Coloring Jordan regions and curves

Coloring Jordan regions and curves

Wouter Cames van Batenburg Department of Mathematics, Radboud University Nijmegen, The Netherlands w.camesvanbatenburg@math.ru.nl Louis Esperet Univ. Grenoble Alpes, CNRS, G-SCOP, Grenoble, France louis.esperet@grenoble-inp.fr  and  Tobias Müller Utrecht University, Utrecht, The Netherlands t.muller@uu.nl
Abstract.

A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most elements of (with sufficiently large), then we show that the elements of can be colored with at most colors so that intersecting Jordan regions are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a problem of Hliněný (1998) on the chromatic number of contact systems of strings.

We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.

Louis Esperet and Tobias Müller were partially supported by ANR Project Stint (anr-13-bs02-0007) and LabEx PERSYVAL-Lab (anr-11-labx-0025-01). Wouter Cames van Batenburg was supported by NWO (613.001.217) and a bilateral project PHC Van Gogh 2016 (35513NM). Tobias Müller’s research was partially supported by an NWO VIDI grant

1. Introduction

In this paper, a Jordan region is a subset of the plane that is homeomorphic to a closed disk. A family of Jordan regions is touching if their interiors are pairwise disjoint. If any point of the plane is contained in at most Jordan regions of , then we say that is -touching. If any two elements of intersect in at most one point, then is said to be simple. All the families of Jordan regions and curves we consider in this paper are assumed to have a finite number of intersection points. The first part of this paper is concerned with the chromatic number of simple -touching families of Jordan regions, i.e. the minimum number of colors needed to color the Jordan regions, so that intersecting Jordan regions receive different colors. This can also be defined as the chromatic number of the intersection graph of , which is the graph with vertex set in which two vertices are adjacent if and only if the corresponding elements of intersect. Recall that the chromatic number of a graph , denoted by , is the least number of colors needed to color the vertices of , so that adjacent vertices receive different colors. The chromatic number of a graph is at least the clique number of , denoted by , which is the maximum number of pairwise adjacent vertices in , but the difference between the two parameters can be arbitrarily large (see [8] for a survey on the chromatic and clique numbers of geometric intersection graphs).

The following question was raised by Reed and Shepherd [10].

Problem 1.1.

[10] Is there a constant such that for any simple touching family of Jordan regions, ? Can we take ?

Our main result is the following (we made no real effort to optimize the constant 490, which is certainly far from optimal, our main concern was to give a proof that is as simple as possible).

Theorem 1.2.

For , any simple -touching family of Jordan regions is -colorable.

Note that apart from the constant 490, Theorem 1.2 is best possible. Figure 1 depicts two examples of simple -touching families of Jordan regions of chromatic number .

Figure 1. Two simple -touching families of Jordan regions with chromatic number .

It was proved in [2] that every simple -touching family of Jordan regions is -colorable (their result is actually stated for -touching families of strings, but it easily implies the result on Jordan regions). We obtain the next result as a simple consequence.

Corollary 1.3.

Any simple -touching family of Jordan regions is -colorable.

Proof.

Let be a simple -touching family of Jordan regions. If then can be colored with at most colors by the result of [2] mentioned above. If , then is also -touching, and it follows from Theorem 1.2 that can be colored with at most colors. Finally, if , Theorem 1.2 implies that can be colored with at most colors. ∎

Observe that for a given simple touching family of Jordan regions, if we denote by the least integer so that is -touching, then , since Jordan regions intersecting some point of the plane are pairwise intersecting. Therefore, we obtain the following immediate corollary, which is a positive answer to the problem raised by Reed and Shepherd.

Corollary 1.4.

For any simple touching family of Jordan regions, (and if ).

Note that the bound is also best possible (as shown by Figure 1, right).

