# The χ-Binding Function of *d*-Directional Segment Graphs

Lech Duraj, Ross Kang, Xuan Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald

► To cite this version:

Lech Duraj, Ross Kang, Xuan Hoang La, Jonathan Narboni, Filip Pokrývka, et al. The χ-Binding Function of *d*-Directional Segment Graphs. Discrete and Computational Geometry, 2025, 74 (3), pp.758-770. 10.1007/s00454-025-00737-2. hal-05406159

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Submitted on 9 Dec 2025

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# The $\chi$-binding function of $d$-directional segment graphs

Lech Duraj*

Ross J. Kang†

Hoang La‡

Jonathan Narboni*

Filip Pokrývka§

Clément Rambaud¶

Amadeus Reinald||

February 6, 2025

## Abstract

Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in $\mathbb{R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $\omega$ that the chromatic number $\chi(G)$ of $G$ is at most $d\omega$. We show for every even value of $\omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the $\chi$-binding function of $d$-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega - 1) + 1$ for $\omega$ odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case $d=2$.

## 1 Introduction

In structural graph theory, a fundamental task is to characterise the complex, global parameter of chromatic number $\chi$ in terms of the simpler, more local parameter of clique number $\omega$. For a given graph class, the question of whether this task is even well-defined belongs to the theory of $\chi$-boundedness, an area systematically initiated by Gyárfás in the 1980s [Gyá87]. A graph class $\mathcal{G}$ is called $\chi$-bounded if there is some function $f_{\mathcal{G}}$ such that $\chi(G) \le f_{\mathcal{G}}(\omega(G))$ for any $G \in \mathcal{G}$, and, if that is so, the optimal choice of $f_{\mathcal{G}}$ is called the $\chi$-binding function of $\mathcal{G}$. Note that the triangle-free graphs of arbitrarily large chromatic number give rise to many interesting graph classes that are not $\chi$-bounded. This area of mathematics is deep and active, but despite many recent advances, there are many important classes for which the question of $\chi$-boundedness is difficult and remains open; for a nice recent account of the state of the art, see [SS20]. Suffice it to say that determination of the $\chi$-binding function for a nontrivial (non-perfect) class is generally considered a rarity.

Since even before Gyárfás’s early work (see [AG60]), much attention has been focused on intersection classes. These are graph classes defined by taking some natural collection of sets, usually geometrically-defined, and forming for every finite subcollection of those sets an auxiliary graph in which each vertex corresponds to a set, and two vertices are connected by an edge if and only if the corresponding sets have a nontrivial intersection. Such a restriction of focus is not too confining, not only because many intersection classes are fundamental to the structural understanding of

<footnote>*Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland. Partially supported by National Science Center of Poland grant 2019/34/E/ST6/00443.</footnote>

<footnote>†Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands. Partially supported by a Vidi grant (639.032.614) of the Dutch Research Council (NWO) and the Gravitation programme NETWORKS (024.002.003) of the Dutch Ministry of Education, Culture and Science (OCW). r.kang@uva.nl</footnote>

<footnote>‡Theoretical Computer Science Department, Jagiellonian University, Kraków, Poland. Partially supported by a Polish National Science Center grant BEETHOVEN; UMO-2018/31/G/ST1/03718. hoang.la.research@gmail.com</footnote>

<footnote>§Faculty of Informatics, Masaryk University, Brno, Czech Republic. xpokryvk@fi.muni.cz</footnote>

<footnote>¶Université Côte d'Azur, CNRS, Inria, I3S, Sophia-Antipolis, France. Supported by the ANR project DIGRAPHS (ANR-19-CE48-0013). clement.rambaud@inria.fr</footnote>

<footnote>||LIRMM, CNRS, Université de Montpellier, France. Supported by the ANR project DIGRAPHS (ANR-19-CE48-0013). amadeus.reinald@lirmm.fr</footnote>
graphs (consult, e.g. [MM99, Spi03] for more context), but also because intersection classes are more than rich enough for its $\chi$-boundedness theory to contain many fascinating challenges.

As one prominent (and pertinent) example, consider the collection of straight line segments in the plane $\mathbb{R}^2$ (or more formally, the collection of those closed intervals drawn between some pairs of points in $\mathbb{R}^2$). Its intersection class is called the segment (intersection) graphs. If all the segments happen to lie in parallel, then the resulting segment graph is an interval graph, and thus perfect; that is, it has equality between $\chi$ and $\omega$. One should not expect this to remain true when we allow the segment slopes to vary, but it is reasonable to ask if there might remain a good relationship between $\chi$ and $\omega$, that is, are segment graphs $\chi$-bounded? Indeed, Erdős asked this in the 1970s (see [Gyá87, PKK$^+$14] for more details of the problem's provenance).

Despite sustained attention and nice partial results (see e.g. [KN95, McG00, Suk14]), Erdős's innocent-looking but very difficult challenge was not resolved until 2014, and in the negative. Work of Pawlik, Kozik, Krawczyk, Lasoń, Micek, Trotter, and Walczak [PKK$^+$14] provided a strikingly elegant construction of triangle-free segment graphs of arbitrarily large chromatic number. Interestingly, the graphs produced by the construction are isomorphic to the intersection graphs that were exhibited by Burling [Bur65] to show that the intersection class for axis-aligned boxes in $\mathbb{R}^3$ is not $\chi$-bounded.

