on the complexity of finite
subgraphs of the curve graph
We say a graph has property when it is an induced subgraph of the curve graph of a surface of genus with punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to . We introduce a new invariant of a graph, the nested complexity length, which provides a novel obstruction to . For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is , i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large ‘half-graphs’ do not have , and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have . We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large multipartite subgraphs cannot have . This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine .
1. Statement of results
Let indicate a hyperbolizable surface of genus with punctures (i.e. ). The curve graph of , denoted , is the infinite graph whose vertices are isotopy classes of simple closed curves on and whose edges are given by pairs of curves that can be realized disjointly. Let indicate the property that a graph is an induced subgraph of the curve graph . We are concerned with the following motivating question:
Which finite graphs have ? When is obstructed?
The low complexity cases , , and are trivial, so we assume further that . See §2 for details and complete definitions.
Property has been considered in different guises in the literature [EET, CW1, CW2, Kob, KK1, KK2, KK3]. It is not hard to see that every finite graph has for large enough , though it is remarkable that there exist finite graphs which do not have for any [EET, §2].111The authors thank Sang-Hyun Kim for bringing this reference to our attention. Question 1 above is especially salient when and are fixed, and we adopt this point of view in everything that follows.
There are few known obstructions to a graph having property . The simplest is the presence of a clique of that is too large, as the size of a maximal complete subgraph of is . A more subtle obstruction follows from a surprising fact proved by Bestvina, Bromberg, and Fujiwara: the graph has finite chromatic number [BBF, KK1].
We introduce an invariant of a graph which we call the ‘nested complexity length’ that controls the topological complexity of any surface whose curve graph contains as an induced subgraph (see §4 for a precise definition). The following is our main result, providing a new obstruction to . In fact, our calculation applies equally well to the clique graph of , whose vertices are multicurves and with edges for disjointness.
We have .
The nested complexity length of a graph is obtained via a supremum over all nested complexity sequences, and while this definition is useful in light of the theorem above, we know of no non-exhaustive algorithm that computes the nested complexity length of a finite graph. Thus it is natural to ask:
What is an algorithm to compute the nested complexity length of a finite graph? How can one find effective upper bounds?
As a starting point toward this question, Proposition 23 gives an upper bound for the nested complexity length of a graph which is exponential in the maximal size of a complete bipartite subgraph of .
We describe several corollaries of Theorem 2 below. The first concerns half-graphs, a family of graphs that has attracted study in combinatorics and model theory.
Given an integer and a graph , we say that is a half-graph of height if there is a partition of the vertices of such that the edge occurs if and only if . The unique bipartite half-graph of height is denoted .
If is a half-graph of height , with , then does not have .
Since is 2-colorable and triangle-free, Corollary 5 implies:
Chromatic number and maximal clique size are not sufficient to determine if a finite graph has .
In fact, a more dramatic illustration of Corollary 6 can be made quantitatively. By definition, is a hereditary graph property (i.e. closed under isomorphism and induced subgraphs). Asymptotic enumeration of hereditary graph properties has been studied by many authors, resulting in a fairly precise description of possible ranges for growth rates [BBW, Theorem 1]. Combining Corollary 5 with a result of Alon, Balogh, Bollobás, and Morris [ABBM], we obtain an upper bound on the asymptotic enumeration of . The argument for the following is in §4. Given , let denote the class of graphs with vertex set satisfying .
There is an such that, for large , .
The set of graphs on satisfying the clique and chromatic tests for includes all -colorable graphs on , and thus this set has size . In particular, the upper bound above cannot be obtained from the clique or chromatic number obstructions to . This also strengthens the statement of Corollary 6: among the graphs on satisfying the clique and chromatic tests, the probability of possessing tends to as .
For monotone graph properties (i.e. closed under isomorphism and subgraphs), even more is known concerning asymptotic structure [BBS]. However, we can use Corollary 5 to show that in most cases, is not monotone.
If then is not a monotone graph property.