It turns out that our main result also implies a positive answer to a question raised by Hliněný in 1998 [6]. A string is the image of some continuous injective function from to , and the interior of a string is the string minus its two endpoints. A contact systems of strings is a set of strings such that the interiors of any two strings have empty intersection. In other words, if is a contact point in the interior of a string , all the strings containing distinct from end at . A contact system of strings is said to be one-sided if for any contact point as above, all the strings ending at leave from the same side of (see Figure 2, left). Hliněný [6] raised the following problem:

Problem 1.5.

[6] Let be a one-sided contact system of strings, such that any point of the plane is in at most strings, and any two strings intersect in at most one point. Is it true that has chromatic number at most ? (or even , for some constant ?)

Figure 2. Turning a one-sided contact system of strings into a simple touching family of Jordan regions.

The following simple corollary of Theorem 1.2 gives a positive answer to Problem 1.5.

Corollary 1.6.

Let be a one-sided contact system of strings, such that any point of the plane is in at most strings, and any two strings intersect in at most one point. Then has chromatic number at most (and at most if ).

Proof.

Assume first that . It was proved in [2] that has chromatic number at most , so in this case at most , as desired. Assume now that . Let be obtained from by thickening each string of , turning into a (very thin) Jordan region (see Figure 2, from left to right). Since is one-sided, each intersection point contains precisely the same elements in and , and therefore and are equal, while is a simple -touching family of Jordan regions. If , then is also -touching and it follows from Theorem 1.2 that has chromatic number at most . Finally, if , then by Theorem 1.2, has chromatic number at most , as desired. ∎

A Jordan curve is the boundary of some Jordan region of the plane. We say that a family of Jordan curves is touching if for any two Jordan curves , the curves and do not cross (equivalently, either the interiors of the regions bounded by and are disjoint, or one is contained in the other). Moreover, if any point of the plane is on at most Jordan curves, we say that the family is -touching. Note that unlike above, the families of Jordan curves we consider here are not required to be simple (two Jordan curves may intersect in several points). Note that previous works on intersection of Jordan curves have usually considered the opposite case, where every two curves that intersect also cross (see for instance [7] and the references therein).

Let be a -touching family of Jordan curves. For any two intersecting Jordan curves , let be the set of Jordan curves distinct from such that the (closed) region bounded by contains exactly one of . The cardinality of is called the distance between and , and is denoted by . Note that since is -touching, any two intersecting Jordan curves are at distance at most .

Given a -touching family , the average distance in is the average of , over all pairs of intersecting Jordan curves . We conjecture the following.

Conjecture 1.7.

For any -touching family of Jordan curves, the average distance in is at most .

It was proved by Fox and Pach [4] that each -touching family of strings is -colorable, which directly implies that each -touching family of Jordan curves is -colorable (note that ). We show how to improve this bound when the average distance is at most , for some .

Theorem 1.8.

Let be a -touching family of Jordan curves, such that the average distance in is at most , for some constant . Then the chromatic number of is at most , where for and .

Note that . Theorem 1.8 has the following direct corollary.

Corollary 1.9.

Let be a -touching family of Jordan curves, such that the average distance in is at most . Then is -colorable, where

By Corollary 1.9, a direct consequence of Conjecture 1.7 would be that every -touching family of Jordan curves is -colorable.

For any -touching family of Jordan curves, the average distance is at most . Theorem 1.8 implies that every family of Jordan curves is -colorable, which is the bound of Fox and Pach [4] (without the ). To understand the limitation of Theorem 1.8 it is interesting to consider the case . Then tends to , and we obtain in this case that is -colorable. A particular case is when . This is equivalent to say that any two intersecting Jordan curves are at distance 0, and therefore the family of Jordan curves can be turned into a -touching family of Jordan regions (here and everywhere else in this manuscript, it is crucial that the curves are pairwise non-crossing). Note that it was proved in [1] (see also [2]) that -touching families of Jordan regions are -colorable.

In order to motivate Conjecture 1.7 and give it some credit, we then prove the following weaker version.

Theorem 1.10.