One might consider this a coda, but permit us to prolong the narrative, in particular by parameterising according to the number of segment slopes. We first note that the construction of Pawlik et al. must have arbitrarily many slopes. Suppose to the contrary that the segments admit at most $d$ slopes. By the same observation as above, each slope induces a perfect graph $H$ having $\chi(H)$ equal to $\omega(H) \le 2$. By colouring the slopes disjointly from one another, we can conclude that $\chi$ is at most $2d$ for the entire graph, a contradiction.

More generally, given a positive integer $d$, we refer to the $d$-directional segment graphs as those intersection graphs formed from a collection of line segments in $\mathbb{R}^2$ having at most $d$ distinct slopes. Call the class of such graphs $d$-DIR. (Take note of the subtlety that this is not *per se* an intersection class, but rather a union of many of them.) What we just argued is that $\chi(G) \le d\omega(G)$ for any $G$ in $d$-DIR. Thus, in a simple fashion, we can conclude that the graph class $d$-DIR is $\chi$-bounded.

The question we would like to address here is, what is the $\chi$-binding function of $d$-DIR? This question was raised first around the turn of the century by Kostochka and Nešetřil [KN02] and again more recently by Bhattacharya, Dvořák and Noorizadeh [BDN23]. In both works, it was proved that there are 2-directional segment graphs of clique number $2t$ and chromatic number $4t$, for all $t \in \mathbb{N}$, which attains the bound of the above simple argument in this special case. Bhattacharya et al. moreover conjectured that that simple bound is optimal in general, that is, that $\omega \mapsto d\omega$ is the $\chi$-binding function of $d$-DIR for all $d > 2$.

We succeeded in completely resolving this question as follows.

**Theorem 1.** The $\chi$-binding function of $d$-DIR is

$$ \omega \mapsto \begin{cases} d\omega & \text{if } \omega \text{ is even, or} \\ d(\omega - 1) + 1 & \text{if } \omega \text{ is odd.} \end{cases} $$

One can interpret this statement as a graceful generalisation of perfection in the $d=1$ case. We note that Kostochka and Nešetřil [KN02] had already established the $d=2$ case of Theorem 1, unbeknownst to the authors in [BDN23]. This had already refuted the conjecture in [BDN23] in this special case.

The crux in Theorem 1 is to prove the lower bound in the even $\omega$ case. Here is a rough sketch of that construction, which at a high level combines aspects of both the construction of Pawlik et al. and that of Bhattacharya et al., and indeed strengthens/refines both constructions. We start by fixing $\omega = 2$ and proceed by induction on $d$, exhibiting triangle-free graphs with $t$-fold chromatic number at least $2td$. We refine the induction used by Pawlik et al. by controlling, at every step, the growth of the number of slopes required for achieving a given $t$-fold chromatic number. We then obtain the result for all even $\omega > 2$ by blowing up each segment by a factor $\omega/2$.
The structure of the paper is as follows. In Section 2, we lay the ground for our proof by introducing the notions and tools used throughout the paper. Then we settle the even $\omega$ case in Section 3. The lower bound for the odd $\omega$ case in Section 4 is a mild adaptation of the even $\omega$ construction. We give a short proof of the upper bound for the odd $\omega$ case in Section 5. We end with some open questions for future research in Section 6.

# 2 Preliminaries

## 2.1 t-fold colourings

For all positive integers $a$ and $t$, a *t-fold a-colouring* of a graph $G$ is a function $\varphi: V(G) \to \binom{\{1,\dots,a\}}{t}$ such that for every edge $uv \in E(G)$, $\varphi(u) \cap \varphi(v) = \emptyset$. For $t=1$, we will often refer to $\varphi$ simply as an *a-colouring*. For any $t$-fold colouring $\varphi$ of a graph $G$, if $X$ is a set of vertices of $G$, we denote by $\varphi(X)$ the set of colours used to colour the vertices in $X$, i.e. $\varphi(X) = \bigcup_{v \in X} \varphi(v)$. We will also use the notation $\varphi|_X$ to denote the restriction of the function $\varphi$ to the set $X$.

## 2.2 Segments

A *segment* $S$ with endpoints $A = (x_1, y_1)$ and $B = (x_2, y_2)$ in $\mathbb{R}^2$ is the set $\{A + \lambda(B - A) \mid \lambda \in [0, 1]\}$. The *length of a segment* is its Euclidean norm. A *rectangle* $R$ is defined as the Cartesian product of two segments $[a, c] \times [b, d]$ with $a, b, c, d \in \mathbb{R}$ and $c > a$ and $d > b$. A rectangle always has non-zero area. If the two intervals have the same length, we say that $R$ is a *square*. We naturally define the *left*, *right*, *top* and *bottom sides* of a rectangle $R$ as the four segments whose union is the boundary of $R$. The left and right side of $R$ are the vertical sides of $R$, and the top and bottom sides of $R$ are the horizontal sides of $R$. The length of a vertical side is the *height* of $R$, the length of a horizontal side is the *width* of $R$. The aspect ratio of $R$ is the height divided by the width of $R$. A segment $S$ crosses a rectangle $R$ vertically or horizontally if $S$ intersects the two horizontal or vertical sides of $R$, respectively. We say that a rectangle $R_1$ crosses a rectangle $R_2$ vertically (respectively, horizontally) if the left and right sides (respectively, the top and bottom sides) of $R_1$ cross the rectangle $R_2$ vertically (respectively, horizontally). If $R_1$ crosses $R_2$ vertically, then $R_2$ crosses $R_1$ horizontally. The *slope* of a segment in $\mathbb{R}^2$ with endpoints $(x_1, y_1)$, $(x_2, y_2)$ is $rac{y_2 - y_1}{x_2 - x_1}$ if $x_1 \neq x_2$, and $\infty$ otherwise. The *slope number* of a finite family $S$ of segments is the cardinality of the set of the slopes of segments in $S$.

