Topology of Hom complexes and test graphs for bounding chromatic number
The complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on chromatic number. In this paper we introduce new methods for understanding the topology of complexes, mostly in the context of -actions on graphs and posets (for some group ). We view the and complexes as functors from graphs to posets, and introduce a functor from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of complexes in terms of spaces of equivariant poset maps and -twisted products of spaces. When is the face poset of a simplicial complex , this provides a useful way to control the topology of complexes. These constructions generalize those of the second author from  as well as the calculation of the homotopy groups of complexes from .
Our foremost application of these results is the construction of new families of test graphs with arbitrarily large chromatic number - graphs with the property that the connectivity of provides the best possible lower bound on the chromatic number of . In particular we focus on two infinite families, which we view as higher dimensional analogues of odd cycles. The family of spherical graphs have connections to the notion of homomorphism duality, whereas the family of twisted toroidal graphs lead us to establish a weakened version of a conjecture (due to Lovász) relating topological lower bounds on chromatic number to maximum degree. Other structural results allow us to show that any finite simplicial complex with a free action by the symmetric group can be approximated up to -homotopy equivalence as for some graph ; this is a generalization of the results of Csorba from  for the case of . We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.
thmTheorem LABEL:#1 \newrefformatpropProposition LABEL:#1 \newrefformatcorCorollary LABEL:#1 \newrefformateq(LABEL:#1) \newrefformatit(LABEL:#1) \newrefformateqn(LABEL:#1) \newrefformatsecSection LABEL:#1 \newrefformatdefnDefinition LABEL:#1 \newrefformatdefDefinition LABEL:#1 \newrefformatlemLemma LABEL:#1 \newrefformatlemmaLemma LABEL:#1 \newrefformatremRemark LABEL:#1 \newrefformatexExample LABEL:#1 \newrefformatfigFigure LABEL:#1 \newrefformatqQuestion LABEL:#1 \newrefformatconjConjecture LABEL:#1
1.1 Some background
In his 1978 proof of the Kneser conjecture, Lovász  showed that the chromatic number of a graph is bounded below by the connectivity of (a complex later shown to be homotopy equivalent to) , a space of homomorphisms from the edge into . Some 25 years later, Babson and Kozlov  were able to show that the connectivity of provided the next natural bound on the chromatic number of (here is an odd cycle), answering in the affirmative a conjecture of Lovász. When is a loopless graph (as is the case when we consider the chromatic number), both and are naturally free -spaces, and one can consider a list of numerical invariants that measure the complexity of this action. The now standard proof of the Lovász criterion gives the following result.
Theorem 1 (Lovász).
For every graph ,
Here, for a free -space , is the smallest dimension of a sphere with the antipodal action that maps into equivariantly. Since , this implies the original result of Lovász from .
As a way to take advantage of the -topology, Babson and Kozlov introduced the use of characteristic classes to the study of complexes. In  they proposed and partially (according to the parity of ) proved the following result incorporating the -action on the complex.
For every graph ,
Here, for a free -space , is the highest nonvanishing power of the first Stiefel-Whitney class of the bundle (called the height or sometimes the cohomological index of ). Again, since X, this implies the connectivity bound that they did succeed in proving completely.
The first complete proof of \prettyrefthm:strong was given by the second author in . More recently, in , the same author was able to prove the following statement which not only implies \prettyrefthm:strong but also provides insight into the structure of the complexes and suggests extensions that are the theme to this paper.
Theorem 3 (Schultz).
For every graph ,
The direct system that defines the colimit is obtained by applying the functor to the system . Here denotes the -sphere with the -antipodal action on the left and the -reflection action on the right (which in total can be considered an action of ). One then observes that , where the action on the first space is induced by the nonidentity automorphism of and by the reflection of that flips an edge. The relevance of in this context is exhibited in the following result, a proof of which can be found in .
If is a -space and , then .
One can then combine these observations to obtain the following corollary which, when combined with \prettyrefthm:Lovtheorem, implies \prettyrefthm:strong.