If then contains a pair of disjoint incompressible subsurfaces that support essential nonperipheral simple closed curves. It follows that the complete bipartite graph has property for all . However, the half-graph is a subgraph of . ∎
It is also worth observing that, for , this result thwarts the possibility of using the Robertson-Seymour Graph Minor Theorem [RS] to characterize by a finite list of forbidden minors. Of course this would also require to be closed under edge contraction, which is already impossible just from the clique number restriction. Indeed, edge contraction of the graphs produces arbitrarily large complete graphs.
Following Kim and Koberda, Question 1 is closely related to the problem of which right-angled Artin groups (RAAGs) embed in the mapping class group of [Kob, KK1, KK2, KK3]. If the graph has , then is a RAAG subgroup of [Kob, Theorem 1.1]. The converse is false in general [KK3, Theorem 3], but a related statement holds: if the RAAG embeds as a subgroup of , the graph is an induced subgraph of the clique graph [KK1, Lemma 3.3].
By exploiting a construction by Erdős of graphs with arbitrarily large girth and chromatic number, Kim and Koberda produce ‘not very complicated’ (precisely, cohomological dimension two) RAAGs that do not embed in [KK1, Theorem 1.2]. Rephrasing Theorem 2 in this context, the graphs provide such examples which are ‘even less complicated’.
If is a RAAG subgroup of , then
Moreover, for any and there exist bipartite graphs so that does not embed in .
The nested complexity length of a graph is closely related to the centralizer dimension of , i.e. the longest chain of nontrivially nested centralizers in the group (this algebraic invariant is discussed elsewhere in the literature [MS]). In particular, it is straightforward from the definitions that the centralizer dimension of is at least . The possible centralizers of an element of a RAAG have been classified by Servatius [Ser], and the characterization there would seem to suggest that equality does not hold in general. Of course, this is impossible to check in the absence of a method to compute nested complesity length, and Question 3 arises naturally.
Our next corollary concerns the model theoretic behavior of . We focus on stability, one of the most important and well-developed notions in modern model theory. Given a first-order structure, stability of its theory implies an abstract notion of independence and dimension for definable sets in that structure (see Pillay [Pil] for details). Stability can also be treated as a combinatorial property obtained from half-graphs. Given an integer , a graph is -edge stable if it does not contain any half-graph of height as an induced subgraph. We can thus rephrase Corollary 5.
is -edge stable for .
When considering the first-order theory of in the language of graphs, this corollary implies that the edge formula (and thus any quantifier-free formula) is stable in the model theoretic sense [TZ, Theorem 8.2.3]. Whether arbitrary formulas are stable remains an intriguing open question, and would likely require some understanding of quantifier elimination for the theory of in some suitable expansion of the graph language. This aspect of the nature of curve graphs remains unexplored, and stability is only one among a host of natural questions about their first-order theories to pursue.
On the other hand, edge-stability of alone has strong consequences for the structure of large finite subgraphs of , via Szemerédi’s regularity lemma [KS, Sze]. In particular, Malliaris and Shelah show that if is a -edge-stable graph, then the regularity lemma can be strengthened so that in Szemerédi partitions of large induced subgraphs of , the bound on the number of pieces is significantly improved, there is no need for irregular pairs, and the density between each pair of pieces is within of or [MS]. Thus a consequence of our work is that the class of graphs with enjoys this stronger form of Szemerédi regularity.
Next we consider an explicit family of multipartite graphs.
Given integers , let denote the complete -partite graph in which each piece of the partition has size .
In Example 25, we show . Combined with the Erdős-Stone Theorem, this implies a general relationship (Proposition 28) between nested complexity length and the upper density of an infinite graph , i.e. the supremum over real numbers such that contains arbitrarily large finite subgraphs of edge density . In the case of the curve graph , we show in Lemma 30 that is obstructed from having for large . From this we obtain the following exact calculation of the upper density of the curve graph, which is proved in §6.
The upper density is equal to .
The exceptional cases and are again remarkable in that they imply in Theorem 13. Thus any family of graphs with and in these exceptional cases must satisfy
Given or , does there exist such that, for any family of graphs with and one has ?
Finally, we use the analysis of for to extend Corollary 6.
