Exact distance coloring in trees

TL;DR

This paper determines the growth rate of the chromatic number for a class of graphs derived from trees with edges between vertices at a fixed distance, revealing it grows proportionally to d log q / log d.

Contribution

It proves the chromatic number of these graphs grows as Θ(d log q / log d), answering an open question about distance coloring in trees and planar graphs.

Findings

Chromatic number grows with d, specifically as Θ(d log q / log d)

Negative answer to a problem on coloring planar graphs at fixed distances

Provides sharp bounds for interval coloring in bounded degree trees

Abstract

For an integer $q\ge 2$ and an even integer $d$, consider the graph obtained from a large complete $q$-ary tree by connecting with an edge any two vertices at distance exactly $d$ in the tree. This graph has clique number $q+1$, and the purpose of this short note is to prove that its chromatic number is $\Theta\big(\tfrac{d \log q}{\log d}\big)$. It was not known that the chromatic number of this graph grows with $d$. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant $C$ such that for any odd integer $d$, any planar graph can be colored with at most $C$ colors such that any pair of vertices at distance exactly $d$ have distinct colors. Finally, we study interval coloring of trees (where vertices at distance at least $d$ and at most $cd$, for some real $c>1$, must be assigned distinct colors), giving a sharp upper bound in the case of bounded degree trees.

Figures

Equations

$$\frac{d\log (q-1)}{4\log (d/2)+4\log (q-1)}\le \chi (T_q,d)\le (2+o(1))\frac{d\log (q-1)}{\log d},$$

$$q(q-1{)}^{\lfloor cd/2\rfloor -\lfloor d/2\rfloor }\le \chi (T_q,[d,cd])\le (q-1{)}^{\lfloor cd/2+1\rfloor }(\lfloor cd\rfloor +1).$$

$$\frac{d\log (q-1)}{\log d}+\sqrt{\frac{d\log (q-1)}{\log d}}+\frac{d\log (q-1)}{\log d-\log \log d+\log \log (q-1)-\log (q-1)}+1=(2+o(1))\frac{d\log (q-1)}{\log d}$$

$$\chi (U_3^{d+1},d)\ge \frac{(d+1)\log (2)}{4\log ((d+1)/2)+4\log (2)}-1\text{mbox}and\chi (Q_3^{d+1},d)\ge {\log }_2(d+8)-3.$$

$$q(q-1{)}^{\lfloor cd/2\rfloor -\lfloor d/2\rfloor }\le \chi (T_q,[d,cd])\le (q-1{)}^{\lfloor cd/2+1\rfloor }(\lfloor cd\rfloor +1).$$

Full text

The authors were partially supported by ANR Projects STINT (anr-13-bs02-0007) and GATO (anr-16-ce40-0009-01) and LabEx PERSYVAL-Lab (anr-11-labx-0025-01) and LabEx CIMI

1. Introduction

Given a metric space $X$ and some real $d>0$, let $\chi(X,d)$ be the minimum number of colors in a coloring of the elements of $X$ such that any two elements at distance exactly $d$ in $X$ are assigned distinct colors. The classical Hadwiger-Nelson problem asks for the value of $\chi(\mathbb{R}^{2},1)$, where $\mathbb{R}^{2}$ is the Euclidean plane. It is known that $5\leq\chi(\mathbb{R}^{2},1)\leq 7$ [1] and since the Euclidean plane $\mathbb{R}^{2}$ is invariant under homothety, $\chi(\mathbb{R}^{2},1)=\chi(\mathbb{R}^{2},d)$ for any real $d>0$. Let $\mathbb{H}^{2}$ denote the hyperbolic plane. Kloeckner [3] proved that $\chi(\mathbb{H}^{2},d)$ is at most linear in $d$ (the multiplicative constant was recently improved by Parlier and Petit [6]), and observed that $\chi(\mathbb{H}^{2},d)\geq 4$ for any $d>0$. He raised the question of determining whether $\chi(\mathbb{H}^{2},d)$ grows with $d$ or can be bounded independently of $d$. As noticed by Kahle (see [3]), it is not known whether $\chi(\mathbb{H}^{2},d)\geq 5$ for some real $d>0$. Parlier and Petit [6] recently suggested to study infinite regular trees as a discrete analog of the hyperbolic plane. Note that any graph $G$ can be considered as a metric space (whose elements are the vertices of $G$ and whose metric is the graph distance in $G$), and in this context $\chi(G,d)$ is precisely the minimum number of colors in a vertex coloring of $G$ such that vertices at distance $d$ apart are assigned different colors. Note that $\chi(G,d)$ can be equivalently defined as the chromatic number of the exact $d$-th power of $G$, that is, the graph with the same vertex-set as $G$ in which two vertices are adjacent if and only if they are at distance exactly $d$ in $G$.