If is a graph with at least one edge, then
thm:limitSch is similar in spirit to the results and constructions involved in the first author’s paper , where it is shown that the homotopy groups of a related space can be determined by certain graph theoretic closed paths in a (pointed) graph . In this context the role of a closed path (a circle) is played by , a cycle of length with loops on all the vertices (the looped 1-skeleton of a triangulated circle). One constructs a graph which parameterizes graph homomorphisms from cycles of arbitrary length. In particular it is shown that the space of (pointed) maps from a circle into can be recovered as path components of , which in turn can be approximated by the spaces .
1.2 New results
In this paper, we generalize the constructions discussed above by showing that one can take graphs obtained as the 1-skeleton of topologically ‘desirable’ spaces and apply them directly to complexes. In this way we unify existing results regarding complexes as well as provide new theorems and constructions. In our applications, the spaces of interest will primarily be spheres with a -action given by the antipodal/reflection maps. One obtains a graph by taking a looped 1-skeleton of a triangulation of the space, which can then be utilized in the context of the complex. More generally, if is a poset we obtain a graph (see \prettyrefdef:1) whose vertices are the atoms of with adjacency if and only if there exists a such that and . In the case that is the face poset of a triangulation of the 1-sphere, we recover the graphs discussed above. The graphs obtained as have loops on every vertex and hence do not admit homomorphisms to graphs with finite chromatic number. However, our results show the -product of such graphs with a given -graph (for example ) interacts well with the relevant complexes. Our two main structural results are the following. All necessary definitions are provided in the next section.
Let be a finite group, and suppose is a poset with a left -action and is a graph with a right -action. Then for any graph there is a natural homotopy equivalence
Let be a finite group, a graph, a graph with a right -action, and a poset with a free left -action. Let be the minimal diameter of a spanning tree in . If the induced action on the graph is -discontinuous we have a natural homotopy equivalence
For the notion of -discontinuity see \prettyrefdef:discont. If the action on is free, then by \prettyreflem:discont-subdiv the action on the poset is -discontinuous. We will usually apply \prettyrefthm:secondentry in situations where we have . For example, \prettyrefcor:ldismantlable gives sufficient conditions.
When we refer to a homotopy equivalence between posets and as in \prettyrefthm:firstentry, we will mean a homotopy equivalence such that the homotopy equivalence as well as its homotopy inverse are induced by poset maps. In \prettyrefsec:enrich we suggest a category such that could refer to an isomorphism in that category.
Loosely speaking \prettyrefthm:firstentry says that if is a topological space with a -action (in the form of its face poset), one can describe the space of -equivariant maps from into the complex in terms of the space of graph homomorphisms from the graph into . This provides a basic link between equivariant topology and the existence of graph homomorphisms and explains our interest in the graphs . \prettyrefthm:secondentry then allows us to study the space of graph homomomorphisms to such a graph, and also describes the (equivariant) topology of certain fiber bundles involving the spaces in terms of complexes from into these twisted graph products.
thm:firstentry and \prettyrefthm:secondentry lead us to a number of applications. We provide the details for these in \prettyrefsec:testgraphs, \prettyrefsec:furtherapps and \prettyrefsec:univ, but wish to briefly describe the ideas here. Our foremost application of \prettyrefthm:firstentry in \prettyrefsec:testgraphs will be related to the construction of new test graphs. Following , we say that a graph is a homotopy test graph if for all graphs we have
Here is the topological connectivity of the space . A graph with a -action that flips an edge is called a Stiefel-Whitney test graph if for all with we have
In this language the results of Lovász, Babson-Kozlov, and the second author say that the edge and the odd cycles are Stiefel-Whitney test graphs. We point out that the constant is best possible since if we take we get that non-empty and hence ()-connected.
To build new test graphs, we take in \prettyrefthm:firstentry to be the face poset of a (properly subdivided) -sphere. We then recover the space of equivariant maps from the -sphere into the complex as a colimit of the complexes (details are below). As a consequence we see that if is a Stiefel-Whitney test graph with certain additional properties (satisfied for example by and ), then so is ; in addition, certain topological invariants (e.g. connectivity) of are closely related to those of . As discussed above, the odd cycles form a directed family of test graphs with the property that the topology of can be related to that of . We view our results as a generalization of this phenomenon, with the graphs serving as ‘higher-dimensional’ analogues of the odd cycles.
