Conflict-Free Coloring of Intersection Graphs¹¹1This work was partially supported by the DFG Research Unit ”Controlling Concurrent Change”, funding number FOR 1800, project FE407/17-2, ”Conflict Resolution and Optimization”.

Coloring the vertices of a graph is one of the fundamental problems in graph theory, both scientifically and historically. The notion of proper graph coloring can be generalized to hypergraphs in several ways. One natural generalization is conflict-free coloring, which asks to color the vertices of a hypergraph such that every hyperedge has at least one uniquely colored vertex. This problem has applications in wireless communication, where “colors” correspond to different frequencies.

The notion of conflict-free coloring can be brought back to simple graphs, e.g., by considering as hyperedges the neighborhoods of the vertices of $G$. The resulting problem arises in certain variants of frequency assignment problems if one is not interested in achieving signal coverage for all points in a region, but only at certain points of interest. For an illustration, consider a scenario in which one has a given set of nodes in the plane and wants to establish a communication network between them. Moreover, assume that constructing nodes at new locations is either very expensive or forbidden, and one can only “upgrade” any existing node to a wireless base station.

Conflict-free coloring also plays a role in robot navigation, where different beacons are used for providing direction. To this end, it is vital that in any given location, a robot is adjacent to a beacon with a frequency that is unique among the ones that can be received.

Both in the frequency assignment setting and in the robot navigation setting, one typically wants to avoid placing unnecessary base stations or beacons. Abstractly speaking, this corresponds to leaving some vertices uncolored, yielding the following formalization of conflict-free coloring of graphs. For any vertex $v\in V$ of a simple graph $G=(V,E)$, the neighborhood $N[v]$ consists of all vertices adjacent to $v$ and $v$ itself. A conflict-free $k$-coloring of $G$ assigns one of $k$ colors to a (possibly proper) subset $S\subseteq V$ of vertices such that every vertex $v\in V$ has a uniquely colored neighbor. The conflict-free chromatic number $\chi_{CF}(G)$ of $G$ is the smallest $k$ for which a conflict-free coloring exists. Depending on the situation it may also be more natural to consider open neighborhood conflict-free coloring, where each vertex $v$ must have a uniquely colored neighbor in its open neighborhood $N(v)$ not including $v$.

Conflict-free coloring has received an increasing amount of attention. Because of the motivation arising from frequency assignment, it is natural to investigate the conflict-free coloring of intersection graphs, in particular, of simple shapes such as disks or squares. In addition, previous work has considered either general graphs and hypergraphs (e.g., see [27]) or other geometric scenarios in the presence of obstacles (e.g., see [20]); we give a more detailed overview further down. This adds to the relevance of conflict-free coloring of intersection graphs, which lie in the intersection of general graphs and geometry.

There is a spectrum of different scientific challenges when studying conflict-free coloring. What are worst-case bounds on the necessary number of colors? When is it NP-hard to determine the existence of a conflict-free $k$-coloring? We address these questions for the case of intersection graphs.

Our contribution. We present the following results.

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  We demonstrate that $n$ geometric objects without fatness properties and size restrictions may induce intersection graphs with conflict-free chromatic number in $\Omega(\log n/\log\log n)$.

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  We prove that non-unit square and disk graphs may require $\Omega(\sqrt{\log n})$ colors. Deciding conflict-free $k$-colorability is NP-hard for any $k$ for these graph classes.

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  It is NP-complete for unit-disk intersection graphs to decide the existence of a conflict-free coloring with one color. The same holds for intersection graphs of unit squares and other shapes.

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  Six colors are always sufficient for conflict-free coloring of unit disks. This uses an algorithm that colors unit disk graphs contained in a strip of restricted height with two colors.

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  Using a similar argument, we prove that four colors are always sufficient for conflict-free coloring of unit squares.

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  As a corollary, we get a tight worst-case bound of two on the conflict-free chromatic number of interval graphs.

Related work. In the geometric context, motivated by frequency assignment problems, the study of conflict-free coloring of hypergraphs was initiated by Even et al. [15] and Smorodinsky [28]. For disk intersection hypergraphs, Even et al. [15] prove that $\mathcal{O}(\log n)$ colors suffice. For disk intersection hypergraphs with degree at most $k$, Alon and Smorodinsky [4] show that $\mathcal{O}(\log^{3}k)$ colors are sufficient. Cheilaris et al. [10] prove that $\mathcal{O}(\log n)$ colors suffice for the case of hypergraphs induced by planar Jordan regions with linear union complexity. If every edge of a disk intersection hypergraph must have $k$ distinct unique colors, Horev et al. [21] prove that $\mathcal{O}(k\log n)$ suffice. Moreover, for unit disks, Lev-Tov and Peleg [24] present an $\mathcal{O}(1)$-approximation algorithm for the conflict-free chromatic number. Abam et al. [1] consider the problem of making a conflict-free coloring robust against removal of a certain number of vertices, and prove worst-case bounds for the number of colors required. The online version of the problem has been studied as well; see, e.g., the work of Chen et al. [12] that presents, among other results, a randomized online algorithm using $\mathcal{O}(\log n)$ colors to maintain a conflict-free coloring of a set of intervals, or the work of Bar-Nov et al. [6] that presents an online algorithm using $\mathcal{O}(k\log n)$ colors for some classes of $k$-degenerate hypergraphs.

