A median-type condition for graph tiling
Komlós [Komlós: Tiling Turán theorems, Combinatorica, 2000] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree.
1 Introduction and Results
A classical question in extremal graph theory is to study which density condition forces a graph to contain a certain fixed subgraph. Let us cite Turán theorem [Tur41], which gives the average degree forcing a clique, or the Erdős–Stone theorem [ES46], which essentially determines the average degree condition guaranteeing the containment of a fixed non-bipartite graph . This approach can be generalised by seeking the density condition forcing the containment of several copies of . One might consider edge-disjoint copies of , a so-called packing of , or vertex-disjoint copies of , known as an -tiling. The latter concept generalises the one of a matching, where we take to be an edge, and this is the direction we follow in this paper.
Definition 1.1 (Tiling, perfect tiling, tiling number).
Let be a fixed graph. We say that a graph has an -tilling of size , if there are vertex-disjoint copies of in .
An -tiling is perfect if it has size .
The maximum size of an -tiling, where the maximum runs through all possible -tilings of , is called the -tiling number of (or just tiling number, when is clear from the context).
The concept is not new and researchers have obtain many results in this direction. The Hajnal-Szemerédi theorem on equitable colouring [HS70] (formulated for the complement of the host graph) determines the minimum degree condition guaranteeing a perfect clique tiling. One can deduce from their result the minimum degree condition needed to force a partial clique-tiling of any given size. For the corresponding question of determining the average degree condition forcing a clique tiling of a given size, the current state of knowledge is rather poor. The only two known cases for which the average degree condition has been determined correspond to -tilings and -tilings, given by Erdős and Gallai [EG59] and Allen et al. [ABHP15], respectively. The case of tiling cliques of higher order is wide open and no conjecture has even been formulated.
Generalising the concept of clique tilings to the one of tilings with an arbitrary graph , Komlós [Kom00] extended the Hajnal-Szemerédi theorem.
Theorem 1.2 (Komlós tiling theorem).
Let . If is an -colourable graph with colour class sizes and is an -vertex graph with minimum degree at least , then contains an -tiling of size at least .
As may have many -colourings, in order to apply Komlós’ theorem with the weakest hypothesis, we need to fix an -colouring of which minimises the size of its smallest colour class. For the corresponding value of and for each , Komlós constructs graphs with minimum degree and -tiling number . This shows that his result is asymptotically optimal. Recently, Hladký, Hu, and Piguet [HHPa] proved stability of this result. Komlós tiling theorem was complemented by Kühn and Osthus [KO09], who established the optimal minimum degree condition forcing a perfect -tiling. Considering average degree rather than minimum degree, Grosu and Hladký [GH12] extended the Erdős–Gallai theorem, by asymptotically establishing the average degree condition forcing the containment of an -tiling, for a fixed bipartite graph .
A finer approach to extremal problems is to take into account more information encoded in the degree sequence of the host graph. Let us give an example in the area of tree containment. It is trivial to see that a minimum degree of ensures a copy of any tree of size . However, Loebl, Komlós, and Sós conjecture that only half of the vertices need to have this degree to guarantee the same assertion. Taking this perspective in the area of tilings, Treglown asymptotically determined an optimal degree sequence condition forcing a perfect -tiling [Tre16]. In this paper, we inquire what portion of vertices need to meet the degree bound in Komlós tiling theorem in order to guarantee an -tiling of a given size. Our main result is the following.
Let be an -colourable graph with colour class sizes and let . Then for any there is an such that for any and for the following holds.
Any -vertex graph with at least vertices of degree at least contains an -tiling of size at least .
Theorem 1.3 strengthens Komlós tiling theorem: the degree bound is the same but we do not require all the vertices of the host graph to meet this bound, but rather only roughly a fraction of them. Note that as ranges from to , ranges from to 1.