In particular this gives us a general inductive procedure for constructing new test graphs of arbitrary chromatic number: one starts with a test graph and repeatedly applies the construction , for the face poset of a properly divided -sphere. In this paper we focus our attention on two new infinite families of test graphs, each parameterized by a pair . The parameter is related to the chromatic number of the test graph, whereas is a measure of its ‘fineness’. Details are provided in \prettyrefsec:testgraphs but we wish to give a brief description of these families here.
The collection of spherical graphs, denoted , are obtained as follows. We let denote the th barycentric subdivision of the boundary of the regular -dimensional cross polytope, and let denote its face poset. We then define to be the graph obtained by taking the twisted product of with the reflexive graph of the 1-skeleton of . For each , the map of posets induces graph homomorphisms . We will see in \prettyrefsec:testgraphs that for each the graphs are Stiefel-Whitney test graphs with chromatic number , and in addition they will play a role in a generalized notion of homomorphism duality discussed in \prettyrefsec:duality.
We obtain the twisted toroidal graphs, denoted , by repeatedly taking twisted products with graphs obtained from subdivisions of a circle. In this case the relevant posets have a simple combinatorial description, and to emphasize this we introduce some new notation. For we let denote the face poset of a -gon; it will be these posets that we use for in \prettyrefthm:firstentry. The graphs form a linear direct system which, for each , again leads to a collection of Stiefel-Whitney test graphs with chromatic number . The graphs have the property that their maximum degree is independent of (analogous to the fact that all odd cycles have maximum degree 2), and this leads to partial progress towards a conjecture of Lovász regarding bounds on chromatic number in terms of connectivity of test graphs of bounded degree.
We conclude \prettyrefsec:testgraphs with a study of a family of graphs obtained from the generalized Mycielski construction. In particular we use our methods to show that the graphs obtained this way provide another family of test graphs with arbitrarily large chromatic number.
In \prettyrefsec:univ we discuss our primary application of \prettyrefthm:secondentry, namely the notion of -universality for complexes. In  Csorba shows that any finite simplicial complex with a free -action can be approximated up to -homotopy equivalence as a complex for an appropriate choice of graph (see also  for an independent proof of this). We describe how his construction fits into our set-up, and we generalize his result to establish the following.
[\prettyrefthm:univn] Let be a finite simplicial complex with a free -action for . Then there exists a loopless graph and -homotopy equivalence
where acts on the left hand side as the automorphism group of .
In our set-up the desired graph is constructed as , where is a the face poset of the given complex , sufficiently subdivided. When , we show how this recovers the construction of Csorba.
The rest of the paper is organized as follows. In \prettyrefsec:defs we review relevant definitions and notation. In \prettyrefsec:testgraphs we describe explicitly our methods for the construction of new test graphs, and in particular the spherical and twisted toroidal graphs mentioned above. In \prettyrefsec:furtherapps we discuss other applications of these results in the context of homomomorphism duality and graph-theoretic interpretations of complexes, as well as the -universality of complexes. \prettyrefsec:proofs is devoted to the proofs of the main theorems as well as some technical lemmas. We conclude in \prettyrefsec:enrich with some comments regarding the categorical content of our constructions, in particular in the context of enriched category theory.
Acknowledgments. The authors would like to thank Eric Babson for useful conversations, as well as the anonymous referee for helpful comments and corrections. The first author was supported by the Deutscher Akademischer Austausch Dienst (DAAD) and by a postdoctoral fellowship from the Alexander von Humboldt Foundation. Both authors would like to thank the organizers of the MSRI Program on Computational Applications of Algebraic Topology in Fall 2006, where many of these ideas were developed.