Let be a family of -touching Jordan curves. Then the average distance in is at most .

An immediate consequence of Theorems 1.8 and 1.10 is the following small improvement over the bound of Fox and Pach [4] in the case of Jordan curves.

Corollary 1.11.

Any -touching family of Jordan curves is -colorable.

An interesting connection between the chromatic number of -touching families of Jordan curves and the packing number of directed cycles in directed planar graphs was observed by Reed and Shepherd in [10]. In a planar digraph , let be the maximum number of vertex-disjoint directed cycles. This quantity has a natural linear relaxation, where we seek the maximum for which there are weights in on each directed cycle of , summing up to , such that for each vertex of , the sum of the weights of the directed cycles containing is at most 1. It was observed by Reed and Shepherd [10] that for any there are integers and such that and contains a collection of pairwise non-crossing directed cycles (counted with multiplicities) such that each vertex is in at most of the directed cycles. If we replace each directed cycle of the collection by its image in the plane, we obtain a -touching family of Jordan curves. Assume that this family is -colorable, for some constant . Then the family contains an independent set (a set of pairwise non-intersecting Jordan curves) of size at least . This independent set corresponds to a packing of directed cycles in . As a consequence, , and then . The following is therefore a direct consequence of Corollaries 1.9 and 1.11.

Theorem 1.12.

For any planar directed graph , . Moreover, if Conjecture 1.7 holds, then

This improves a result of Reed and Shepherd [10], who proved that for any planar directed graph , . The same result with a constant factor of essentially followed from the result of Fox and Pach [4] (and the discussion above). Using classical results of Goemans and Williamson [5], Theorem 1.12 also gives improved bounds on the ratio between the maximum packing of directed cycles in planar digraphs and the dual version of the problem, namely the minimum number of vertices that needs to be removed from a planar digraph in order to obtain an acyclic digraph.

Organization of the paper. The proofs of Theorem 1.2, 1.8 and 1.10 are given in Sections 2, 3 and 4, respectively. Section 5 concludes the paper with some remarks and open problems.

2. Proof of Theorem 1.2

In the proof below we will use the following parameters instead of their numerical values (for the sake of readability): , , and .

The proof proceeds by contradiction. Assume that there exists a counterexample , and take it with a minimum number of Jordan regions.

We will construct a bipartite planar graph from as follows: for any Jordan region of we add a vertex in the interior of (such a vertex will be called a disk vertex), and for any contact point (i.e. any point on at least two Jordan regions), we add a new vertex at (such a vertex will be called a contact vertex). Now, for every Jordan region and contact point on , we add an edge between the disk vertex corresponding to and the contact vertex corresponding to .

We now start with some remarks on the structure of .

Claim 2.1.

is a connected bipartite planar graph.

Proof.

The fact that is planar and bipartite easily follows from the construction. If is disconnected, then itself is disconnected, and some connected component contradicts the minimality of . ∎

Claim 2.2.

All the faces of have degree (number of edges in a boundary walk counted with multiplicity) at least 6.

Proof.

Note that by construction, the graph is simple (i.e. there are no parallel edges). Assume for the sake of contradiction that has a face of degree 4. Then either bounds three vertices (and consists of two Jordan curves intersecting in a single point, in which case the theorem trivially holds), or the face corresponds to two Jordan regions of sharing two distinct points, which contradicts the fact that is simple. Since is bipartite, it follows that each face has degree at least 6. ∎

Two disk vertices having a common neighbor are said to be loose neighbors in (this corresponds to intersecting Jordan regions in ).

Claim 2.3.

Every disk vertex has at least loose neighbors in .

Proof.

Assume that some disk vertex has at most loose neighbors in . Then the corresponding Jordan region of intersects at most other Jordan regions in . By minimality of , the family is -colorable, and any -coloring easily extends to , since intersects at most other Jordan regions. We obtain a -coloring of , which is a contradiction. ∎

Claim 2.4.

has minimum degree at least 2, and each contact vertex has degree at most .