When is chosen according to the colouring minimising the size of the smallest colour class, Theorem 1.3 is asymptotically optimal for each value of . To show this, we construct an -vertex graph with vertices of degree that does not contain an -tiling of size more than . The vertex set of is partitioned into four sets, , , , and . The sets , and are independent, and the set induces a balanced complete -partite graph. There is a complete bipartite graph between and and a complete bipartite graph between and , and no other edge. The graph is depicted in Figure 1.1. Note that the vertices in the set meet our degree assumption and we have . We claim that each copy of in must have at least vertices in . Suppose it is not the case. We can then recolour so that the smallest colour class has less than vertices, as follows . Give the vertices embedded in colour , the ones embedded in colour , and distribute the other colours to vertices embedded in . In this colouring, the smallest colour class has at most as many vertices as colour has, which is less than . Since , we conclude that there cannot be an -tiling of size more than .
The proof of Theorem 1.3 relies on the regularity method and on the generalisation of the LP-duality between fractional matching and fractional cover. Such a generalisation of the LP-duality has already been used in [MJ17] for the particular case of fractional clique-tiling. However, we think that this method deserves more attention and is exposed here in the more general form for fractional -tiling111For a precise definition of fractional tiling, see Section 2.2.. Similar use of LP-duality for graph-tiling was used in [DH, HHPa, HHPb] but in the context of graphons.
2 Notation and Preliminaries
2.1 The regularity lemma and the embedding lemma
Given two disjoint subsets , we denote by the number of edges between vertices in and vertices in , that is,
Definition 2.1 (Regular pairs, bipartite density).
For a given , a pair of disjoint sets is called an -regular pair if for every , with , we have that , where the (bipartite) density between two disjoint sets and is defined as . If the pair is not -regular, then we call it -irregular.
Lemma 2.2 (Slicing lemma, Fact 1.5 in [Ks96]).
Let be a graph and be an -regular pair of density in . Let and such that and . Then, is an -regular pair of density at least , where .
For completeness, let us next state the fundamental regularity lemma.
Lemma 2.3 (Szemerédi regularity lemma, Theorem 7.4.1 in [Die16]).
For all and there exist such that for every and every -vertex graph there exists a partition of , , with the following properties:
For every we have , and .
All but at most pairs , , , are -regular.
The partition of allows to define a cluster graph (with parameters , and ) as follows: and if and only if is an -regular pair such that .
Given a cluster graph , let the blow-up graph denote the graph defined as follows: Every vertex of is replaced by a set of vertices, which we call the clone set of , and every edge by a complete bipartite graph between the corresponding clone-sets.
There is a natural association between the clusters of , the vertices of , and the clone-sets of : The cluster of is associated to the vertex of and, in , this vertex is replaced by the clone-set of , which we denote by
Lemma 2.4 (Lemma 7.5.2 in [Die16] (Embedding lemma)).
For any and , there exists an such that the following holds: If and the parameters of satisfy and , then
2.2 Fractional hom-tiling, fractional hom-cover, and LP-duality
From Lemma 2.4 it is clear that a copy of in a cluster graph implies a copy of in the original graph ; actually using Lemma 2.4 it can be shown that it generates up to many disjoint copies of . Going through all (not-necessarily disjoint) copies of in , we could use the notion of fractional tiling to find a large -tiling in . A fractional tiling of in is a function giving weights in to copies of in such that for each vertex the sum of the weights of all copies containing is at most . Hence an -tiling in (Definition 1.1) is an integral tiling, i.e., a fractional tiling whose weights are either or .
But not only copies of in may generate copies of in . For example, if there is a triangle in , we know there are many copies of in . Therefore, instead of looking for copies of in , we shall seek copies of , where is some homomorphic image of . A homomorphism is a mapping from to such that implies . Hence, we shall generalise the concept of fractional tiling of in , by enriching it by the homomorphism images of .
Definition 2.5 (Fractional hom-tiling).
A function is a fractional hom-tiling in of size if it satisfies the following two properties:
For any vertex , we have , where the sum runs over all homomorphisms .