2 Definitions and conventions
For us a graph is a finite set of vertices with a symmetric adjacency relation ; hence our graphs are undirected without multiple edges, but possibly with loops. If and are vertices of such that then we will often say that and are adjacent and denote this . A graph homomorphism is a vertex set map that preserves adjacency: if in then in . The complete graph has vertices and all possible non-loop edges. Given a graph , we define , the chromatic number of , to be the minimum such that there exists a graph homomorphism .
Let and be graphs. The categorical product is the graph with vertex set and with adjacency given by if and . The exponential graph is the graph on the vertex set with adjacency if for all in .
A graph is called reflexive if the adjacency relation is reflexive, i.e. if all of the vertices have loops. A graph is called loopless if there are no loops on any of the vertices. The graph is defined to be the (reflexive) graph with a single looped vertex. Note that there are natural isomorphisms and .
Let be a graph with a given equivalence relation on its vertices. The quotient graph is the graph with vertices and with adjacency if there exists and such that in .
For our applications, the equivalence relation will most often be given by the orbits of some group action. Recall that if is a group, and and are spaces with (respectively) a right and a left -action, then acts diagonally on the product according to . The space is then defined to be the orbit space under this action, so that , where . Similarly, we have the following construction for graphs.
Let be a graph with a left -action and a graph with a right -action. Define to be the graph with vertices given by the orbits of the diagonal -action on , with adjacency given by if there exists representatives in with .
Let be a graph with a left -action for some group . For an integer , we say that the action is -discontinuous if for each vertex , the neighborhood of radius around has the property that for all nonidentity .
We next come to the construction of the complex. We point out that our definition is slightly different from the one given in  in the sense that the complex we define here is the face poset of the polyhedral complex given in . Since the geometric realization of the face poset of a regular cell complex is homeomorphic to the complex itself, the underlying spaces of both complexes are the same.
Let and be graphs. We define to be the poset whose elements are all set maps with the condition that if in then for all and . The partial order is given by if for all .
For any graph , is a functor from graphs to posets. We will also need the following construction as a way to obtain a (reflexive) graph from a poset.
Let be a poset. We define to be the reflexive graph with vertices given by the atoms of , and with adjacency given by if there exists with and .
Note that the atoms of are precisely the homomorphisms . We will sometimes refer to (arbitrary) elements of as multihomomorphisms. The poset is ranked according to , for . We will often speak about topological properties of the complexes, and in this context we will be referring to the (geometric realization of the) order complex of the poset . We will use the notation to emphasize the distinction but will also use simply when the context is clear.
The complex is functorial in both entries, and in particular carries an action by , the automorphism group of the graph . If has an involution that flips an edge, this then induces a free -action on the space for any loopless graph (see for instance ). The examples discussed in the introduction arise from taking to be an edge with the nonidentity involution, or to be an odd cycle on the vertices with the reflection given by .
If is a space with a (free) -action, there are several invariants used to measure the complexity of the action. We collect some of these notions in the next definition.
Let be a space with a free -action, and let denote the -sphere endowed with the antipodal action. We define the index and coindex of as follows:
where denotes a -equivariant map. The height of , denoted , is defined to be the highest nonvanishing power of the first Stiefel-Whitney class of the bundle .
We refer to  for further discussion of these invariants, and especially their use in combinatorial applications. One can check that if a free -space, these values are related in the following way:
Finally, we collect a couple notions from the theory of posets.
Let be a poset. Define to be the poset whose elements are the nonempty chains of , with the relation given by containment.
Let and be posets. Then is the poset of all order preserving maps , with the relation if for all . If and are both equipped with actions by some group , we let denote the subposet of consisting of all equivariant poset maps.
3 Constructing new test graphs
In this section we provide details regarding our primary application of \prettyrefthm:firstentry, namely the construction of new test graphs for topological bounds on chromatic number. We begin with a brief discussion regarding the definition and history of such graphs, as well as our general approach to their construction.