The dual problem in which one has to color a given set of points such that each region contains a uniquely colored point has also received some attention. Har-Peled and Smorodinsky [19] prove that for families of pseudo-disks, every set of points can be colored using $\mathcal{O}(\log n)$ colors. For rectangles, Ajwani et al. [3] show that $\mathcal{O}(n^{0.382})$ colors suffice, whereas Elbassioni and Mustafa [14] show that it is possible to add a sublinear number of points such that sublinearly many colors suffice. For coloring points on a line with respect to intervals, Cheilaris et al. [9] present a 2-approximation algorithm, and a $\left(5-\frac{2}{k}\right)$-approximation algorithm when every interval must contain $k$ uniquely colored points.

In general hypergraphs, i.e., hypergraphs not necessarily arising from a geometric context, Ashok et al. [5] prove that maximizing the number of conflict-freely colored hyperedges is FPT with respect to the number of conflict-freely colored hyperedges in the solution.

Conflict-free coloring also arises in the context of the conflict-free variant of the chromatic Art Gallery Problem, which asks to guard a polygon using colored guards such that each point sees a uniquely colored guard. Fekete et al. [16] prove that computing the chromatic number is NP-hard in this context. On the positive side, Hoffman et al. [20] prove $\Theta(\log\log n)$ colors are sometimes necessary and always sufficient for the conflict-free chromatic art gallery problem under rectangular visibility in orthogonal polygons. For straight-line visibility, Bärtschi et al. [8] prove that $\mathcal{O}(\log n)$ colors suffice for orthogonal and monotone polygons and $\mathcal{O}(\log^{2}n)$ colors suffice for simple polygons. This is generalized by Bärtschi et al. [7], who prove that $\mathcal{O}(\log n)$ colors suffice for simple polygons.

There also has been work regarding the scenario where the hypergraph is induced by the neighborhoods of vertices of a simple graph. Except for the need to color all vertices, this corresponds to the scenario considered in this work. This does not change the asymptotic number of colors required, since it suffices to insert one additional color to color all vertices that would otherwise remain uncolored. In this situation, Pach and Tardos [27] prove that the conflict-free chromatic number of an $n$-vertex graph is in $\mathcal{O}(\log^{2}n)$. Glebov et al. [18] extend this result by proving that almost all $G(n,\omega(1/n))$-graphs have conflict-free chromatic number $\mathcal{O}(\log n)$. Moreover, they show that the upper bound of Pach and Tardos [27] is tight by giving a randomized construction for graphs having conflict-free chromatic number $\Theta(\log^{2}n)$. In more recent work, Gargano and Rescigno [17] show that finding the conflict-free chromatic number for general graphs is NP-complete, and prove that the problem is FPT w.r.t. vertex cover or neighborhood diversity number. In our work with Abel et al. [2], we consider conflict-free coloring of general and planar graphs and proved a conflict-free variant of Hadwiger’s conjecture, which implies that planar graphs have conflict-free chromatic number at most three. Most recently, Keller and Smorodinsky [23] consider conflict-free coloring on intersection graphs of geometric objects, in a scenario very similar to ours. Among other results, they prove that $\mathcal{O}(\log n)$ colors suffices to color a family $\mathcal{F}$ of pseudodisks in a conflict-free manner. With respect to open neighborhoods (also known as pointed neighborhoods), they prove that this is tight; for closed neighborhoods as studied in this paper, the tightness of this bound is not proven and remains open. They also consider the list coloring variant of the problem.

Conflict-free coloring has also been studied for other graph-based hypergraphs. For instance, Cheilaris and Tóth [11] consider the case of hypergraphs induced by the paths of a graph. If the input is the graph, they prove that it is coNP-complete to decide whether a given coloring is conflict-free.

Conflict-free coloring is not the only type of coloring for which unit disk graphs have been found to require a bounded number of colors. In their recent work, McDiarmid et al. [25] consider clique coloring of unit disk graphs, in particular with regard to the asymptotic behavior of the clique chromatic number of random unit disk graphs. They also prove that every unit disk graph in the plane can be colored with nine colors, while three colors are sometimes necessary. Similar to the present paper, they prove this by cutting the plane into strips of height $\sqrt{3}$; for each of these strips it is then proven that three colors suffice.

In the following, $G=(V,E)$ denotes a graph on $n:=|V|$ vertices. For a vertex $v$, $N(v)$ denotes its open neighborhood and $N[v]=N(v)\cup\{v\}$ denotes its closed neighborhood. A conflict-free $k$-coloring of a graph $G=(V,E)$ is a coloring $\chi:V^{\prime}\to\{1,\ldots,k\}$ of a subset $V^{\prime}\subseteq V$ of the vertices of $G$, such that each vertex $v$ has at least one conflict-free neighbor $u\in N[v]$, i.e., a neighbor $u$ whose color $\chi(u)$ occurs only once in $N[v]$. The conflict-free chromatic number $\chi_{CF}(G)$ is the minimum number of colors required for a conflict-free coloring of $G$.