Let $T_{q}$ denote the infinite $q$-regular tree. Parlier and Petit [6] observed that when $d$ is odd, $\chi(T_{q},d)=2$ and proved that when $d$ is even, $q\leq\chi(T_{q},d)\leq(d+1)(q-1)$. A similar upper bound can also be deduced from the results of Van den Heuvel, Kierstead, and Quiroz [2], while the lower bound is a direct consequence of the fact that when $d$ is even, the clique number of the exact $d$-th power of $T_{q}$ is $q$ (note that it does not depend on $d$). In this short note, we prove that when $q\geq 3$ is fixed,

[TABLE]

where the asymptotic $o(1)$ is in terms of $d$. A simple consequence of our main result is that for any even integer $d$, the exact $d$-th power of a complete binary tree of depth $d$ is of order $\Theta(d/\log d)$ (while its clique number is equal to 3).

The following problem (attributed to Van den Heuvel and Naserasr) was raised in [4] (see also [2] and [5]).

Problem 1.1** (Problem 11.1 in [4]).**

Is there a constant $C$ such that for every odd integer $d$ and every planar graph $G$ we have $\chi(G,d)\leq C$?

We will show that our result on large complete binary trees easily implies a negative answer to Problem 1.1. More precisely, we will prove that the graph $U_{3}^{d}$ obtained from a complete binary tree of depth $d$ by adding an edge between any two vertices with the same parent gives a negative answer to Problem 1.1 (in particular, for odd $d$, the chromatic number of the exact $d$-th power of $U_{3}^{d}$ grows as $\Theta(d/\log d)$). We will also prove that the exact $d$-th power of a specific subgraph $Q_{3}^{d}$ of $U_{3}^{d}$ grows as $\Omega(\log d)$. Note that $U_{3}^{d}$ and $Q_{3}^{d}$ are outerplanar (and thus, planar) and chordal (see Figure 2).

Kloeckner [3] proposed the following variant of the original problem: For a metric space $X$, an integer $d$ and a real $c>1$, we denote by $\chi(X,[d,cd])$ the smallest number of colors in a coloring of the elements of $X$ such that any two elements of $X$ at distance at least $d$ and at most $cd$ apart have distinct colors. Considering as above the natural metric space defined by the infinite $q$-regular tree $T_{q}$, Parlier and Petit [6] proved that

[TABLE]

We will show that $\chi(T_{q},[d,cd])\leq\tfrac{q}{q-2}(q-1)^{\lfloor cd/2\rfloor-d/2+1}+cd+1$, which implies that the lower bound of Parlier and Petit [6] (which directly follows from a clique size argument) is asymptotically sharp.

2. Exact distance coloring

Throughout the paper, we assume that the infinite $q$-regular tree $T_{q}$ is rooted in some vertex $r$. This naturally defines the children and descendants of a vertex and the parent and ancestors of a vertex distinct from $r$. In particular, given a vertex $u$, we define the ancestors $u^{0},u^{1},\ldots$ of $u$ inductively as follows: $u^{0}=u$ and for any $i$ such that $u^{i}$ is not the root, $u^{i+1}$ is the parent of $u^{i}$. With this notation, $u^{d}$ can be equivalently defined as the ancestor of $u$ at distance $d$ from $u$ (if such a vertex exists). For a given vertex $u$ in $T_{q}$, the depth of $u$, denoted by $0pt(u)$, is the distance between $u$ and $r$ in $T_{q}$. For a vertex $v$ and an integer $\ell$, we define $L(v,\ell)$ as the set of descendants of $v$ at distance exactly $\ell$ from $v$ in $T_{q}$.

We first prove an upper bound on $\chi(T_{q},d)$.