We have that , where the sum runs through all homomorphisms .
So in a fractional hom-tiling we not only assign weights to isomorphic copies of in , but to homomorphic copies of , as well.
In order to prove Theorem 1.3, we shall use LP-duality. So, similarly as above, we need to generalise the notion of fractional -cover in a graph to consider also homomorphic copies of , which will be the dual notion of fractional hom-tiling (Definition 2.6). We shall prove in Proposition 2.8 that those two notions are indeed dual.
Definition 2.6 (Fractional hom-cover).
A function is a fractional hom-cover in of size if it satisfies the following two properties:
For any subgraph and any homomorphism , we have , where the sum runs through all vertices .
We have , where the sum runs through all vertices .
Definition 2.7 (Fractional tiling/cover number).
The fractional hom-tiling number of a graph is the maximum of the sizes of all its fractional hom-tilings.
The fractional hom-cover number of a graph is the minimum of the sizes of all its fractional hom-covers.
Proposition 2.8 (LP-duality for hom).
For any graph , its fractional hom-tiling number equals its fractional hom-cover number.
Sketch of the proof.
The proof is a straightforward generalisation of the LP-duality between the fractional matching and the fractional vertex-cover. To attain the fractional hom-tiling number is equivalent to the following instance of linear programming:
where and are all-one vectors, is the vector of variables (to be determined) corresponding to a fractional hom-tiling, and is a matrix, where each column corresponds to a homomorphism with image , each row corresponds to a vertex of , and . Since this optimisation problem clearly has an optimal solution, the Strong LP-duality theorem says that it is equivalent to the following dual problem:
where is the vector of variables (to be determined) corresponding to a fractional hom-cover. ∎
3 Proof of Theorem 1.3
Let be a fixed -colourable graph with colour classes and let be fixed. Assume we are given . Notice that we can assume that . Set . Since and , we have that . Lemma 2.4 with input and outputs an . We define
Notice that , which implies that .
The regularity lemma (Lemma 2.3) with input and outputs . Set
Let and let be a -vertex graph with at least vertices of degree at least . We apply Lemma 2.3 on (with parameters and ) and obtain an equitable -regular partition with . We erase all edges within clusters , in irregular pairs, in regular pairs of density smaller than , and edges incident to the cluster . Slightly abusing notation, we still call this subgraph . Let be the corresponding cluster graph with parameters , (), and .
After erasing edges as described in the previous paragraph, has at least vertices of degree at least .
By erasing all edges within clusters, we remove at most edges. Notice that
By erasing all edges between irregular pairs, we remove at most edges. By erasing all edges between regular pairs of density smaller than , we remove at most edges. Finally, by erasing all edges incident to cluster , we remove at most edges. In total, the number of edges that have been removed from is at most
If the statement of the claim were not true, after the transformations, more than vertices of would have had their degree decreased by more than . This would imply that the number of edges removed from is at least
which yields a contradiction.
has at least vertices of degree at least .
For every vertex of , we denote by the cluster of the regular partition containing .
Let a vertex of with degree at least . Given a distinct vertex of , we have that implies that . Since has at most neighbours in every cluster , we obtain that has degree at least
Similarly, let have degree at least . At most vertices of of degree at least belong to cluster . Therefore, the number of vertices of with degree at least is at least
If the fractional hom-tiling number of is at least , then the -tiling number of is at least .
Let be a fractional hom-tiling of of maximum size. By hypothesis,
where the sum runs through all homomorphisms .
Using the embedding lemma (Lemma 2.4), we will sequentially find copies of in , as long as unused vertices are left in each cluster. We denote by the vertices of used by the copies of . At the beginning . We define an auxiliary graph as follows. Let be the smallest number of free vertices in a cluster. For each let be an arbitrary set of size in . The graph is the subgraph of induced by . Observe that as long as , we have by Lemma 2.2 that the pairs are -regular with density at least , where and .