Chromatic number, maximal clique size, and nested complexity length are not sufficient to determine if a finite graph has .
The authors thank Tarik Aougab, Ian Biringer, Josh Greene, Sang-Hyun Kim, and Thomas Koberda for helpful comments and discussions.
2. Notation and conventions
A graph consists of a set of vertices and an edge set which is a subset of unordered distinct pairs from ; we denote the edge between vertices and by . A subgraph is induced if and implies that . The closed neighborhood of a vertex is the set of vertices
Given a graph the right-angled Artin group corresponding to , denoted , is defined by the following group presentation: the generators of are given by and there is a commutation relation for every edge .
Recall that we are concerned with the hyperbolizable surface of genus with punctures, so we assume that . The mapping class group of , denoted , is the group .
By a curve we mean the isotopy class of an essential nonperipheral embedded loop on , and we refer to the union of curves which can be made simultaneously disjoint as a multicurve. The curve graph is the graph consisting of a vertex for each curve on , and so that a pair of curves span an edge when the curves can be realized disjointly. The clique graph of the curve graph is the graph obtained as follows: The vertices of correspond to cliques of (i.e. multicurves), and two cliques are joined by an edge when they are simultaneously contained in a maximal clique (i.e can be realized disjointly). The curve graph is the subgraph of the clique graph induced by the one-cliques.
We strengthen the assumption on and above to . When the curve graph as defined above has no vertices. When or , the curve graph has no edges, and the matter of deciding if a graph has in these cases is trivial. The common alteration of the definition of these curve graphs yields the Farey graph. We make no comment on induced subgraphs of the Farey graph, though a comprehensive classification can be made.
Whenever we refer to a subsurface we make the standing assumption that is a disjoint union of closed incompressible homotopically distinct subsurfaces with boundary of , i.e. the inclusion induces an injection on the fundamental groups of components of . We write for the isotopy class of .
Given a subsurface , we say that a curve is supported on if it has a representative which is either contained in an annular component of , or is a nonperipheral curve in a (necessarily non-annular) component. The curve is disjoint from if it is has a representative disjoint from . We say that is transverse to , written , if it is not supported on and not disjoint from . A multicurve is supported on (resp. disjoint from) if each of its components is supported on (resp. disjoint from) , and is transverse to if at least one of its components is. Each of these above definitions applies directly for curves and isotopy classes of subsurfaces.
We note that the definitions above may be nonstandard, as they are made with our specific application in mind. For example, in our terminology the core of an annular component of is both supported on and disjoint from the subsurface .
Given a pair of subsurfaces and , we write when every curve supported on is supported on . If and (denoted ), we say that and are nontrivially nested.
Given a collection of curves , we let indicate the isotopy class of the minimal subsurface of , with respect to the partial ordering just defined, that supports the curves in . Concretely, a representative of can be obtained by taking the union of regular neighborhoods of the curves in (for suitably small regular neighborhoods that depend on the realizations of the curves in ) and filling in contractible complementary components of the result with disks.
A set of curves supported on a subsurface fills the subsurface if . Concretely, fills if the core of each annular component of is in , and if the complement in of a realization of the remaining curves consists of peripheral annuli and disks.
Given a subsurface , we write for the number of components of a maximal multicurve supported on .
3. Topological complexity of subsurfaces
Since the ambient surface is fixed throughout, it should not be surprising that any nontrivially nested chain of subsurfaces of has bounded length. We make this explicit below, using to keep track of ‘how much’ of the surface has been captured by subsurfaces from the chain. In fact, this is not quite enough to notice nontrivial nesting of subsurfaces, as annuli can introduce complications. We keep track of this carefully in Lemma 20 below. Throughout this section, and refer to a pair of subsurfaces of .
If , then is isotopic to a subsurface of .
The reader is cautioned that the converse is false: consider a one-holed torus subsurface and let indicate the disjoint union of with an annulus isotopic to . Though is isotopic to a subsurface of , the curve is supported on but not on . Thus .
