Tiling tripartite graphs with -colorable graphs: The extreme case
There is a sufficiently large such that the following holds. If is a tripartite graph with vertices in each vertex class such that every vertex is adjacent to at least vertices in each of the other classes, then can be tiled perfectly by copies of . This extends work by two of the authors [Electron. J. Combin, 16(1), 2009] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that in our result can not be replaced by and that if is divisible by , then we can replace it with the value and this is tight.
Let be a graph on vertices, and let be a graph on vertices. An -tiling of is a subgraph of which consists of vertex-disjoint copies of and a perfect -tiling, or -factor, of is an -tiling consisting of copies of . The celebrated Hajnal-Szemerédi Theorem  says that each -vertex graph with contains a -factor. (Corrádi and Hajnal  proved the case .) Using Szemerédi’s regularity lemma , Alon and Yuster [1, 2] obtained results on -tiling for arbitrary . Their results were improved substantially [16, 17, 18, 25], in particular, Kühn and Osthus  determined the minimum degree threshold for -factors for arbitrary up to an additive constant, see the survey  for details.
In this paper, we consider multipartite tiling, which restricts to be an -partite graph. For , this is an immediate consequence of the König-Hall Theorem (e.g. see ). Wang  considered -factors in bipartite graphs for all ; Zhao  gave the best possible minimum degree condition for this problem. With the exception of one case, Hladký and Schacht  found best possible minimum degree conditions for -factors in bipartite graphs with ; the last case was settled by Czygrinow and DeBiasio . Later, Bush and Zhao  considered tiling bipartite graphs with an arbitrary graph .
For a tripartite graph , the graphs induced by , and are called the natural bipartite subgraphs of . Let be the family of -partite graphs with vertices in each partition set. Such a graph is called balanced because the number of vertices in each partition set is the same. In an -partite graph , stands111In , was used in place of . for the minimum degree over all natural bipartite subgraphs of .
In addition to the bipartite results discussed above, there have also been a number of results on multipartite graphs with , many of which were inspired by a conjecture of Fischer. Fischer  conjectured that if satisfies , then contains a -factor. However, if and are odd integers, then Catlin  had earlier given an example of a graph where . In , Magyar and Martin proved that, for large , this graph is a unique counterexample to Fisher’s conjecture for by showing that if is a sufficiently large odd multiple of 3, the so-called blow-up graph is the unique graph with and no -factor. The conjecture of Fischer can be modified to exclude this case. The blow-up graph for , which is obtained by replacing each edge of with a copy of and replacing each non-edge by an bipartite graph with no edges, is shown in Figure 1.
This gives the following Corrádi-Hajnal-type result.
Theorem 1 ()
Let have . If for some absolute constant , then contains a -factor or for an odd integer.
Martin and Szemerédi  proved a quadripartite version of the Hajnal-Szemerédi Theorem. Han and Zhao  reproved the results of [21, 23] by using the absorbing method. An approximate version of the multipartite Hajnal-Szemerédi Theorem was given by Csaba and Mydlarz . Keevash and Mycroft  and independently Lo and Markstrom  confirmed Fischer’s conjecture asymptotically, and finally Keevash and Mycroft  proved the modified Fischer conjecture exactly for any sufficiently large graph. More recently, an asymptotic multipartite version of the the Alon-Yuster Theorem was proved by Martin and Skokan .
It was shown [24, Theorem 1.2] that in the tripartite case, is the correct coefficient of required to have a -factor.
Theorem 2 ()
For any positive real number and any positive integer , there is such that the following holds. Given an integer such that is divisible by , if is a tripartite graph with vertices in each vertex class such that every vertex is adjacent to at least vertices in each of the other classes, then contains a -factor.
Let be the smallest value for which there exists a sufficiently large such that if is a balanced tripartite graph on vertices, and each vertex is adjacent to at least vertices in each of the other classes, then contains a -factor. Our main result is the following more precise theorem.