Proof.

The fact that each contact vertex has degree at least two and at most directly follows from the definition of a -touching family. If contains a disk vertex of degree at most one, then since contact vertices have degree at most , has at most loose neighbors in , which contradicts Claim 2.3. ∎

Claim 2.5.

For any edge , at least one of has degree at least 3.

Proof.

Assume that a disk vertex of degree 2 is adjacent to a contact vertex of degree 2. Then has at most loose neighbors, which contradicts Claim 2.3. ∎

A -vertex (resp. -vertex, -vertex) is a vertex of degree (resp. at most , at least ). A -vertex is also said to be a big vertex. A vertex that is not big is said to be small.

Claim 2.6.

Each disk vertex of degree at most 7 has at least one big neighbor.

Proof.

Assume that some disk vertex of degree at most 7 has no big neighbor. It follows that all the neighbors of have degree at most , and so has at most loose neighbors, which contradicts Claim 2.3. ∎

We now assign to each vertex of a charge , and to each face of a charge (here the function refers to the degree of a vertex or a face). By Euler’s formula, the total charge assigned to the vertices and edges of is precisely . We now proceed by locally moving the charges (while preserving the total charge) until all vertices and faces have nonnegative charge. In this case we obtain that , which is a contradiction. The charges are locally redistributed according to the following rules (for Rule (R2), we need the following definition: a bad vertex is a disk 3-vertex adjacent to two contact 2-vertices , such that the three faces incident to have degree 6 and the neighbors of and have degree 3).

  1. For each big contact vertex and each sequence of three consecutive neighbors of in clockwise order around , we do the following. If has a unique big neighbor (namely, ), then gives to . Otherwise gives 1 to , and to each of and .

  2. Each big contact vertex gives to each bad neighbor.

  3. Each small contact vertex of degree at least 4 gives to each neighbor.

  4. Each contact 3-vertex adjacent to some -vertex gives to each neighbor of degree 2.

  5. Each disk vertex of degree at least 4 gives to each neighbor of degree at most 3.

  6. For each disk vertex of degree 3 and each neighbor of with , we do the following. If either has degree 3, or has degree two and the neighbor of distinct from has degree at least 4, then gives to . Otherwise, gives 1 to .

  7. Each face of degree at least 8 gives to each disk vertex incident with .

We now analyze the new charge of each vertex and face after all these rules have been applied.

By Claim 2.2, all faces have degree at least 6. Since faces of degree 6 start with a charge of 0, and do not give any charge, their new charge is still 0. Let be a face of degree . Then starts with a charge of and gives at most by Rule (R7). The new charge is then at least , as desired.

We now consider disk vertices. Note that these vertices receive charge by Rules (R1–4) and (R7), and give charge by Rules (R5–6). Consider first a disk vertex of degree . Then starts with a charge of and gives at most (by Rule (R5)), so the new charge of is at least (since ).

Assume now that is a disk vertex of degree . Then by Claim 2.6, has at least one big neighbor. The vertex starts with a charge of , receives at least by Rule (R1), and gives at most by Rule (R5). The new charge of is then at least (since ).

We now consider a disk vertex of degree 3. Again, it follows from Claim 2.6 that has at least one big neighbor. The vertex starts with a charge of 0, and since has at least one big neighbor, receives at least from its big neighbors by Rule (R1). Let be a big neighbor of , and assume first that at least one of the two neighbors of distinct from (call them ) is not a 2-vertex adjacent to two 3-vertices. Then by Rule (R6), gives at most to (recall that by Claim 2.5, no two vertices of degree are adjacent in ). In this case the new charge of is at least , as desired. Assume now that both have degree two and their neighbors all have degree 3. In this case gives 1 to each of and the new charge of is at least . If is incident to a face of degree at least 8, receives at least from such a face, and its new charge is at least , as desired. So we can assume that all the faces incident to are faces of degree 6. In other words, is a bad vertex. Then gives an additional charge of to by Rule (R2), and the new charge of in this last case is at least , as desired.