Recall from  that a graph is called a homotopy test graph if for every graph , we have the following inequality:
The results of Lovász and Babson, Kozlov imply that the complete graphs , , and the odd cycles are homotopy test graphs. For some time it was an open question whether all graphs were homotopy test graphs, but Hoory and Linial showed that this was not the case in  by constructing a graph with such that is connected. In fact there are very few graphs that are known to be test graphs (see  and  for some discussion regarding this).
Now suppose is graph with a -action that flips an edge. Also from , we say that a graph is a Stiefel-Whitney test graph if for all with we have
We point out that it is enough to restrict graphs in the second coordinate to the set of all complete graphs with . Indeed, we have and whenever . Hence for any we get
Also, in  Kozlov insists on equality in the formulation involving complete graphs, but for our purposes the inequality will suffice. Note that the existence of an equivariant coloring (as defined in \prettyreflem:colorTkm) will in fact imply such an equality, since such a coloring induces an equivariant map which implies
Also note that every Stiefel-Whitney test graph (in our sense) is also a homotopy test graph since for a -space . Hence if is a Stiefel-Whitney test graph and is a graph with we have
We next describe our method for constructing new Stiefel-Whitney test graphs. As mentioned above, we will obtain these graphs by possibly repeated applications of the construction, where is a symmetric triangulation of . The basic result which makes this possible is the following.
Let be a graph with a right -action and let be a graph. For , let be the face poset of a -triangulation of . If then we have
Hence if is a Stiefel-Whitney test graph, we have , and for any graph such that we get
thm:firstentry yields an equivalence and hence in particular (recalling our convention laid down in \prettyrefrem:homotopy) a poset map
Since , we obtain a continuous map
The first inequality now follows from \prettyrefprop:Sk. If is assumed to be a Stiefel-Whitney test graph then the last inequality follows from \prettyrefeq:testgraph. Finally, setting we obtain
so that . ∎
In \prettyrefsec:spherical and \prettyrefsec:toroidal we use \prettyrefprop:main to construct infinite families of test graphs, with special attention paid to the spherical and twisted toroidal graphs.
As the reader may have noticed in the proof of \prettyrefprop:main, the full strength of \prettyrefthm:firstentry is not needed to establish the desired bounds on . In fact, to establish upper bounds on for some -graph , it is enough to construct -maps
where is a -graph (often ) for which upper bounds on are known. \prettyrefprop:Sk then yields
This method was used in  in the context of odd cycles in the first coordinate of the complexes. In \prettyrefsec:mycielski we will employ this method to show that the generalized Mycielski graphs provide another family of test graphs with arbitrarily high chromatic number. The full strength of \prettyrefthm:firstentry will be used in the context of spherical and twisted toroidal graphs to establish the existence of graph homomorphisms from these graphs, as in \prettyrefprop:coind and \prettyrefcor:mapsfrom.
3.1 Spherical graphs
The most natural application of \prettyrefprop:main comes from setting to obtain the graph , where is the face poset of a ()-triangulation of . As we have seen, the graph is a Stiefel-Whitney test graph, and hence from \prettyrefprop:main we get
It now follows that the graph is a test graph if its chromatic number is and \prettyrefprop:main also tells us that . On the other hand, need not be -colorable in general (for example a triangulation of as a -gon yields ), but it is if the triangulation is fine enough. We describe concrete colorings in \prettyrefprop:col-n-compl, but it also follows more abstractly from the following result, which also gives a graph theoretical interpretation of the coindex of the space , for a graph with an involution that flips an edge.
Let be a graph with an involution that flips an edge. For each , suppose is a sequence of symmetrically (with respect to the antipodal action) triangulated -spheres such the maximal diameter of a simplex of tends to zero when tends to infinity (e.g. take to be the th barycentric subdivision of the boundary of the regular -dimensional cross polytope). Let be the reflexive graph given by the 1-skeleton of . Then
If there is a graph homomorphism then by \prettyrefthm:firstentry
and hence there is an equivariant map
which means that .
On the other hand, if , then by simplicial approximation there is an equivariant simplicial map from to the barycentric subdivision of , whenever the simplices of are small enough, and therefore for almost all . Such a simplicial map induces a poset map by sending a simplex to the maximum of the images of its vertices. This shows that . ∎
Let and be as above. Then there is an such that for all the equality holds and hence is a Stiefel-Whitney test graph.