Given a set $S$ of segments, the *intersection graph* $G(S)$ of $S$ is the graph with vertex set $S$ and with edges consisting of all pairs $SS'$ with $S, S' \in S$ such that $S \cap S' \neq \emptyset$.

A *t-blowup* of a *segment* is a multiset of $t$ copies of the same segment. A *t-blowup* of a *set of segments* is the multiset of the $t$-blowup of the segments. A *t-blowup* of a *graph* is the graph obtained from $G$ by replacing every vertex by a copy of $K_t$, and every edge by a copy of $K_{t,t}$. Observe that a $t$-fold colouring of $G$ corresponds to a colouring of the $t$-blowup of $G$.

## 2.3 Configurations

We now introduce the notion of probes, first used in [PKK$^+$14]. Let $S$ be a family of segments in the unit square $U = [0, 1] \times [0, 1]$. Let $P = [a, c] \times [b, d]$ be a rectangle included in $U$. We denote by $S(P)$ the set of segments in $S$ intersecting $P$. The *right-extension* of $P$ is the rectangle $[a, 1] \times [b, d]$. The rectangle $P = [a, c] \times [b, d]$ is a *probe* for $S$ if the following hold:

* $c=1$, that is, the right side of $P$ lies on the boundary of $U$;

* all segments in $S(P)$ cross $P$ vertically; and

* all segments in $S(P)$ are pairwise disjoint.

Given a probe $P$, a *root* of $P$ is a rectangle $[a, c'] \times [b, d]$ where $c'$ is a real number such that $[a, c'] \times [b, d]$ is disjoint from every segment in $S$.
A configuration is a pair $\mathcal{C} = (\mathcal{S}, \mathcal{P})$ with $\mathcal{S}$ a set of segments in $\mathcal{U}$ and $\mathcal{P}$ a family of pairwise disjoint probes for $\mathcal{S}$.

For convenience, we define the intersection graph $G(\mathcal{C})$ of a configuration $\mathcal{C} = (\mathcal{S}, \mathcal{P})$ as $G(\mathcal{S})$. We say that $\mathcal{C}$ is triangle-free if $G(\mathcal{S})$ is a triangle-free graph.

Let $R$ be a square inside $\mathcal{U}$ and let $\psi$ be the positive homothety mapping $\mathcal{U}$ to $R$. The $R$-scaled copy of $\mathcal{C}$ is the configuration $\mathcal{C}' = (\mathcal{S}', \mathcal{P}')$ where $\mathcal{S}'$ is the set of images of the segments in $\mathcal{S}$ by $\psi$ and $\mathcal{P}'$ is the set of the right-extensions of the images by $\psi$ of the probes in $\mathcal{P}$. Observe that $\mathcal{S}'$ has the same set of slopes as $\mathcal{S}$.

## 2.4 Copying configurations

Our refinement on the construction of [PKK+14] is obtained at the cost of adding significantly many more copies of the construction achieving $\chi$ colours when building the graph requiring $\chi + 1$ colours. Here, we describe this operation. For every configuration $\mathcal{C}$ and for every positive integer $k$, we define the new configuration $\mathcal{C}^{(k)}$ with segments set $\mathcal{S}^{(k)}$ and probes set $\mathcal{P}^{(k)}$ as “a way of joining $k$ independent copies of $\mathcal{C}$ together”. With every probe $P \in \mathcal{P}^{(k)}$, we associate a set of $k$ sub-rectangles of $P$ that we call the *pillars* of $P$. The pillars have the same height as $P$, are pairwise disjoint, and no pillar starts from the left side of $P$ so that there is a root of $P$ not belonging to any pillar. Informally, each pillar will correspond to a copy of $\mathcal{C}$.

Formally, $\mathcal{C}^{(k)} = (\mathcal{S}^{(k)}, \mathcal{P}^{(k)})$ is defined as follows:

* If $k=1$, then $\mathcal{C}^{(1)} = \mathcal{C}$. For every probe $P$, we define its only pillar $I_0^P$ as the minimal rectangle containing $P$ minus a root. Every segment of $\mathcal{S}(P)$ is a segment of $\mathcal{S}(I_0^P)$ and it also crosses $I_0^P$ vertically.

* If $k \ge 2$, then for every probe $P \in \mathcal{P}^{(k-1)}$ we choose a square $R_P$ in a root of $P$ disjoint from its pillars and insert an $R_P$-scaled copy $\mathcal{C}_P = (\mathcal{S}_P, \mathcal{P}_P)$ of $\mathcal{C}$. We define $\mathcal{S}^{(k)} = \mathcal{S}^{(k-1)} \cup \bigcup_{P \in \mathcal{P}^{(k-1)}} \mathcal{S}_P$ and $\mathcal{P}^{(k)} = \bigcup_{P \in \mathcal{P}^{(k-1)}} \mathcal{P}_P$. In other words, instead of $P$, we take new probes, generated in $R_P$ by the new copy of $\mathcal{C}$, extended to the right. These new probes are then wholly contained in $P$.