A graph $G$ is a disk graph iff $G$ is the intersection graph of disks in the plane. $G$ is a unit disk graph iff $G$ is the intersection graph of disks with fixed radius $r=1$ in the plane, and a unit square graph iff $G$ is the intersection graph of axis-aligned squares with side length 1 in the plane. A unit disk (square) graph is of height $h$ iff $G$ can be modeled by the intersection of unit disks (squares) with center points in $(-\infty,\infty)\times[0,h]$. In the following, when dealing with intersection graphs, we assume that we are given a geometric model. In the case of unit disk and unit square graphs, we identify the vertices of the graph with the center points of the corresponding geometric objects in this model.

Intersection graphs of geometric objects can generally contain cliques of arbitrary size, so their chromatic number may be unbounded. However, cliques do not require a large number of colors in a conflict-free coloring, so it is not immediately clear whether the intersection graphs for a family of geometric objects have bounded conflict-free chromatic number.

If the intersecting objects can be scaled down arbitrarily, i.e., if every representable graph can be represented using arbitrarily small area, we can make use of the following lemma to prove lower bounds on the number of colors required.

For general objects like freely scalable ellipses or rectangles, it is possible to model a complete graph $K_{n}$ of arbitrary size $n$, such that the following conditions hold: (1) For every object $v$, there is some non-empty area of $v$ not intersecting any other objects. (2) For every pair of objects $v,w$, there is a non-empty area common to these objects not intersecting any other objects. This can be seen by choosing $n$ intersecting lines such that no three lines intersect in a common point. These lines can then be approximated using sufficiently thin objects to achieve the desired configuration.

In this case, the conflict-free chromatic number is unbounded, because we can inductively build a family $G_{n}$ of intersection graphs with $\chi_{C}(G_{n})=n$ as follows. Starting with $G_{1}=(\{v\},\emptyset)$ and $G_{2}=C_{4}$ (a four-vertex cycle), we construct $G_{n}$ by starting with a $K_{n}$ modeled according to conditions (1) and (2). For every object $v$, we place two scaled-down non-intersecting copies of $G_{n-1}$ into an area covered only by $v$; Figure 1 depicts the construction of $G_{5}$ for ellipses. According to Lemma 1, these gadgets enforce that every vertex of the underlying $K_{n}$ is colored. For every pair of objects $v,w$, we place two scaled-down non-intersecting copies of $G_{n-2}$ into an area covered only by $v$ and $w$. Using an argument similar to that used in the proof of Lemma 1, these gadgets enforce that $v$ and $w$ have to receive different colors. Thus the resulting graph requires $n$ colors.

The number of vertices used by this construction satisfies the recurrence

To estimate the growth of $|G_{n}|$, let $\bar{G}_{n}=|G_{n}|$ for $n\leq 2$ and $\bar{G}_{n}=3n\bar{G}_{n-1}+n(n-1)\bar{G}_{n-2}$; clearly, $\bar{G}_{n}\geq|G_{n}|$ for all $n$. The recurrence $\bar{G}_{n}$ has the closed-form solution

implying that the number of colors required in geometric intersection graphs on $n$ vertices may be $\Omega(\frac{\log n}{\log\log n}).$

We summarize.

The best upper bound on the number of colors required in this scenario that we are aware of is $\mathcal{O}(\log^{2}n)$, which holds for general graphs and is due to Pach and Tardos [27].

Due to their fatness, squares and disks do not allow us to construct an arbitrarily big clique $K_{n}$ such that condition (2) of Section 3 holds. However, we can still prove that there is no constant bound on their conflict-free chromatic number. The proof is based on Lemma 1, which enables us to reduce the conflict-free coloring problem on intersection hypergraphs to our problem.

In [15], Even et al. prove that $\Theta(\log n)$ colors are always sufficient and sometimes necessary to color a disk intersection hypergraph in a conflict-free manner. This implies that $\mathcal{O}(\log n)$ colors are sufficient in our case, leaving a gap of $\mathcal{O}(\sqrt{\log n})$.

The construction used in the previous section hinges on high aspect ratios of the intersecting shapes. In the setting of frequency assignment for radio transmitters, it is natural to only consider fat objects with bounded aspect ratio, such as unit disks and unit squares. As it turns out, their intersection graphs have conflict-free chromatic number bounded by a small constant; on the other hand, even deciding the existence of a conflict-free coloring with a single color is NP-complete.

The constructions of the previous section can also be applied to the case of squares; for interval graphs, we get a tight worst-case bound.

There are various directions for future work. In addition to closing the worst-case gap for unit disks (and proving Conjecture 6), the worst-case conflict-free chromatic number for unit square graphs also remains open. Other questions include a tight bound for disk (or square) intersection graphs, and a necessary criterion for a family of geometric objects to have intersection graphs with unbounded conflict-free chromatic number.