Theorem 2.1**.**

For any integer $q\geq 3$, any even integer $d$, and any integer $k\geq 1$ such that $k(q-1)^{k-1}\leq d$, we have $\chi(T_{q},d)\leq(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}+\tfrac{d}{k}+1$. In particular, $\chi(T_{q},d)\leq d+q+1$, and when $q$ is fixed and $d$ tends to infinity, $\chi(T_{q},d)\leq(2+o(1))\tfrac{d\log(q-1)}{\log d}$.

Proof.

A vertex of $T_{q}$ distinct from $r$ and whose depth is a multiple of $k$ is said to be a special vertex. Let $v$ be a special vertex. Every special vertex $u$ distinct from $v$ such that $u^{k}=v^{k}$ is called a cousin of $v$. Note that $v$ has at most $q(q-1)^{k-1}-1$ cousins (at most $(q-1)^{k}-1$ if $v^{k}\neq r$). A special vertex $u$ is said to be a relative of $v$ if $u$ is either a cousin of $v$, or $u$ has the property that $u$ and $v^{k}$ have the same depth and are at distance at most $k$ apart in $T_{q}$. Two vertices $a,b$ at distance at most $k$ apart and at the same depth must satisfy $a^{\lfloor k/2\rfloor}=b^{\lfloor k/2\rfloor}$, and so the number of vertices $u$ such that $u$ and $v^{k}$ have the same depth and are at distance at most $k$ apart in $T_{q}$ is $(q-1)^{\lfloor k/2\rfloor}$. It follows that if $v^{k}=r$, then $v$ has at most $q(q-1)^{k-1}-1$ relatives and otherwise $v$ has at most $(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}-1$ relatives.

The first step is to define a coloring $C$ of the special vertices of $T_{q}$. This will be used later to define the desired coloring of $T_{q}$, i.e. a coloring such that vertices of $T_{q}$ at distance $d$ apart are assigned distinct colors (in this second coloring, the special vertices will not retain their color from $C$).

We greedily assign a color $C(v)$ to each special vertex $v$ of $T_{q}$ as follows: we consider the vertices of $T_{q}$ in a breadth-first search starting at $r$, and for each special vertex $v$ we encounter, we assign to $v$ a color distinct from the colors already assigned to its relatives, and from the set of ancestors $v^{ik}$ of $v$, where $2\leq i\leq\tfrac{d}{k}+1$ (there are at most $\tfrac{d}{k}$ such vertices). Note that if $v^{k}=r$, the number of colors forbidden for $v$ is at most $q(q-1)^{k-1}-1$ and if $v^{k}\neq r$ the number of colors forbidden for $v$ is at most $(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}+\tfrac{d}{k}-1$. Since $k(q-1)^{k-1}\leq d$, in both cases $v$ has at most $(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}+\tfrac{d}{k}-1$ forbidden colors, therefore we can obtain the coloring $C$ by using at most $(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}+\tfrac{d}{k}$ colors.

For any special vertex $v$, the set of descendants of $v$ at distance at least $d/2-k$ and at most $d/2-1$ from $v$ is denoted by $K(v,k)$. We now define the desired coloring of $T_{q}$ as follows: for each special vertex $v$, all the vertices of $K(v,k)$ are assigned the color $C(v)$. Finally, all the vertices at distance at most $d/2-1$ from $r$ are colored with a single new color (note that any two vertices in this set lie at distance less than $d$ apart). The resulting vertex-coloring of $T_{q}$ is called $c$. Note that $c$ uses at most $(q-1)^{k}+(q-1)^{\lfloor k/2\rfloor}+\tfrac{d}{k}+1$ colors, and indeed every vertex of $T_{q}$ gets exactly one color.