For every such probe $P'$, we define $k$ different pillars of $P'$. There are $k-1$ pillars $I_0^{P'}$, $I_1^{P'}$, ..., $I_{k-2}^{P'}$ which are intersections between the pairwise disjoint pillars $I_0^P$, $I_1^P$, ..., $I_{k-2}^P$ of $P$ and $P'$ and one pillar $I_{k-1}^{P'}$ that is the minimal rectangle containing $R_P \cap P'$ minus a root of $P'$. Since $R_P$ is contained in a root of $P$, $I_{k-1}^{P'}$ is also disjoint from $I_0^{P'}$, ..., $I_{k-2}^{P'}$. Furthermore, for every $i \in \{0, \dots, k-2\}$, since segments in $\mathcal{S}^{(k-1)}(I_i^P)$ cross $I_i^P$ vertically, they also cross $I_i^{P'}$ vertically. Therefore, $\mathcal{S}^{(k-1)}(I_i^P) \subseteq \mathcal{S}^{(k)}(I_i^{P'})$.

Note that $G(\mathcal{C}^{(k)})$ is the union of disjoint copies of $G(\mathcal{C})$, and in particular $\mathcal{C}^{(k)}$ is triangle-free if $\mathcal{C}$ is triangle-free. See Figure 1.
<figure><img src="image_2.png" /><figcaption>Figure 1: Building $\mathcal{C}^{(2)}$ from $\mathcal{C}^{(1)} = \mathcal{C}$.</figcaption></figure>

**Lemma 2.** Let $\mathcal{C} = (\mathcal{S}, \mathcal{P})$ be a configuration and let $t$ be a positive integer. If there exists a nonnegative integer $s$ such that, for every $t$-fold colouring $\varphi$ of $G(\mathcal{C})$, there exists a probe $P \in \mathcal{P}$ such that $|\varphi(\mathcal{S}(P))| \ge s$, then, for every $t$-fold colouring $\varphi$ of $G(\mathcal{C}^{(k)})$, there exists a probe $P'$ such that, for every pillar $I$ of $P'$, $|\varphi(\mathcal{S}(I))| \ge s$.

*Proof.* The proof is an induction on $k$.

Suppose that $k=1$. Observe that $\mathcal{C}^{(1)} = \mathcal{C}$. Let $\varphi$ be a $t$-fold colouring of $G(\mathcal{C}^{(1)})$. There exists a probe $P$ such that $|\varphi(\mathcal{S}(P))| \ge s$ by assumption on $\mathcal{C}$. Since every segment of $\mathcal{S}(P)$ crosses the only pillar $I$ of $P$ vertically, $|\varphi(\mathcal{S}(I))| \ge s$. Therefore, $P' = P$ satisfies the conclusion of Lemma 2.

Suppose that $k \ge 2$. Let $\varphi$ be a $t$-fold colouring of $G(\mathcal{C}^{(k)})$. By the inductive hypothesis, there exists a probe $P$ of $\mathcal{C}^{(k-1)}$ such that for every $i \in \{0, \dots, k-2\}$, every pillar $I_i^P$ of $P$
satisfies $|\varphi|_{S^{(k-1)}}(S^{(k-1)}(I_i^P))| \ge s$. By the definition of $C^{(k)}$, there exists a square $R_P$ in a root of $P$ with an $R_P$-scaled copy $C_P = (S_P, P_P)$ of $C$. By assumption on $C$, there exists a probe $P' \in P_P \subseteq P^{(k)}$ such that $|\varphi|_{S_P}(S_P(P'))| \ge s$. Since every segment of $S_P(P')$ crosses $I_{k-1}^{P'}$ vertically, $|\varphi(S^{(k)}(I_{k-1}^{P'}))| \ge |\varphi|_{S_P}(S_P(I_{k-1}^{P'}))| \ge |\varphi|_{S_P}(S_P(P'))| \ge s$. For every $i \in \{0, \dots, k-2\}$, every segment of $S^{(k-1)}(I_i^P) \subseteq S^{(k)}(I_i^P)$ crosses the pillar $I_i^{P'}$ vertically. Therefore, we also have $|\varphi(S^{(k)}(I_i^{P'}))| \ge |\varphi|_{S^{(k-1)}}(S^{(k-1)}(I_i^P))| \ge s$. $\square$

### 3 Lower bound for even $\omega$

In this section, we show the lower bound for the $\chi$-binding function of $d$-DIR for even clique numbers. First, we exhibit, for any $t$ and $d$, a triangle-free $d$-DIR configuration with $t$-fold chromatic number at least $2td$. Then, a $t$-blowup of this construction yields one with clique number $2t$ and chromatic number at least $2td$, establishing the even case in Theorem 1.

**Theorem 3.** For all positive integers $t$ and $d$, there exists a triangle-free configuration $C_{t,d} = (S_{t,d}, P_{t,d})$ with slope number at most $d$ such that for every $t$-fold colouring $\varphi$ of $G(C_{t,d})$, there exists a probe $P \in P_{t,d}$ such that $|\varphi(S_{t,d}(P))| \ge 2td$.

*Proof.* We proceed by induction on $d$. For $d=0$, the configuration $(\emptyset, \{U\})$ satisfies the statement of the theorem.

Suppose that $d > 0$ and that $C_{t,d-1}$ is already constructed. Let $\mathcal{H} = C_{t,d-1}^{(4t+1)} = (S_{\mathcal{H}}, P_{\mathcal{H}})$ be the copy of $4t+1$ of these configurations as defined in Subsection 2.4. Let $\Gamma$ be the set of the slopes of the segments in $S_{t,d-1}$. We define the slope $\gamma$ as the minimum aspect ratio over all probes of $P_{\mathcal{H}}$, divided by $4t$.