Assume now that is a disk vertex of degree two. Then the vertex starts with a charge of . By Claim 2.6, has a big neighbor, call it . By Claim 2.5, the neighbor of distinct from , call it , has degree at least 3. If is big then receives a charge of by Rule (R1) and its new charge is thus at least , so we can assume that is small (in particular, receives from by Rule (R1)). If has degree at least 4, then gives a charge of to by Rule (R3) and the new charge of is then at least . If lies on a face of degree at least 8, then receives from this face by Rule (R7), and its new charge is then at least . So we can assume that has degree 3 and all the faces containing have degree 6. If is adjacent to some -vertex, then gives to by Rule (R4), and in this case the new charge of is at least . So we can further assume that all the neighbors of are 2-vertices. Call the neighbors of distinct from , and for let be the neighbor of distinct from . Since has degree 3, it follows from Claim 2.6 that and are big. Let (resp. ) be the neighbor of immediately succeeding (resp. preceding) in clockwise order around . The faces containing have degree 6, and since is bipartite with minimum degree at least 2 (by Claims 2.1 and 2.4), each of these two faces is bounded by 6 vertices. As a consequence, we can assume that is adjacent to and is adjacent to (see Figure 3). It follows that each of has at least two big neighbors. Therefore, by Rule (R1), received from (in addition to the that were taken into account earlier) . So the new charge of is at least , as desired.

Figure 3. Contact vertices are depicted with white squares and disk vertices are depicted with black dots.

We now study the new charge of contact vertices. Note that contact vertices give charge by Rules (R1–4) and receive charge by Rules (R5–7). Consider a contact vertex of degree two. Then starts with a charge of . By Claim 2.5, the two neighbors of (call them an ) have degree at least 3. If they both have degree at least 4, then they both give to by Rule (R5), and the new charge of is at least . If one of has degree at least 4 and the other has degree 3, then receives by Rule (R5) and by Rule (R6). In this case the new charge of is at least . Finally, if and both have degree 3, then they both give 1 to by Rule (R6), and the new charge of is at least , as desired.

Consider a contact vertex of degree 3. Then starts with a charge of 0, and only gives charge if Rule (R4) applies. In this case, gives a charge of to at most two of its neighbors. However, if Rule (R4) applies, then by definition, has a neighbor of degree at least 3. Then receives at least from such a neighbor by Rules (R5–6). In this case, the new charge of is at least (since ).

Assume now that is a contact vertex of degree . Then starts with a charge of . If is small, then gives at most by Rule (R3), and the new charge of is then at least . Assume now that is big. In this case, applications of Rule (R1) cost no more than charge. We claim the following.

Figure 4. Illustration of the proof of Claim 2.7. The four removed vertices are circled.
Claim 2.7.

For every big contact vertex of degree , applications of Rule (R2) cost no more than charge.

Proof.

We will show that never gives a charge of to three consecutive neighbors of , which implies the claim. Assume for the sake of contradiction that gives a charge of to three consecutive neighbors of (in clockwise order around ). Assume that the neighbors of are (in clockwise order around ), and the neighbors of are (in clockwise order around ). Recall that by the definition of a bad vertex, each of has degree two, and all the faces incident to or have degree 6. Let be the neighbors of distinct from , and let be the neighbors of distinct from . By the definition of a bad vertex, each of has degree 3, and since all the faces incident to or have degree 6, and the vertices have a common neighbor, which we call . Again, by the definition of a bad vertex, the neighbor of distinct from has degree two and is adjacent to (see Figure 4). Let be the family obtained from by removing the disks corresponding to . By minimality of , has a -coloring , which we seek to extend to (by a slight abuse of notation we identify a disk vertex of with the corresponding disk of ). Note that and have at most colored neighbors, while and have at most colored neighbors. Since colors are available, it follows that each of has a list of at least 2 available colors, while each of has a list of at least 3 available colors. We must choose a color in each of the four lists such that each pair of vertices among , except the pair , are assigned different colors. This is equivalent to the following problem: take to be the complete graph on 4 vertices minus an edge, assign to each vertex of an arbitrary list of at least colors, and then choose a color in each list such that adjacent vertices are assigned different colors. It follows from a classical result of Erdős, Rubin and Taylor [3] that this is possible for any 2-connected graph distinct from a complete graph and an odd cycle (and in particular, this holds for ). Therefore, the -coloring of can be extended to to obtain a -coloring of , which is a contradiction. This proves Claim 2.7. ∎