We now prove that vertices at distance $d$ apart in $T_{q}$ are assigned distinct colors in $c$. Assume for the sake of contradiction that two vertices $x$ and $y$ at distance $d$ apart were assigned the same color. Then the depth of both $x$ and $y$ is at least $d/2$. We can assume by symmetry that the difference $t$ between the depth of $x$ and the depth of $y$ is such that $0\leq t\leq d$ since otherwise they would be at distance more than $d$. Let $u$ be the unique (special) vertex of $T_{q}$ such that $x\in K(u,k)$ and $v$ be the unique (special) vertex such that $y\in K(v,k)$. By the definition of our coloring $c$, we have $C(u)=C(v)$. Note that $u$ and $v$ are distinct; indeed, otherwise $x$ and $y$ would not be at distance $d$ in $T_{q}$. Assume first that $u$ and $v$ have the same depth. Then since $u$ and $x$ (resp. $v$ and $y$) are distance at least $d/2-k$ apart, $u$ and $v$ are cousins (and thus, relatives), which contradicts the definition of the vertex-coloring $C$. We may, therefore, assume that the depths of $u$ and $v$ are distinct. Moreover, since $u$ and $v$ are special vertices, we may assume that their depths differ by at least $k$. In particular, $u$ lies deeper than $v$ in $T_{q}$.

First assume that the depths of $u$ and $v$ differ by at least $2k$. Then $v$ is not an ancestor of $u$ in $T_{q}$. Indeed, for otherwise we would have $v=u^{ik}$ for some integer $2\leq i\leq\tfrac{d}{k}+1$, which would contradict the definition of $C$. This implies that the distance between $x$ and $y$ is at least $d/2-k+d/2-k+2k+2=d+2$, which is a contradiction. Therefore, we can assume that the depths of $u$ and $v$ differ by precisely $k$. Since $v$ is not a relative of $u$, we have that $v\neq u^{k}$ and the distance between $u^{k}$ and $v$ is more than $k$. Moreover, since $u$ and $x$ (resp. $v$ and $y$) are at distance at least $d/2-k$ apart, this implies that the distance between $x$ and $y$ is more than $d/2-k+k+k+d/2-k=d$, a contradiction.

Thus, $c$ is a proper coloring.

By taking $k=1$ we obtain a coloring $c$ using at most $(q-1)^{1}+(q-1)^{\lfloor 1/2\rfloor}+\tfrac{d}{1}+1=q+d+1$ colors, and by taking $k=\lfloor\tfrac{\log d-\log\log d+\log\log(q-1)}{\log(q-1)}\rfloor$, we obtain a coloring $c$ using at most

[TABLE]

colors. ∎

For $k=1$, the proof above can be optimized to show that $\chi(T_{q},d)\leq q+\tfrac{d}{2}$ (by simply noting that vertices at even depth and vertices at odd depth can be colored independently). Since we are mostly interested in the asymptotic behaviour of $\chi(T_{q},d)$ (which is of order O\big{(}\tfrac{d}{\log d}\big{)}), we omit the details.

We now prove a simple lower bound on $\chi(T_{q},d)$. Let $T_{q}^{d}$ be the rooted complete $(q-1)$-ary tree of depth $d$, with root $r$. Note that each node has $q-1$ children, so this graph is a subtree of $T_{q}$.

Theorem 2.2**.**

For any integer $q\geq 3$ and any even $d$, $\chi(T_{q}^{d},d)\geq\log_{2}(\tfrac{d}{4}+q-1)$.

Proof.

Consider any coloring of $T_{q}^{d}$ with colors $1,2,\ldots,C$, such that vertices at distance precisely $d$ apart have distinct colors. For any vertex $v$ at depth at most $\tfrac{d}{2}+1$ in $T_{q}^{d}$, the set of colors appearing in $L(v,\tfrac{d}{2}-1)$ is denoted by $S_{v}$. Observe that if $v$ and $w$ have the same parent, then $S_{v}$ and $S_{w}$ are disjoint since for any $x\in L(v,\tfrac{d}{2}-1)$ and $y\in L(w,\tfrac{d}{2}-1)$, $x$ and $y$ are at distance $d$.

Fix some vertex $u$ at depth at most $\tfrac{d}{2}$ in $T_{q}^{d}$ and some child $v$ of $u$. We claim that:

Claim 2.3**.**

For any integer $1\leq k\leq\tfrac{0pt(u)}{2}$, there is a color of $S_{u^{2k-1}}$ that does not appear in $S_{v}$.

To see that Claim 2.3 holds, observe that in the subtree of $T_{q}^{d}$ rooted in $u^{k}$, there is a vertex of $L(u^{2k-1},\tfrac{d}{2}-1)$ at distance $d$ from all the elements of $L(v,\tfrac{d}{2}-1)$. The color of such a vertex does not appear in $S_{v}$, therefore Claim 2.3 holds.