We will construct a family of triangle-free configurations $\mathcal{H}_i = (S_i, P_i)$ for all $i \ge 0$ such that

1. for every segment $S \in S_i$, the slope of $S$ is in $\Gamma \cup \{\gamma\}$, and

2. for every $t$-fold colouring $\varphi$ of $G(\mathcal{H}_i)$, there is a probe $P \in P_i$ such that $|\varphi(S_i(P))| \ge 2t(d-1) + 2t(1 - \frac{1}{2t})$.

For $i = \lfloor \log(2t) \rfloor + 1$, Item 2 implies that $|\varphi(S_i(P))| \ge 2td$, and so it suffices to take $C_{t,d} = \mathcal{H}_i$.

We build $\mathcal{H}_i$ as follows. For $i=0$, we take $\mathcal{H}_0 = C_{t,d-1}$. Item 1 holds by definition of $\mathcal{H}_0$ and Item 2 holds by the inductive hypothesis on $C_{t,d-1}$.

Now, assume that $i > 0$. We will build $S_i$ and $P_i$ iteratively. Initialise $S_i$ to $S_{i-1}$ and $P_i$ to $\emptyset$. For each probe $P \in P_{i-1}$, let $R$ be a square in a root of $P$, and add an $R$-scaled copy $\mathcal{H}_P$ of $\mathcal{H}$ in $R$. At this point, the configuration is still triangle-free as it is a disjoint union of $\mathcal{H}_{i-1}$ and copies of $\mathcal{H}$. For every probe $Q$ in $\mathcal{H}_P$, we divide $Q$ vertically into $4t$ rectangles of equal height with the same width as $Q$. We call these rectangles the *layers* of $Q$. Let $(L_j(Q))_{j \in \{0, \dots, 4t-1\}}$ be the layers of $Q$ starting from the top.

First, we add the top layer $L_0(Q)$ to $P_i$ as a single probe. Then, for every $j \in \{1, \dots, 4t-1\}$, we add two segments in the interior of $L_j(Q)$ to $S_i$, with slope $\gamma$, that intersect at exactly one point: $D_1^j$ crossing exactly the pillars $I_{j+1}^Q, \dots, I_{4t}^Q$ horizontally, and $D_2^j$ crossing $I_j^Q$ horizontally and no other pillars. Since the slope of the diagonal of a layer is at least $\gamma$, such segments are well-defined. Moreover, the configuration is still triangle-free. On the one hand, any $D_1^j$ or $D_2^j$ intersects a set of disjoint segments over all pillars. On the other, any pillar, and thus any segment crossing it vertically, cannot intersect both $D_1^j$ and $D_2^j$. See Figure 2.
<figure><img src="image_3.png" /><figcaption>Figure 2: Layers of a probe P drawn in green, with new segments $D_1^j$ and $D_2^j$. Example for $t=1$.</figcaption></figure>

Finally, we add to $\mathcal{P}_i$ two probes $P_{D_1^j}, P_{D_2^j}$, with $P_{D_1^j}$ crossed vertically by exactly $D_1^j$ and $I_0^Q, \dots, I_j^Q$, and $P_{D_2^j}$ crossed vertically by exactly $D_2^j$ and $I_0^Q, \dots, I_{j-1}^Q$. See Figure 3.

<figure><img src="image_4.png" /><figcaption>Figure 3: The two new probes $P_{D_1^j}$ and $P_{D_2^j}$ inside the layer $L_j(P)$.</figcaption></figure>