Hence, if is a big contact vertex of degree , then the new charge of is at least . Since is big, and so the new charge of is at least (since ). It follows that the new charge of all vertices and faces is nonnegative, and then the total charge (which equals ) is nonnegative, which is a contradiction. This concludes the proof of Theorem 1.2.

3. Proof of Theorem 1.8

We start with a simple lemma showing that in order to bound the chromatic number of -touching families of Jordan curves, it is enough to bound asymptotically the number of edges in their intersection graphs.

Lemma 3.1.

Assume that there is a constant and a function such that for any integers and any -touching family of Jordan curves, the graph has at most edges. Then for any integer , any -touching family of Jordan curves is -colorable.

Proof.

Let be a -touching family of Jordan curves, and let denote the number of edges of . For some integer , replace each element by concentric copies of , without creating any new intersection point (i.e., any portion of Jordan curve between two intersection points is replaced by parallel portions of Jordan curves). Let denote the resulting family. Note that is -touching, contains elements, and contains edges. Hence, we have . Therefore, , and contains a vertex of degree at most . This holds for any , and since the degree of a vertex is an integer and , indeed contains a vertex of degree at most . We proved that -touching families of Jordan curves are -degenerate, and therefore -colorable. ∎

We will also need the following two lemmas.

Lemma 3.2.

For any integers such that , and for any ,

Proof.

For fixed and we write . Note that . Furthermore, for all reals , So for all integers . ∎

Lemma 3.3.

For any reals and , we have .

Proof.

We clearly have . To see that the second part of the inequality holds, observe first that for any real , we have and thus the desired inequality holds for .

Assume now that . Note that for any real , we have . Thus,

with the rightmost inequality holding since . It follows that , as desired. ∎

We are now ready to prove Theorem 1.8.

Proof of Theorem 1.8. Let be a -touching family of Jordan curves, with average distance at most , and let be as provided by Theorem 1.8. Note that since , we have . We denote by the edge-set of , and by the cardinality of . We will prove that . Using Lemma 3.1, this implies that the chromatic number of any -touching family of Jordan curves with average distance at most is at most . Note that the chosen value of minimizes the value of . In the remainder of the proof, we will only use the fact that .

As observed in [2], we can assume without loss of generality that each Jordan curve is a polygon (this is a simple consequence of the fact that any simple plane graph can be drawn with straight-line edges).

We recall that for two intersecting Jordan curves , is the set of Jordan curves distinct from such that the (closed) region bounded by contains exactly one of , and the cardinality of (which is called the distance between and ) is denoted by . For each edge , we choose an arbitrary point in the intersection of the Jordan curves corresponding to and . Observe that since the curves are pairwise non-crossing, is contained in all the curves of . We now select each Jordan curve of uniformly at random, with probability . Let be the obtained family. The expectation of the number of Jordan curves in is . For any pair of intersecting Jordan curves , we denote by the probability that the set of Jordan curves of containing satisfies

  1. has size at most 3,

  2. , and

  3. if , then the Jordan curve of distinct from and is not an element of .

Observe that

where denotes the number of Jordan curves containing in .

We say that an edge is good if satisfy (1), (2), and (3) above. It follows from Lemmas 3.2 and 3.3 that the expectation of the number of good edges is

since .