For every , we shall construct vertex disjoint copies of in . Any such copy will satisfy that, for every vertex , its image is contained in the cluster of corresponding to . So assume that we have a homomorphism and let . We first construct a copy of in as follows: Let ; then each of the vertices in is embedded to a distinct vertex in the clone-set of . Since is a homomorphism, no two vertices of this group are adjacent in . By repeating this procedure for all vertices in , we obtain an embedding of the vertices of in . By construction of and the fact that homomorphisms map edges of to edges of , we obtain that the edges of are also correctly embedded in . Due to the choices of and given in equations (3.1) and (3.2), we can use Lemma 2.4 with , , , and to find an embedding of in such that for all of the vertices in are embedded in the cluster of . We put the vertices involved in this copy of in in the set . We repeat the process above times, and obtain vertex-disjoint copies of in . This can be done using Lemma 2.4 (with the same parameters as above) as long as .
We next repeat this process for the remaining homomorphisms . For this to be possible, we need to argue that . Let be a vertex in . The number of vertices in is at most the number of vertices used in copies of in so far. This is always at most , where the sum runs through all homomorphisms . In the next formula, the sums always run through all homomorphisms :
Consequently, the number of vertices in is at least .
It only remains to bound the size of the -tiling of obtained by the above procedure. As we have seen, each homomorphism yields vertex-disjoint copies of in . Hence, the total size of the tiling is
where the sum runs through all homomorphisms , and where we have used that . Notice that this holds because . ∎
From Proposition 2.8, we infer that it is enough to show that the fractional hom-tiling number of is at least . In order to do this, we prove the following proposition:
Let be an -colourable graph with colour class sizes
and let . Set
Then any -vertex graph with at least vertices of degree at least has fractional hom-cover number least .
Before proving the proposition, we argue that this concludes the proof of Theorem 1.3. Indeed, we apply Proposition 3.4 to the cluster graph , where the value used in the proposition is . For this to be valid, we need to argue that , and that has the appropriate number of vertices of the appropriate degree. Since (see (3.1) and the paragraph afterwards), we have that . Regarding the degree condition, by Claim 3.2, has at least
vertices of degree at least
Thus, we can use Proposition 3.4 with the desired parameters, which ensures a fractional hom-cover number of at least in . Proposition 2.8 (LP-duality for hom) then implies that there is a fractional hom-tiling of size at least in . Finally, Claim 3.3 ensures an -tiling in of size at least .
Proof of Proposition 3.4.
Let be any fractional -cover of , and let denote its size. In order to prove the proposition, we want to show that .
Let denote the set of vertices of having degree at least , and let denote the set containing the remaining vertices of . By hypothesis,
For any vertex of , we denote the set of neighbours of in and by and , respectively. We start by selecting vertices in the following way.
Let be a vertex in with smallest . If , and the selection is done. Otherwise, since , we have that . Among all the elements in , let be a vertex with smallest . We continue selecting vertices in an analogous way. More precisely, suppose that we have already selected vertices in . We define:
Among all the elements in , we select a vertex with smallest . In order to do so, we need to argue that .
It is well-known that, given a set of size and subsets of ,
Note that Inequality (3.6) holds also in the case when . Since
and , we have that .
After having selected vertices , we set to be a vertex with the smallest , and one with the smallest . It may happen that or does not exist, but at least one of them does. For , we define . Let , if exists, and set , otherwise. Similarly, let , if exists, and set , otherwise. Let . We then have:
Notice that or/and form a clique in . Thus, there exists a homomorphism that sends all vertices in the -sized color class of to (for ), and all vertices in the -sized color class of to or . Since is a fractional hom-cover of , we obtain the following relation:
As , we have
We consider two cases:
In CASE 1, we have
In CASE 2, we get
We now have the necessary ingredients to prove that .
By (3.7), we have that , for and that . Therefore,
Using Claim 3.5, we get