We only have to show , and this follows from the preceding proposition and . ∎
Let be the second barycentric subdivision of a regular -dimensional cell complex with a free cellular -action, and let denote its face poset. Then .
Let be the face poset of the complex of which is the second barycentric subdivision. Then and we identify the vertices of with the elements of . We now choose one element of each orbit of the free -action on and call the set of all chosen elements . For a face we denote its dimension by . We now define a function
For with we show that . Assume that . Then and and hence . Now let . Then there is a with . Since , we have . This shows that the map
is a graph homomorphism, since is a neighbor of if and only if or . ∎
prop:coind gives us several candidates for families of test graphs (depending on which triangulations of spheres that we choose). We wish to fix the following as the family of spherical graphs.
Definition 5 (Spherical graphs).
Let and to be the th barycentric subdivision of the boundary of the regular -dimensional cross polytope. Then we define a loopless graph with a right -action by
For and the graphs are Stiefel-Whitney test graphs with .
Since the boundary of the -dimensional cross polytope is the barycentric subdivision of a free -cell complex with two cells in each dimension from to , \prettyrefprop:col-n-compl implies . We have already seen that the rest is a consequence of \prettyrefeq:snk-test. ∎
One might ask if for example it would have been enough to know the connectivity of to establish . Indeed, for this is sufficient, since in this case a Theorem of Csorba (here \prettyrefthm:univ2, see the remark there) applies and yields .
3.2 Twisted toroidal graphs
In this section we let be a )-triangulation of a 1-sphere, and consider the case of repeatedly applying the construction. In this case, we have a particularly simple description of these graphs in mind. We define to be the face poset of a -gon with vertex set , with the antipodal left action given by (mod ) and the reflection right action by (mod ). This yields . In \prettyrefthm:firstentry and \prettyrefthm:secondentry we take a quotient of the product of graphs in the context of the -action on . In the case , we will want to consider the graphs of the sort (see \prettyreffig:K2C3 for the case of with the nontrivial -action).
Iterating this construction gives the following family of ‘twisted toroidal’ graphs.
Definition 8 (Twisted toroidal graphs).
For integers , we define the graph
The example in \prettyreffig:K2C3 is . We point out that each is a graph without loops, since has no loops. As was the case with the spherical graphs in the previous section, \prettyrefprop:main gives us the following.
Let be a graph with a (right) -action and let be a graph. Assume that . Then for we have
In particular if for some and , we have
To show that the are indeed Stiefel-Whitney test graphs we have to show that they have the desired chromatic number. To establish this we will show that under certain additional assumptions, the construction raises the chromatic number by exactly one. These additional assumptions will be fulfilled for example when we pass from to .
Let be a graph with a right -action and , . If there is an equivariant graph homomorphism
where the -action on is given by exchanging the vertices and , and leaving all other vertices fixed, then there is an equivariant graph homomorphism
with the -action on as that on .
First note that for any we have a -equivariant homomorphism given by , for , , , for , and . The looped vertices of induce a 6-cycle and can be identified with , and hence we have a -homomorphism . We also have a homomorphism given by extending any element of to for any vertex (this is equivariant since both actions are trivial on these vertices). Now, assume that and that we have an equivariant coloring . This gives us
a sequence of -equivariant homomorphisms and hence an equivariant coloring
as desired. ∎
For all and we have .
Induction on . ∎
With this machinery in place we can state our main result of this section.
Let , . Then is a Stiefel-Whitney test graph of chromatic number . In particular we have,
for every graph such that .
We have and, again by \prettyrefprop:mainprop1, for and . ∎
The graphs also have interesting properties relating their maximum degree and odd girth.
Let and , . Then has chromatic number , odd girth , and every vertex has degree .
The equality has already been established.
Every vertex of has degree and every vertex of has degree , it follows inductively that every vertex of has degree .
Consider a cycle of length less than in