Fix a positive integer . If and with , then
So, the result is tight when divides , almost tight unless is an odd multiple of and, in the worst case, the upper and lower bounds differ by . We are not sure whether the upper or lower bounds of Theorem 3 are correct in the cases when they are not equal.
Clearly the complete tripartite graph can itself be perfectly tiled by any 3-colorable graph on vertices. Since whenever is divisible by , we have the following corollary.
Let be a -colorable graph of order . There exists a positive integer such that if and divisible by , then every with contains an -factor.
Fix a positive integer . There exists an such that
if , and is divisible by , then there is a graph with no -factor and ; and
if , and is not divisible by , then there is a graph with no -factor and .
As to the upper bound, we use Theorem 7 (Theorem 1.4 from ) to take care of the main case. For vertex sets and , let denote the density of and . Before we can prove the main case, we need the following definition.
Given , we say that is -extremal when there are three sets such that , for all and for .
If is -extremal and , then for , the pair is a very dense bipartite graph. Thus, we expect most members of our -factor with vertices in to have vertices in and the remaining vertices in , where .
Theorem 7 ()
Given any positive integer and any , there exists an and an integer such that whenever , and divides , the following occurs: If satisfies , then either contains a -factor or is -extremal.
Hence, for the upper bound, it suffices to assume that is -extremal. The proof, given in Section 3, is detailed and involves a case analysis. Moreover, it requires the definition of a particular structure we call the very extreme case, which we deal with in Section 3.5. This definition is given below, but roughly, it means that the graph looks like .
A balanced tripartite graph on vertices is in the very extreme case if the following occurs: First, there are integers such that . Second, there are sets for , each with size at least , such that if then is nonadjacent to at most vertices in whenever the pair corresponds to an edge in the graph with respect to the usual correspondence.
Note that we use different language for -extremal and the very extreme case because the definition of -extremal requires a parameter, whereas the very extreme case does not.
Now that we have defined the very extreme case, we can formally state the upper bound theorem as follows:
Fix . Let be sufficiently large and assume . If , then has a -factor or is in the very extreme case. If is in the very extreme case and , then has a -factor.
2 Lower bound
First, we need a lemma (Lemma 2.1 in ) which permits sparse tripartite graphs with no triangles and with no quadrilaterals in its natural bipartite subgraphs:
For each integer , there exists an such that, if , there exists a balanced tripartite graph, on vertices such that each of the natural bipartite subgraphs are -regular with no and has no .
Finally, we prove the lower bound given in Proposition 5.
Construction 2: Let and so that, in this case, . Let be defined such that (the notation emphasizes that it is a disjoint union of sets) in which column is defined to be the triple of the form . Let the graph in column 1 be where , the graph in column 2 be and the graph in column 3 be . If two vertices are in different columns and different vertex-classes, then they are adjacent. It is easy to verify that . Suppose, by way of contradiction, that has a -factor.
If a copy of has vertex classes , then for some . Since there are no triangles in any column and no ’s in the natural bipartite subgraphs of a column, the intersection of a copy of with a column is either a star, with all leaves in the same vertex-class, or a set of vertices in the same vertex-class. So each copy of has at most vertices in column 1 and at most vertices in each of column 2 and column 3.
There are three cases for a copy of . Case 1 has vertices in each column. Case 2 has vertices in column 1, vertices in column 2 and vertices in column 3. Case 3 has vertices in column 1, vertices in column 2 and vertices in column 3.
In Cases 1 and 2, since contains no in column 3, having vertices of a in column implies that all of them are in the same vertex class. In Case 3, since has no in column 2, having vertices in column 2 means that all are in the same vertex-class. Since vertices in column 1 means that they form a star, the remaining vertices in column 3 must be in the same vertex-class (the same vertex-class as the center of the star). Hence, the intersection of any copy of with column 3 is contained within a single vertex-class. Therefore, the number of copies of in the -factor of is at least , a contradiction because the factor has exactly copies of .