Figure 5. The point is depicted by a white dot, and the newly created points are depicted by black dots.

Let be obtained from by slightly modifying the Jordan curves around each intersection point as follows. If , for some good edge , then we do the following. Note that by (1) and (2), and is contained in at most one Jordan curve of distinct from and . Assume that such a Jordan curve exists, and call it . By (3), and are at distance 0 in . We then slightly modify in a small disk centered in so that for any , if and are at distance 0 in , then they remain at distance 0 in . Moreover, the point and the newly created points are 2-touching in (see Figure 5). If (resp. ) is also a good edge with (resp. ), then note that the conclusion above also holds with replaced by (resp. ). Now, for any other intersection point of Jordan curves of , that is not equal to for some good edge , we make the Jordan curves disjoint at . It follows from the definition of a good edge that the family obtained from after these modifications is 2-touching, and for any good edge , and are at distance 0 in . Note that is planar, since is 2-touching, and its expected number of edges is . Since the number of edges of a planar graph is less than three times its number of vertices, we obtain:

As a consequence,

as desired. This concludes the proof of Theorem 1.8.

4. Proof of Theorem 1.10

The following is an easy variation of the main result of Fox and Pach [4]. Consider three Jordan curves such that is outside the region bounded by , is inside the region bounded by , and intersects . Then we say that the pair is -crossing.

Lemma 4.1.

Let be a Jordan curve, and let be a family of Jordan curves such that is -touching and all the elements of intersect . Then the number of -crossing pairs in is at most .

Proof.

Let be the number of -crossing pairs in . For each -crossing pair in , we consider an arbitrary point in . We now select each Jordan curve of uniformly at random with probability . Let be the resulting family. A -crossing pair in is good if contains and , but does not contain any other Jordan curve of containing . Note that the probability that a given -crossing pair is good is at least , and therefore the expectation of the number of good -crossing pairs is at least . For any intersection point of Jordan curves of , that is not equal to for some good -crossing pair , we make the Jordan curves disjoint at (this is possible since the Jordan curves are pairwise non-crossing). Let be the obtained family. Observe that is 2-touching and each intersection point contains one Jordan curve lying outside the region bounded by and one Jordan curve lying inside the region bounded by . The graph is therefore planar and bipartite. The expectation of the number of vertices of is and the expectation of the number of edges of is at least . Since any planar bipartite graph on vertices contains at most edges, it follows that . Since , we obtain that , as desired. ∎

Some planar quadrangulations can be represented as 2-touching families of Jordan curves intersecting a given Jordan curve (so that each edge of the quadrangulation corresponds to a -crossing pair of Jordan curves). Therefore, the bound cannot be decreased (by more than an additive constant) in the proof of Lemma 4.1. Furthermore, the possibly near-extremal example in Figure 6 shows that the bound in Lemma 4.1 cannot be improved to less than .

Figure 6. A family of Jordan curves that all intersect a fixed Jordan curve , such that is touching. Each Jordan curve in the interior of touches each Jordan curve in the exterior of . In the interior, there are only two sets of concentric Jordan curves. The remaining Jordan curves are in the exterior of . As goes to infinity, the number of crossing pairs divided by converges to .

We are now ready to prove Theorem 1.10.

Proof of Theorem 1.10. Let denote the edge-set of and let . Let . Note that the average distance in is .

Fix some , and set . For any edge with , we do the following. Note that there is a unique ordering of the elements of , such that for any , the distance between and is . Then the edge gives a charge of 1 to each of the elements . Let be the total charge given during this process. Note that

We now analyze how much charge was received by an arbitrary Jordan curve . Let denote the neighborhood of , and let (resp. ) denote the set of neighbors of lying outside (resp. inside) the region bounded by . Observe that if received a charge of 1 from some edge , then without loss of generality we have , , and both and are at distance at most from . Let denote the set of neighbors of that are at distance at most from . Then the charge received by is at most the number of -crossing pairs in the subfamily of induced by , which is at most by Lemma 4.1.