In particular, Claim 2.3 implies that all the sets $\{S_{u^{2k-1}}\,|\,1\leq k\leq d/4\}$ and $\{S_{w}\,|\,w\mbox{ is a child of }u\}$ are pairwise distinct. Since there are $\tfrac{d}{4}+q-1$ such sets, we have $\tfrac{d}{4}+q-1\leq 2^{C}$ and therefore $C\geq\log_{2}(\tfrac{d}{4}+q-1)$, as desired. ∎

It was observed by Stéphan Thomassé that the proof of Theorem 2.2 only uses a small fraction of the graph $T_{q}^{d}$. Consider for simplicity the case $q=3$, and define $P_{3}^{d}$ as the graph obtained from a path $P=v_{0},v_{1},\ldots,v_{d}$ on $d$ edges, by adding, for each $1\leq i\leq d$, a path on $i$ edges ending at $v_{i}$ (see Figure 1). This graph is an induced subgraph of $T_{q}^{d}$ and the proof of Theorem 2.2 directly shows the following111Stéphan Thomassé noticed that this can also be deduced from the fact that the vertices at depth at least $\tfrac{d}{2}$ and at most $d$ in the exact $d$-th power of $P_{3}^{d}$ induce a shift graph..

Corollary 2.4**.**

For any even integer $d$, $\chi(P_{3}^{d},d)\geq\log_{2}(d+8)-2$.

The proof of Theorem 2.2 can be refined to prove the following better estimate for $T_{q}^{d}$, showing that the upper bound of Theorem 2.1 is (asymptotically) tight within a constant multiplicative factor of 8.

Theorem 2.5**.**

For any integer $q\geq 3$ and every even integer $d\geq 2$, $\chi(T_{q}^{d},d)\geq\tfrac{d\log(q-1)}{4\log(d/2)+4\log(q-1)}$.

Proof.

Consider any coloring of $T_{q}^{d}$ with colors $1,2,\ldots,C$, such that vertices at distance precisely $d$ apart have distinct colors. We perform a random walk $v_{0},v_{1},\ldots,v_{d}$ in $T_{q}^{d}$ as follows: we start with $v_{0}=r$, and for each $i\geq 1$, we choose a child of $v_{i}$ uniformly at random and set it as $v_{i+1}$. Note that the depth of each vertex $v_{i}$ is precisely $i$.

From now on we fix a color $c\in\{1,\ldots,C\}$. For any vertex $v$ of $T_{q}^{d}$, the set of vertices contained in the subtree of $T_{q}^{d}$ rooted in $v$ is denoted by $V_{v}$, and we set $A_{v}=\{\mbox{depth}(u)\,|\,u\in V_{v}\mbox{ and }u\mbox{ has color }c\}$. When $v=v_{i}$, for some integer $0\leq i\leq d$, we write $A_{i}$ instead of $A_{v_{i}}$.

Claim 2.6**.**

Assume that for some even integers $i$ and $j$ with $2\leq i<j\leq d$, and for some vertex $v$ at depth $\tfrac{i+j-d}{2}$, the set $A_{v}$ contains both $i$ and $j$. Then $v$ has precisely one child $u$ such that $A_{u}$ contains $i$ and $j$, and moreover all the children $w$ of $v$ distinct from $u$ are such that $A_{w}$ contains neither $i$ nor $j$.

To see that Claim 2.6 holds, simply note that $\tfrac{i+j-d}{2}<i<j$ and if two vertices $u_{1},u_{2}$ colored $c$ are respectively at depths $i$ and $j$, and their common ancestor is $v$, then they are at distance $d$ in $T_{q}^{d}$ (which contradicts the fact that they were assigned the same color). Indeed, the distance of $u_{1}$ to $v$ is $i-\tfrac{i+j-d}{2}$ and the distance of $u_{2}$ to $v$ is $j-\tfrac{i+j-d}{2}$. This proves the claim.