Choose a collection of curves filling . Since , each curve in is supported on . It follows that there are small enough regular neighborhoods of representatives of the curves in that are contained in . Filling in contractible components of the complement produces a subsurface isotopic to inside . ∎
If and , then .
Fix representatives , using the previous lemma. Let be a collection of curves that fill . Since the curves in can be realized on so that they have regular neighborhoods contained in . Their complement in is a collection of disks and peripheral annuli, so their complement in must also be a collection of disks and peripheral annuli. Hence fills and . ∎
Suppose that . Then one of the following holds:
we have , or
there exists a curve which is the core of an annular component of but not disjoint from .
By Lemma 18 we have . Thus there is a curve supported on that is not supported on . If were the core of an annular component of , then would be immediate. Likewise if were disjoint from then again.
We are left with the case that is transverse to . Choose a boundary component of intersected essentially by . Evidently, this curve must be supported on . Since is supported on , either one has or is supported as well on . In this case, is the core of an annular component of intersected essentially by . If could be made disjoint from then its intersection with would be inessential, so we are done. ∎
Suppose that is a nontrivially nested chain of subsurfaces of . Then .
Choose with , and let . We construct, inductively on , a (possibly empty) multicurve . Evidently, we have , so that by Lemma 19 either , or there is a curve isotopic to the core of an annular component of but not disjoint from . In the second case, add to . Note that in the latter case there is a representative for which is contained in .
We continue inductively. Since we have , Lemma 19 guarantees that either , or there is a curve isotopic to the core on an annular component of but not disjoint from . Suppose that, for some , the curve is another component of supported on . Since has a representative disjoint from , and , has a representative disjoint from . Moreover, because cannot be made disjoint from but can be (since ), the curves and are not isotopic. It follows that the curve may be added to the multicurve so that remains a collection of disjoint curves, and so that its number of components increases by one.
At each step of the chain , either strictly increases, or gains another component. Since and , we have and . The number of components of is also at most , so we conclude that
Finally, we will make use of a straightforward certificate that .
Suppose that . If is a curve disjoint from , is a curve supported on , and and intersect essentially, then .
Because is disjoint from , it has a representative disjoint form . If were supported on , it would have a representative contained in , contradicting the assumption that and intersect essentially. Thus is a curve supported on but not supported on . ∎
4. The nested complexity length of a graph
The topological hypotheses of Lemma 21 suggest a natural combinatorial parallel, which we capture in the definition of nested complexity length.
Let be a graph.
Given , we say that is a nested complexity sequence for if for each there is such that .
The nested complexity length of , denoted , is given by
Note that in the definition of a nested complexity sequence, may be equal to , for ; see Figure 1 for a schematic in which this is not the case. To highlight this subtlety, and as a first step toward an answer to Question 3, we prove an upper bound for in terms of a maximal complete bipartite subgraph of .
Let be a graph. Fix and suppose does not contain a subgraph isomorphic to . Then .
Given , let . Note that . Set . For a contradiction, suppose is a nested complexity sequence for , witnessed by .
We inductively produce values below, such that for all , and for all . With these indices chosen, the set produces a (not necessarily induced) subgraph of isomorphic to , a contradiction.
If then we set . Otherwise, if then are independent vertices in . So , and we set .
Fix , and suppose we have defined as above. If for all then we let . Otherwise, let and suppose for some . We will find such that and for any . By induction, , and so setting finishes the inductive step of the construction.
Suppose no such exists. Then for all , we have for some . Fix . For any , we have , and so . Therefore, for any , and are independent vertices in , which means . In other words, we have shown that for all ,
It follows that are distinct elements of . ∎
We make note of two useful properties of that follow immediately from the definitions.
If is an induced subgraph of , then .
We have .
We also give the following examples, which are heavily exploited in the results outlined in §1.
Let be a half-graph of height . We may partition the vertices as , where if and only if . Then is a nested complexity sequence for , witnessed by . Therefore .
Let be the multipartite graph . Let where for . Set . For let and . Then is a nested complexity sequence for , witnessed by . Combined with Proposition 24(2), we have .
Proof of Corollary 7.