For any , let denote the number of edges such that and are at distance at most . It follows from the analysis above that . Therefore, . Since , we have .

We now study the contribution of an arbitrary edge to the sum . Let be some integer. If , then contributes at most to , and therefore at most to . Note that there are such edges . For each , each edge such that contributes at most to , and there are such edges. Finally, each edge with contributes at most 1 to , and there are such edges. As a consequence,

since for every . As a consequence, we obtain that . Since this holds for any integer , we have and therefore , as desired.

5. Remarks and open questions

Most of the proof of Theorem 1.2 proceeds by finding a Jordan region intersecting at most other Jordan regions (see Claim 2.3). On a single occasion, we use a different reduction (via a list-coloring argument). A natural question is: could this be avoided? Is it true that in any simple -touching family of Jordan regions, if is large enough, then there is a Jordan region which intersects at most other Jordan regions? It turns out to be wrong, as depicted in Figure 7. However, a proof along the lines of that of Theorem 1.2 (but significantly simpler), shows that if is large enough, then there is a Jordan region which intersects at most other Jordan regions. It was pointed out to us by Patrice Ossona de Mendez (after the original version of this manuscript was submitted) that he also obtained this result in 1999 (see [9]). His result and its proof are stated with a completely different terminology, but the ideas are essentially the same. In particular, his result also implies (relatives of) our Corollaries 1.3 and 1.4.

Figure 7. Every Jordan region intersects precisely other Jordan regions.

This can be used to obtain a result on the chromatic number of simple families of -touching Jordan curves (families of -touching Jordan curves such that any two Jordan curves intersect in at most one point). Using the result mentioned above (that if is sufficiently large and the interiors are pairwise disjoint, then there is a Jordan region that intersects at most other Jordan regions), it is not difficult to show that the chromatic number of any simple family of -touching Jordan curves is at most plus a constant. We believe that the answer should be much smaller.

Problem 5.1.

Is it true that for some constant , any simple family of -touching Jordan curves can be colored with at most colors?

It was conjectured in [2] that if is a family of pairwise non-crossing strings such that (i) any two strings intersect in at most one point and (ii) any point of the plane is on at most strings, then is -colorable, for some constant . Note that, if true, this conjecture would give a positive answer to Problem 5.1.

References

  • [1] O. Amini, L. Esperet, and J. van den Heuvel, A unified approach to distance-two colouring of graphs on surfaces, Combinatorica 33(3) (2013), 253–296.
  • [2] L. Esperet, D. Gonçalves, and A. Labourel, Coloring non-crossing strings, Electron. J. Combin. 23(4) (2016), #P4.4.
  • [3] P. Erdős, A.L. Rubin, and H. Taylor, Choosability in graphs, Congr. Numer. 26 (1979), 125–157.
  • [4] J. Fox and J. Pach, Touching strings, manuscript, 2012.
  • [5] M.X. Goemans and D.P. Williamson, Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs, Combinatorica 17 (1997), 1–23.
  • [6] P. Hliněný, The maximal clique and colourability of curve contact graphs, Discrete Applied Math. 81 (1998), 59–68.
  • [7] R.J. Kang and T. Müller, Arrangements of pseudocircles and circles, Discrete Comput. Geom. 51(4) (2014), 896–925.
  • [8] A. Kostochka, Coloring intersection graphs of geometric figures with a given clique number, in: J. Pach (ed.), Towards a Theory of Geometric Graphs, Contemp. Math. 342, Amer. Math. Soc., Providence, RI, 127–138, 2004.
  • [9] P. Ossona de Mendez, The Reduced Genus of a Multigraph, in: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS 99), Trier, Germany, 1999, 16–31.
  • [10] B.A. Reed and F.B. Shepherd, The Gallai-Younger conjecture for planar graphs, Combinatorica 16(4) (1996) 555–566.
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