We now define a family of graphs $(G_{k})_{0\leq k\leq d/2}$ as follows. For any $0\leq k\leq\tfrac{d}{2}$, the vertex-set $V(G_{k})$ of $G_{k}$ is the set $A_{k}\cap 2\mathbb{N}\cap(d/2,d]$, and two (distinct) even integers $i,j\in A_{k}$ are adjacent in $G_{k}$ if and only if $\tfrac{i+j-d}{2}<k$. For each $0\leq k\leq\tfrac{d}{2}$ we define the energy $\mathcal{E}_{k}$ of $G_{k}$ as follows: $\mathcal{E}_{k}=\sum_{i\in V(G_{k})}(q-1)^{\text{deg}(i)}$, where $\text{deg}(i)$ denotes the degree of the vertex $i$ in $G_{k}$.

Note that each graph $G_{k}$ depends on the (random) choice of $v_{1},v_{2},\ldots,v_{k}$.

Claim 2.7**.**

For any $0\leq k\leq\tfrac{d}{2}-1$, $\mathbb{E}(\mathcal{E}_{k+1})\leq\mathbb{E}(\mathcal{E}_{k})$.

Assume that $v_{1},v_{2},\ldots,v_{k}$ (and therefore also $G_{k}$) are fixed. Observe that $G_{k+1}$ is obtained from $G_{k}$ by possibly removing some vertices and adding some edges. Thus, $\mathcal{E}_{k+1}$ can be larger than $\mathcal{E}_{k}$ only if $G_{k+1}$ contains edges that are not in $G_{k}$. Therefore, it suffices to consider the contributions of those pairs of nonadjacent vertices in $G_{k}$ which could become adjacent in $G_{k+1}$ (since these correspond to pairs $i,j$ with $k=\tfrac{i+j-d}{2}$, these pairs are pairwise disjoint), and prove that these contributions are, in expectation, equal to 0. Fix a pair of even integers $i<j$ in $V(G_{k})$ with $k=\tfrac{i+j-d}{2}$ (and note that $i$ and $j$ are not adjacent in $G_{k}$). By Claim 2.6, either $v_{k+1}$ is such that $A_{k+1}$ contains $i$ and $j$ (this event occurs with probability $\tfrac{1}{q-1}$), or $A_{k+1}$ contains neither $i$ nor $j$ (with probability $1-\tfrac{1}{q-1}$). As a consequence, for any $i<j$ in $V(G_{k})$ with $k=\tfrac{i+j-d}{2}$, with probability $\tfrac{1}{q-1}$ we add the edge $ij$ in $G_{k+1}$ and with probability $1-\tfrac{1}{q-1}$ we remove vertices $i$ and $j$ from $G_{k+1}$. This implies that for any $i,j\in V(G_{k})$, $i<j$, with $k=\tfrac{i+j-d}{2}$, with probability $\tfrac{1}{q-1}$ we have contribution at most $(q-1)^{\text{deg}(i)+1}+(q-1)^{\text{deg}(j)+1}-(q-1)^{\text{deg}(i)}-(q-1)^{\text{deg}(j)}=(q-2)((q-1)^{\text{deg}(i)}+(q-1)^{\text{deg}(j)})$ to $\mathcal{E}_{k+1}$ (where deg refers to the degree in $G_{k}$) and with probability $1-\tfrac{1}{q-1}$ we have a contribution of at most $-(q-1)^{\text{deg}(i)}-(q-1)^{\text{deg}(j)}$ to $\mathcal{E}_{k+1}$. Thus, the expected contribution of such a pair $i,j$ is at most $\tfrac{1}{q-1}(q-2)((q-1)^{\text{deg}(i)}+(q-1)^{\text{deg}(j)})-\tfrac{q-2}{q-1}((q-1)^{\text{deg}(i)}+(q-1)^{\text{deg}(j)})=0.$

Summing over all such pairs $i,j$, we obtain $\mathbb{E}(\mathcal{E}_{k+1})\leq\mathbb{E}(\mathcal{E}_{k}).$ This proves Claim 2.7.