Given , let denote the class of graphs for which there is a partition such that holds if and only if . Any graph in contains an induced half-graph of height (take , where ). By Corollary 5, every graph with omits the class for . The result now follows immediately from Theorem 2 of Alon, et al. [ABBM]. ∎
5. Proof of Theorem 2
Recall that refers to the minimal isotopy class, with respect to inclusion, of a subsurface of that supports the curves in .
Proof of Theorem 2.
As is an induced subgraph of , Proposition 24(1) ensures that . We proceed by showing that is simultaneously a lower bound for and an upper bound for .
For the lower bound, choose a maximal multicurve . For each curve , choose a transversal , i.e. a curve intersecting essentially and disjoint from for . That such collections of curves exist is routine (e.g. a ‘complete clean marking’ in the language of Masur and Minsky [MM, §2.5] is an even more restrictive example, see Figure 2).
For each , let the curves and be given by
It is straightforward to check that forms a nested complexity chain for , with witnessing curves .
Towards the upper bound, suppose is a nested complexity sequence for , so that there exists a vertex in with for each as in Definition 22. For , let , and let .
Choose . Because , we have ; as , is disjoint from ; and since we have that is supported on . Because , and are independent vertices of , and so there are components of the multicurves and that intersect essentially. Lemma 21 now applies, so that . Thus is a nontrivially nested chain of subsurfaces, and Lemma 20 implies that . ∎
The upper bound can be strengthened for under the additional assumption that the antigraphs of the subgraphs induced on the vertices are connected for each (indeed, in this case the negative Euler characteristic of is strictly less than that of ). In particular, if then the bipartite half-graph does not have .
6. Obstructing and the upper density of the curve graph
We turn to and the upper density of curve graphs.
Let be a graph.
If the density of is
If is infinite the upper density of is
In other words, given an infinite graph , is the supremum over all such that contains arbitrarily large finite subgraphs of density at least . Given a fixed , it is easy to verify . It is a consequence of a remarkable theorem of Erdős and Stone that the graphs witness the densities of all infinite graphs. We record this precisely in language most relevant for our application ([Bol, Ch. IV], [ES]).
Fix . For any infinite graph , if then is a subgraph of for arbitrary .
Using this theorem, we obtain the following relationship between density and nested complexity length.
Let be an infinite graph. If then
If this inequality fails then we have . By the Erdős-Stone Theorem, there is such that is a subgraph of for arbitrarily large . Let be the size of the largest finite clique in , which exists since . If we consider a copy of in , it follows that each piece of the partition contains a pair of independent vertices. Therefore is an induced subgraph of . By Proposition 24 and Example 25, , which contradicts the choice of . ∎
Note that if is complete then . Therefore an infinite graph with upper density need not have large nested complexity length.
Finally, we obstruct the multipartite graphs from having for large enough and , and employ this fact in the proof of Theorem 13.
The maximum number of pairwise disjoint subsurfaces of which are not annuli or pairs of pants is .
It suffices to consider a pairwise disjoint sequence of subsurfaces in which each component is a four-holed sphere or a one-holed torus; any more complex subsurface can be cut further without decreasing the number of non-annular and non-pair of pants subsurfaces. Suppose there are one-holed tori and four-holed spheres. The dimension of the homology of requires . Additivity of the Euler characteristic under disjoint union implies that . Maximizing on this polygon is routine, and the solution is and . Moreover, it is straightforward to construct such a collection of subsurfaces of . ∎
This is enough to obstruct for large .
For , has if and only if .
Let . First, suppose is a sequence of pairwise disjoint subsurfaces consisting of tori and four-holed spheres, as guaranteed by Lemma 29. The curves supported on the induce as a subgraph of . Hence is an induced subgraph of for all and . Conversely, suppose towards contradiction that is an induced subgraph of , and let be the partition of its vertices. For each , both curves in are disjoint from the curves in , so the subsurfaces and are disjoint. Moreover, since the curves in intersect, is a connected surface that is not an annulus or a pair of pants. Thus is a sequence of disjoint non-annular and non-pair of pants subsurfaces, contradicting Lemma 29. ∎
We can now prove Theorem 13.
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