It follows from the construction that $\mathcal{P}_i$ is a set of probes for $S_i$, and that every segment in $S_i$ has its slope in $\Gamma \cup \{\gamma\}$ so Item 1 holds. It remains to prove that for every $t$-fold colouring $\varphi$, there is a
probe $P \in \mathcal{P}_i$ such that $|\varphi(S_i(P))| \ge 2t(d-1)+2t(1-\frac{1}{2^i})$. Let $\varphi$ be a $t$-fold colouring of $G(\mathcal{H}_i)$. By the inductive hypothesis, there exists $P \in \mathcal{P}_{i-1}$ such that $|\varphi(S_{i-1}(P))| \ge 2t(d-1)+2t(1-\frac{1}{2^{i+1}})$. By Lemma 2 applied to $\mathcal{H} = C_{t,d-1}^{(4t+1)}$, there exists a probe $Q$ of $\mathcal{H}_P$ such that every pillar $I_j^Q$ satisfies $|\varphi(S_i(I_j^Q))| \ge 2t(d-1)$. For every $j \in \{0, \dots, 4t-1\}$, let $\Phi_j = \varphi(S_i(I_j^Q))$. Recall that the top layer $L_0(Q)$ was added as a probe, and it crosses all the $I_j^Q$ s horizontally. If it contains at least $2t(d-1)+2t(1-\frac{1}{2^i})$ colours, then this probe satisfies Item 2. So, from now on, we may assume that $Q$ contains fewer than $2t(d-1)+2t(1-\frac{1}{2^i})$ colours. Since $|\Phi_0| \ge 2t(d-1)$, we have $2t(d-1)+2t(1-\frac{1}{2^i}) - |\Phi_0| < 2t$. Therefore, at most $2t-1$ indices $j \in \{1, \dots, 4t-1\}$ are such that the set $\Phi_j$ contains a colour not in $\cup_{j' \le j-1} \Phi_{j'}$. Thus, for at least $4t-1-2t+1=2t$ indices $j \in \{1, \dots, 4t-1\}$, $\Phi_j \subseteq \cup_{j' \le j-1} \Phi_{j'}$. Symmetrically, for at least $2t$ indices $j \in \{1, \dots, 4t-1\}$, we have $\Phi_j \subseteq \cup_{j' \ge j+1} \Phi_{j'}$. Thus, there is an index $j_0 \in \{1, \dots, 4t-1\}$ such that $\Phi_{j_0} \subseteq \cup_{j' \ge j_0+1} \Phi_{j'}$ and $\Phi_{j_0} \subseteq \cup_{j' \le j_0-1} \Phi_{j'}$. Let $A = \Phi_{j_0}$, $B = \cup_{j' \le j_0-1} \Phi_{j'}$ and $C = \cup_{j' \ge j_0+1} \Phi_{j'}$. We have $A \subseteq B$ and $A \subseteq C$. Consider the layer $L_{j_0}(Q)$ and the corresponding segments $D_1^{j_0}, D_2^{j_0}$. Since $D_1^{j_0}$ and $D_2^{j_0}$ share a common point, $\varphi(D_1^{j_0}) \cap \varphi(D_2^{j_0}) = \emptyset$. Since $D_1^{j_0}$ crosses $I_{j_0+1}^Q, \dots, I_{4t}^Q$ horizontally, we have $\varphi(D_1^{j_0}) \cap C = \emptyset$. Since $A \subseteq C$, we also have $\varphi(D_1^{j_0}) \cap A = \emptyset$. Moreover, $D_2^{j_0}$ crosses $I_{j_0}^Q$ horizontally, so $\varphi(D_2^{j_0}) \cap A = \emptyset$. As a result, $\varphi(D_1^{j_0}), \varphi(D_2^{j_0})$, and $A$ are pairwise disjoint. Thus, $|\varphi(D_1^{j_0}) \cup \varphi(D_2^{j_0}) \cup A| = |\varphi(D_1^{j_0})| + |\varphi(D_2^{j_0})| + |A| \ge t+t+2t(d-1) = 2td$. Hence, letting $Z_{i-1} = \varphi(S_{i-1}(P))$, we have $|(\varphi(D_1^{j_0}) \cup \varphi(D_2^{j_0}) \cup A) \setminus Z_{i-1}| \ge 2td - |Z_{i-1}|$. Therefore, there is some $\alpha \in \{1, 2\}$ such that $|(\varphi(D_\alpha^{j_0}) \cup A) \setminus Z_{i-1}| \ge \frac{2td - |Z_{i-1}|}{2}$. It follows that

$$|\varphi(D_{\alpha}^{j_0}) \cup A \cup Z_{i-1}| \ge \frac{2td + |Z_{i-1}|}{2} \ge 2t(d-1) + 2t \left(1 - \frac{1}{2^i}\right).$$

Then $\varphi(S_i(P_{D_\alpha^{j_0}}))$ contains at least $|\varphi(D_\alpha^{j_0}) \cup B \cup Z_{i-1}| \ge |\varphi(D_\alpha^{j_0}) \cup A \cup Z_{i-1}| \ge 2t(d-1)+2t(1-\frac{1}{2^i})$ colours. Therefore, Item 2 always holds for $\mathcal{H}_i$. This concludes the proof of the theorem. $\square$

**Corollary 4.** For all positive integers $t$ and $d$, there exists a multiset of segments $S'_{t,d} \subseteq \mathcal{U}$ with slope number at most $d$ such that $\omega(G(S'_{t,d})) = 2t$ and $\chi(G(S'_{t,d})) \ge 2td$.

*Proof.* Let $C_{t,d} = (S_{t,d}, \mathcal{P}_{t,d})$ be a configuration given by Theorem 3. Let $S'_{t,d}$ be a $t$-blowup of $S_{t,d}$, that is a configuration obtained from $S_{t,d}$ by replacing every segment $v \in S_{t,d}$ by $t$ identical copies $v_1, \dots, v_t$. Since the graph $G(S_{t,d})$ is triangle-free, we have $\omega(G(S'_{t,d})) = 2t$. Let $\varphi$ be a colouring of $G(S'_{t,d})$. The colouring $\varphi$ corresponds to a $t$-fold colouring $\varphi_t$ of $G(S_{t,d})$ defined by $\varphi_t(v) = \{\varphi(v_1), \dots, \varphi(v_t)\}$ for all $v \in V(G(S_{t,d}))$. By Theorem 3, there exists a probe $P \in \mathcal{P}_{t,d}$ such that $|\varphi_t(S_{t,d}(P))| \ge 2td$. Hence, by definition of $\varphi_t$, $\varphi$ uses at least $2td$ colours as claimed. $\square$

# 4 Lower bound for odd $\omega$

In this section, we certify the lower bound for the $\chi$-binding function in the case of odd clique numbers. We rely on the construction of $d$-DIR graphs with $(t+1)$-fold chromatic number at least $2(t+1)d$ as guaranteed by Theorem 3. Instead of blowing up each segment by $t+1$ as in the even case, many blowups will be $t$-blowups, ensuring the clique number increases only to $2t+1$. We do so in a way that also ensures the chromatic number is $2td+1$, as desired.

**Theorem 5.** For every nonnegative integer $t$ and every positive integer $d$, there exists a multiset of segments $S''_{t,d}$ with slope number at most $d$ such that $\omega(G(S''_{t,d})) = 2t+1$ and $\chi(G(S''_{t,d})) \ge 2td+1$.