Since $2\leq i<j\leq d$, we have $\tfrac{i+j-d}{2}\leq\tfrac{d}{2}-1$, and in particular it follows that $G_{d/2}$ is a (possibly empty) complete graph, whose number of vertices is denoted by $\omega\geq 0$. Note that the energy $\mathcal{E}$ of a complete graph on $\omega$ vertices is equal to $\omega(q-1)^{\omega-1}$, while the energy $\mathcal{E}_{0}$ of $G_{0}$ is equal to $|A_{0}\cap 2\mathbb{N}\cap(d/2,d]|\leq\tfrac{d}{4}$. For a vertex $u\in L(r,\tfrac{d}{2})$, let $\omega_{u}=|A_{u}\cap 2\mathbb{N}\cap(d/2,d]|$ (this is the number of distinct even depths at which a vertex colored $c$ appears in the subtree of height $\tfrac{d}{2}$ rooted in $u$). It follows from Claim 2.7 that the average of $\omega_{u}(q-1)^{\omega_{u}-1}$, over all vertices $u\in L(r,\tfrac{d}{2})$, is at most $\tfrac{d}{4}$. Let $a$ be the average of $\omega_{u}$, over all vertices $u\in L(r,\tfrac{d}{2})$. By Jensen’s inequality and the convexity of the function $x\mapsto x(q-1)^{x-1}$ for $x\geq 0$, we have that $a(q-1)^{a-1}\leq\tfrac{d}{4}$, and thus $a\leq\tfrac{\log(d/2)}{\log(q-1)}+1$.

Note that $a$ depends on the color $c$ under consideration (to make this more explicit, let us now write $a_{c}$ instead of $a$). Since there are $\tfrac{d}{4}$ even depths between depth $\tfrac{d}{2}$ and depth $d$, there is a color $c\in\{1,\ldots,C\}$ such that $a_{c}\cdot C\geq\tfrac{d}{4}$ and thus, $C\geq\tfrac{d}{4a_{c}}\geq\tfrac{d\log(q-1)}{4\log(d/2)+4\log(q-1)}$, as desired. ∎

We now explain how the results proved above give a negative answer to Problem 1.1. Let $U_{3}^{d}$ (resp. $Q_{3}^{d}$) be obtained from $T_{3}^{d}$ (resp. $P_{3}^{d}$) by adding an edge $uv$ for any pair of vertices $u,v$ having the same parent. Note that for any $d$, $U_{3}^{d}$ and $Q_{3}^{d}$ are outerplanar (and thus, planar) and chordal, and $Q_{3}^{d}$ has pathwidth 2 ($U_{3}^{3}$ and $Q_{3}^{5}$ are depicted in Figure 2) and the original copies of $T_{3}^{d}$ and $P_{3}^{d}$ are spanning trees of $U_{3}^{d}$ and $Q_{3}^{d}$, respectively. In the remainder of this section, whenever we write $T_{3}^{d}$, we mean the original copy of $T_{3}^{d}$ in $U_{3}^{d}$.

Observe that for any two vertices $u$ and $v$ distinct from the root of $T_{3}^{d}$, $u$ and $v$ are at distance $d$ in $T_{3}^{d}$ if and only if they are at distance $d-1$ in $U_{3}^{d}$ (since the depth of $T_{3}^{d}$ is $d$, the fact that $u$ and $v$ differ from the root and are at distance $d$ apart implies that none of the two vertices is an ancestor of the other). The same property holds for $Q_{3}^{d}$ and $P_{3}^{d}$. As a consequence, for any odd integer $d$, $\chi(U_{3}^{d+1},d)$ and $\chi(T_{3}^{d+1},d+1)$ differ by at most one, and $\chi(Q_{3}^{d+1},d)$ and $\chi(P_{3}^{d+1},d+1)$ also differ by at most one. Using this observation, we immediately obtain the following corollary of Theorem 2.5 and Corollary 2.4, which gives a negative answer to Problem 1.1.

Corollary 2.8**.**

For any odd integer $d$,

[TABLE]

The graphs $U_{3}^{d+1}$ and its exact $d$-th power have $n=2^{d+2}$ vertices, and thus the chromatic number of the exact $d$-th power of $U_{3}^{d+1}$ grows as \Omega\big{(}\tfrac{\log n}{\log\log n}\big{)}. The graphs $Q_{3}^{d+1}$ and its exact $d$-th power have $n={d+2\choose 2}$ vertices, and thus the chromatic number of the exact $d$-th power of $Q_{3}^{d+1}$ grows as $\Omega({\log n})$. It is not difficult (using Theorem 2.1 for $U_{3}^{d+1}$) to show that these bounds are asymptotically tight.

It was recently proved by Quiroz [8] that if $G$ is a chordal graph of clique number at most $t\geq 2$, and $d$ is an odd number, then $\chi(G,d)\leq{t\choose 2}(d+1)$. By Corollary 2.8, the graph $U_{3}^{d}$ shows that this is asymptotically best possible (as $d$ tends to infinity), up to a $\log d$ factor.

