of graphs of rank-width at most $p$ is $\chi$-bounded.

Theorem 20** (Dvořák and

Král’ [19]).**

*For each positive integer $p$, the class of graphs of rank-width at most $p$ is $\chi$-bounded. *

We observe that this fact directly generalizes to classes admitting low rank-width colorings.

Proposition 21**.**

*Every hereditary class of graphs that admits low rank-width colorings is $\chi$-bounded. *

Proof.

Let $\mathscr{C}$ be the hereditary class of graphs in question, let $G\in\mathscr{C}$, and let $q=\omega(G)$. Since $\mathscr{C}$ admits low rank-width colorings, there exist functions $N\colon\mathbb{N}\rightarrow\mathbb{N}$ and $R\colon\mathbb{N}\rightarrow\mathbb{N}$, depending only on $\mathscr{C}$, such that for all $p$, $G$ can be colored using $N(p)$ colors so that each induced subgraph of $G$ that receives $i\leq p$ colors has rank-width at most $R(i)$. Let $c$ be such a coloring for $p=1$, and w.l.o.g. suppose that $c$ uses colors $\{1,\ldots,N(1)\}$. By theorem 20, there is function $f_{R(1)}$ such that for every graph $H$ of rank-width at most $R(1)$, we have $\chi(H)\leq f_{R(1)}(\omega(H))$.

For $i\in\{1,\ldots,N(1)\}$, let $G_{i}=G[c^{-1}(i)]$ be the subgraph induced by vertices of color $i$. Since $G_{i}$ is an induced subgraph of $G$, we have that $\omega(G_{i})\leq q$, so $G_{i}$ has a proper coloring $c_{i}$ using $f_{R(1)}(q)$ colors, say colors $\{1,\ldots,f(q)\}$. Then we can take the product coloring $c^{\prime}$ of $G$ defined as $c^{\prime}(u)=(c(u),c_{c(u)}(u))$. Observe that since each coloring $c_{i}$ is proper, $c^{\prime}$ is a proper coloring of $G$. It follows that the chromatic number of $G$ is at most $N(1)\cdot f(q)$, so we can take $f^{\prime}(q)=N(1)\cdot f_{R(1)}(q)$ as the $\chi$-bounding function for $\mathscr{C}$. $\square$

7 Conclusions

In this work we introduced the concept of low rank-width colorings, and showed that such colorings exist on $r$th powers of graphs from any bounded expansion class, for any fixed $r$, as well as on unit interval and bipartite permutation graphs. These classes are non-sparse and have unbounded rank-width. On the negative side, the classes of interval and permutation graphs do not admit low rank-width colorings.

The obvious open problem is to characterise hereditary graph classes which admit low rank-width colorings in the spirit of the characterisation theorem for graph classes admitting low tree-depth colorings. We believe that theorem 14 may provide some insight into this question, as it shows that containing arbitrarily large twisted chain graphs is an obstacle for admitting low rank-width colorings. Is it true that every hereditary graph class that does not admit low rank-width colorings has to contain arbitrarily large twisted chain graphs?

In this work we did not investigate the question of computing low rank-width colorings, and this question is of course crucial for any algorithmic applications. Our proof for the powers of sparse graphs can be turned into a polynomial-time algorithm that, given a graph $G$ from a graph class of bounded expansion $\mathscr{C}$, first computes a low tree-depth coloring, and then turns it into a low rank-width coloring of $G^{r}$, for a fixed constant $r$. However, we do not know how to efficiently compute a low rank-width coloring given the graph $G^{r}$ alone, without the knowledge of $G$. The even more general problem of efficiently constructing an approximate low rank-width coloring of any given graph remains wide open.

Finally, we remark that our proof for the existence of low rank-width colorings on powers of graphs from a class of bounded expansion actually yields a slightly stronger result. Ganian et al. [23] introduced a parameter shrub-depth (or SC-depth), which is a depth analogue of rank-width, in the same way as tree-depth is a depth analogue of tree-width. DeVos, Kwon and Oum [16] later introduced another parameter rank-depth which is tied to the shrub-depth. It can be shown that for constant $r$, the $r$th power of a graph of constant tree-depth belongs to a class of constant shrub-depth, and hence our colorings for powers of graphs from a class of bounded expansion are actually low shrub-depth colorings. In a follow-up of this work [22] we proved that in fact every first-order interpretation of a bounded expansion class (the $r$th power of a graph is a simple first-order interpretation) has low shrub-depth colorings. It remains an important open problem whether the $r$th powers, or more generally first-order interpretations, of nowhere dense graph classes for any $\varepsilon>0$ admit low shrub-depth colorings with $\mathcal{O}(n^{\varepsilon})$ colors.

Acknowledgment. We would like to thank Konrad Dabrowski for pointing out the known constructions similar to twisted chain graphs. We also thank the anonymous reviewers for careful reading and helpful advices to improve the original manuscript.