*Proof.* Let $t$ be a nonnegative integer and let $d$ be a positive integer. By Theorem 3, there is a triangle-free configuration $C_{t+1,d} = (S_{t+1,d}, \mathcal{P}_{t+1,d})$ with slope number at most $d$ such that every $(t+1)$-fold colouring of $G(C_{t+1,d})$ uses at least $2(t+1)d$ colours. Let $\mathcal{S}^1, \dots, \mathcal{S}^d$ be the partition of $S_{t+1,d}$ according to the slopes of the segments. For every $i \in [d]$, $G(\mathcal{S}^i)$ is a triangle-free interval graph, and so it is bipartite. Hence there is a partition $\mathcal{S}^{i,1}, \mathcal{S}^{i,2}$ of $\mathcal{S}^i$ such that the segments in
$S^{a,i}$ are pairwise disjoint for each $a \in \{1,2\}$. Now, let $S_{t,d}''$ be obtained from $S_{t+1,d}$ by blowing up every segment $v$ in $\bigcup_{i \in [d-1]} S^i$ and $S^{d,1}$ into $t$ copies $v_1, \dots, v_t$, and every segment $v$ in $S^{d,2}$ into $t+1$ copies $v_1, \dots, v_{t+1}$.

First, by construction, $\omega(G(S_{t,d}'')) \le 2t+1$. We now show that $\chi(G(S_{t,d}'')) \ge 2td+1$. Let $\varphi$ be a $k$-colouring of $G(S_{t,d}'')$. Let us introduce new colours $(1,i), (2,i)$, for $i \in [d]$, and suppose they are not in the image of $\varphi$. For every $i \in [d-1]$ and every $v \in S^{i,a}$, where $a \in \{1,2\}$, we let $\psi(v) = \{\varphi(v_1), \dots, \varphi(v_t), (a,i)\}$. For every $v \in S^{d,1}$, we let $\psi(v) = \{\varphi(v_1), \dots, \varphi(v_t), (1,d)\}$. Finally, for every $v \in S^{d,2}$, we let $\psi(v) = \{\varphi(v_1), \dots, \varphi(v_{t+1})\}$. Note that $\psi(v)$ has size exactly $t+1$ for every $v \in S_{t+1,d}$, and $\psi$ uses at most $k+2d-1$ colours (since we did not use the colour $(2,d)$). Moreover, as $\varphi$ is proper and we invoke the bipartitions of the $S^i$s, we deduce that $\psi$ is a $(t+1)$-fold colouring of $S_{t+1,d}$. This implies that $k+2d-1 \ge 2(t+1)d$, and so $\varphi$ uses $k \ge 2td+1$ colours. This proves that $\chi(G(S_{t,d}'')) \ge 2td+1$. $\square$

# 5 Upper bound for odd $\omega$

In this section, we upper bound the $\chi$-binding function for $d$-DIR in the case of odd clique numbers, which combined with the result of the previous section settles the odd case in Theorem 1.

Let $G$ be an interval graph and let $\mathcal{I} = (I_v \mid v \in V(G))$ be an interval representation of $G$. Let $\omega$ be an integer with $\omega \ge \omega(G)$. For any $x \in \mathbb{R}$, we define the set $S_x = \{u \in V(G) \mid x \in I_u\}$ of vertices corresponding to intervals containing $x$. A vertex $v$ of $G$ is said to be $\omega$-uncovered if there exists $x \in I_v$ such that $|S_x| \le \frac{\omega-1}{2}$. Observe that $S_x$ consists of a clique of $\omega$-uncovered vertices only. We use the following lemma appearing in [KN02] as a tool to build colourings of $d$-DIR graphs in the case of odd $\omega$.

**Lemma 6** ([KN02, Lemma 1]). Let $G$ be an interval graph with clique number at most $\omega$. There is a colouring $\varphi: V(G) \to \{0, \dots, \omega-1\}$ of $G$ such that $\varphi(v) \ne 0$ for every $\omega$-uncovered vertex $v$.

**Theorem 7.** For any graph $G$ in $d$-DIR, if $\omega(G)$ is odd, then $\chi(G) \le d(\omega(G) - 1) + 1$.

*Proof.* Let $(I_u \mid u \in V(G))$ be a $d$-DIR representation of $G$. Let $V_1, \dots, V_d$ be the partition of $V(G)$ according to the slopes of the segments in its $d$-DIR representation. For every $i \in \{1, 2, \dots, d\}$, $G[V_i]$ is an interval graph and $\omega(G) \ge \omega(G[V_i])$. So by Lemma 6, there is a colouring $\varphi_i$ of $G[V_i]$ using the colours $0, 1, 2, \dots, \omega(G)-1$ and no $\omega(G)$-uncovered vertices of $G[V_i]$ are coloured 0. For every $i \in \{1, \dots, d\}$ and for every $v \in V_i$, let $\varphi(v) = (i, \varphi_i(v))$ if $\varphi_i(v) \ne 0$, and $\varphi(v) = 0$ otherwise. We claim that $\varphi$ is a $(d(\omega(G)-1)+1)$-colouring of $G$. Indeed, if $uv$ is an edge in $G$ such that $\varphi(u) = \varphi(v)$, then $\varphi(u) = \varphi(v) = 0$. In other words, $u$ and $v$ are both not $\omega(G)$-uncovered vertices and are in distinct $V_i$s. Let $i_u, i_v \in \{1, \dots, d\}$ be such that $u \in V_{i_u}$ and $v \in V_{i_v}$. Since $\omega(G)$ is odd, the intersection point $x$ of $I_u$ and $I_v$ is included in at least $\frac{\omega(G)+1}{2}$ intervals corresponding to vertices in $V_{i_u}$ and at least $\frac{\omega(G)+1}{2}$ intervals corresponding to vertices in $V_{i_v}$. This gives a clique in $G$ of order $\omega(G)+1$, a contradiction. $\square$