3. Interval coloring

For an integer $d$ and a real $c>1$, recall that $\chi(T_{q},[d,cd])$ denotes the smallest number of colors in a coloring of the vertices of $T_{q}$ such that any two vertices of $T_{q}$ at distance at least $d$ and at most $cd$ apart have distinct colors. Parlier and Petit [6] proved that

[TABLE]

In this final section, we prove that their lower bound (which is proved by finding a set of vertices of this cardinality that are pairwise at distance at least $d$ and at most $cd$ apart in $T_{q}$) is asymptotically tight.

Theorem 3.1**.**

For any integers $q\geq 3$ and $d$ and any real $c>1$, $\chi(T_{q},[d,cd])\leq\tfrac{q}{q-2}(q-1)^{\lfloor cd/2\rfloor-d/2+1}+cd+1$.

Proof.

The proof is similar to the proof of Theorem 2.1. We consider any ordering $e_{1},e_{2},\ldots$ of the edges of $T_{q}$ obtained from a breadth-first search starting at $r$. Then, for any $i=1,2,\ldots$ in order, we assign a color $c(e_{i})$ to the edge $e_{i}$ as follows. Let $e_{i}=uv$, with $u$ being the parent of $v$, and let $\ell=\lfloor cd/2\rfloor-d/2$. We assign to $uv$ a color $c(uv)$ distinct from the colors of all the edges $xy$ (with $x$ being the parent of $y$) such that $x$ is at distance at most $\ell$ from $u^{k}$ (where $k$ is the minimum of $\ell$ and the depth of $u$), or $x$ is an ancestor of $u$ at distance at most $cd$ from $u$ (and $y$ lies on the path from $u$ to $x$). There are at most $cd+\sum_{j=0}^{\ell}q(q-1)^{j}\leq\tfrac{q}{q-2}(q-1)^{\ell+1}+d-1$ such edges, so we can color all the edges following this procedure by using a total of at most $\tfrac{q}{q-2}(q-1)^{\ell+1}+cd$ colors.

As in the proof of Theorem 2.1, we now define our coloring of the vertices of $T_{q}$ as follows: first color all the vertices at distance at most $\tfrac{d}{2}-1$ from $r$ with a new color that does not appear on any edge of $T_{q}$, then for each vertex $v$ with parent $u$, we color all the vertices of $L(v,\tfrac{d}{2}-1)$ with color $c(uv)$. In this vertex-coloring, at most $\tfrac{q}{q-2}(q-1)^{\ell+1}+cd+1$ colors are used.

Assume that two vertices $s$ and $t$, at distance at least $d$ and at most $cd$ apart, were assigned the same color. This implies that $c(s^{d/2-1}s^{d/2})=c(t^{d/2-1}t^{d/2})$. Assume without loss of generality that the depth of $s$ is at least the depth of $t$, and consider first the case where $t^{d/2-1}$ is an ancestor of $s$. Then $t^{d/2}$ is an ancestor of $s^{d/2}$ at distance at most $cd$ from $s^{d/2}$ (and $t^{d/2-1}$ lies on the path from $s^{d/2}$ to $t^{d/2}$), which contradicts the definition of our edge-coloring $c$. Thus, we can assume that $t^{d/2-1}$ is not an ancestor of $s$. This implies that $t^{d/2-1}t^{d/2}$ lies on the path between $s$ and $t$, and therefore $t^{d/2}$ is at distance at most $\ell=\lfloor cd/2\rfloor-d/2$ from the ancestor of $s^{d/2}$ at distance $\ell$ from $s^{d/2}$ (or simply from $r$, if the depth of $s^{d/2}$ is at most $\ell$). Again, this contradicts the definition of our coloring $c$. We obtained a coloring of the vertices of $T_{q}$ with at most $\tfrac{q}{q-2}(q-1)^{\ell+1}+cd+1$ colors in which each pair of vertices at distance at least $d$ and at most $cd$ apart have distinct colors, as desired. ∎

Acknowledgement

We are very grateful to Lucas Pastor, Stéphan Thomassé, and an anonymous reviewer for their excellent observations and comments.