# 6 Concluding remarks

Recall the following definitions. The *Hall ratio* $\rho(G)$ of a graph $G$ is defined by $\rho(G) = \max_{H \subseteq G} \frac{|V(H)|}{\alpha(H)}$, where $\alpha$ denotes the independence number. The *fractional chromatic number* $\chi_f(G)$ of $G$ is the infimum of $a/t$ as $t \to \infty$ with $a$ being the least integer such that $G$ admits a $t$-fold $a$-colouring. Note that $\omega(G) \le \rho(G) \le \chi_f(G) \le \chi(G)$ for all $G$. Walczak [Wal15] showed that there are triangle-free segment graphs of arbitrarily large $\rho$. Can we find constructions to certify the same lower bounds as in Theorem 1 but for $\chi_f$ or $\rho$ instead of $\chi$? In particular, for every $d$ and even $\omega$ is there a $d$-directional segment graph of clique number $\omega$ with $\chi_f \ge d\omega$ or with $\rho \ge d\omega$?

A possible generalisation of our problem is to consider rectangles instead of segments. In a similar manner, we can introduce the notion of the $d$-directional rectangle graphs: consider a set
$d$ non-parallel lines and any set of rectangles such that each of them has one of the sides parallel to one of the given lines. We want to find the $\chi$-binding function of such graphs.

This problem has been quite extensively studied in the case of axis-parallel rectangles (i.e. $d = 1$) and thanks to Chalermsook and Walczak [CW21] it is known that in this case $O(\omega \log \omega)$ colours always suffice; however, the lower bound remains $\Omega(\omega)$. This implies that for any $d$, no more than $O(d\omega \log \omega)$ colours are needed, and our construction from this paper can be easily adapted (by replacing the segments with “thin enough” rectangles) to show a lower bound of $\omega \cdot (d+o(1))$. The gap between linear and log-linear function remains open, though, in both $d = 1$ and the general case.

**Acknowledgement.** This work was initiated during the *Sparse (Graphs) Coalition* session, $\chi$-boundedness in March 2023, organised by James Davies, Bartosz Walczak, and the second author. Our team thanks the organisers and participants for the conducive working atmosphere. We in particular thank Zdeněk Dvořák for proposing the problem. We are grateful to the reviewers for their careful reading and valuable comments, and especially for pointing us to the work of Kostochka and Nešetřil [KN02] as well as the suggestion of a simpler proof of Theorem 5.

**Open access statement.** For the purpose of open access, a CC BY public copyright license is applied to any Author Accepted Manuscript (AAM) arising from this submission.

## References

[AG60] Edgar Asplund and Branko Grünbaum. On a coloring problem. *Math. Scand.*, 8:181–188, 1960.

[BDN23] Sudatta Bhattacharya, Zdeněk Dvořák, and Fariba Noorizadeh. Chromatic number of intersection graphs of segments with two slopes. In *Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications (EUROCOM23)*, 2023.

[Bur65] James P. Burling. *On coloring problems of families of polytopes*. PhD thesis, University of Colorado, Boulder, 1965.

[CW21] Parinya Chalermsook and Bartosz Walczak. Coloring and maximum weight independent set of rectangles. In *Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)*, pages 860–868. [Society for Industrial and Applied Mathematics (SIAM)], Philadelphia, PA, 2021.

[Gyá87] András Gyárfás. Problems from the world surrounding perfect graphs. In *Proceedings of the International Conference on Combinatorial Analysis and its Applications (Pokrzywna, 1985)*, volume 19, pages 413–441 (1988), 1987.

[KN95] Alexandr Kostochka and Jaroslav Nešetřil. Chromatic number of geometric intersection graphs. In *Proceedings of the Prague Midsummer Combinatorial Workshop (Charles University, Prague)*, volume 95–309, pages 43–45, 1995.

[KN02] Alexandr V. Kostochka and Jaroslav Nešetřil. Colouring relatives of intervals on the plane, ii: Intervals and rays in two directions. *European Journal of Combinatorics*, 23(1):37–41, January 2002.

[McG00] Sean McGuinness. Colouring arcwise connected sets in the plane. I. *Graphs Combin.*, 16(4):429–439, 2000.

[MM99] Terry A. McKee and Fred R. McMorris. *Topics in intersection graph theory*. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.
[PKK$^{+}$14] Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, and Bartosz Walczak. Triangle-free intersection graphs of line segments with large chromatic number. *J. Combin. Theory Ser. B*, 105:6–10, 2014.

[Spi03] Jeremy P. Spinrad. *Efficient graph representations*, volume 19 of *Fields Institute Monographs*. American Mathematical Society, Providence, RI, 2003.

[SS20] Alex Scott and Paul Seymour. A survey of $\chi$-boundedness. *J. Graph Theory*, 95(3):473–504, 2020.

[Suk14] Andrew Suk. Coloring intersection graphs of $x$-monotone curves in the plane. *Combinatorica*, 34(4):487–505, 2014.

[Wal15] Bartosz Walczak. Triangle-free geometric intersection graphs with no large independent sets. *Discrete Comput. Geom.*, 53(1):221–225, 2015.