Next consider the case when and with . Let be defined such that the graph in column 1 is , but all other possible edges in are present. It is easy to verify that . Suppose, by way of contradiction, that has a -factor. The intersection of one copy of with column 1 must be contained within a single vertex class and can contain at most vertices. So at least copies of are needed to cover all of column 1. This is a contradiction, because the factor has exactly copies of .
3 The extreme case
Throughout Section 3, assume that is minimal, i.e., no edge of can be deleted so that the minimum degree condition still holds. As we complete the proof of Theorem 3 by proving Theorem 9, we will develop the usual hierarchy of constants:
There are 4 parts to the proof. Part 1 begins with being -extremal and, if a -factor is not found in , then we move to Part 2 where looks like the graph in Figure 3. If a -factor is not found in when is in Part 2, then we move on to Part 3. If is in Part 3a, then it is approximately . (See Figure 2.)
In general, the graph is an -partite graph with vertices , and such that is adjacent to (denoted ) if and only if and . If is in Part 3b, then it is approximately . (See Figure 1.) The following definition will come into play as we describe the structure of .
For , , a graph and positive integer , we say a graph is -approximately if can be partitioned into nearly-equally sized pieces, each of size or , corresponding to a vertex of so that for vertices with , the parts of corresponding to and have pairwise density less than .
Note that if , we do not require that the parts of corresponding to and have pairwise density close to 1.
We will assume for Parts 1, 2, 3a and 3b (Sections 3.1, 3.2, 3.3 and 3.4, respectively) that . This takes care of everything except for the very extreme case, which we will consider in Section 3.5. For this last part, we will require to complete the proof.
3.1 Part 1: The basic extreme case
For Part 1, we will prove that either a -factor exists in , or is in Part 2.
Let for be the three pairwise sparse sets given by the definition of -extremal and for . Recall that , so . We then define to be the typical vertices with respect to , to be typical with respect to , and are what remain. Formally, for ,
Using these definitions, the fact that is -extremal and the bound on , we get
As a result, we have that and . So, with and , we get the following bounds for and :
Step 1: Adjusting the sizes of the sets
Let with and .
Without loss of generality, assume that . For , define if ; otherwise, . If , then we will move vertices of to by applying Lemma 12 below, which is proved in Section 3.6. It is applied several times throughout this paper to different sets.
Let us be given and a positive integer .
Let be a bipartite graph such that every vertex in is adjacent to at least vertices in . Suppose further that and for .
Provided , there is a family of vertex-disjoint copies of all of whose centers lie in .
Let be a tripartite graph such that every vertex not in is adjacent to at least vertices in , for . Suppose further that and for .
Provided , there is a family of vertex-disjoint copies of all of whose centers lie in and leaves lie in (index arithmetic is modulo 3).
Since , we have . As , we can guarantee that each vertex not in is adjacent to at least vertices in . So we apply Lemma 12(2) to the graph induced by , with , , and . This will construct stars with the property that there are exactly enough centers in such that, when removed, the resulting set has its size bounded above by either or , whichever is required. Place these centers into a set we will call .
If , then we will move vertices of to , unless is -approximately .
For a subgraph , with , define the center to be the vertex that is adjacent to all others. We will refer to the remaining vertices as leaves, although their degree is .
In , we will find vertex-disjoint copies of such that each of copies has its center vertex in for and such that each of copies has its center vertex in otherwise. This will be accomplished with Lemma 13, which is proved in Section 3.6. It is applied several times throughout this paper to slight variations of the sets .
Given , there exists an such that the following occurs:
Let be a tripartite graph on vertices such that for all , each vertex in is adjacent to at least vertices in . Furthermore, .
If contains no copy of with 1 vertex in , and vertices in each of and , then the graph is -approximately .
Lemma 13 can be repeatedly applied to at most times with , and . Each time, either a is found and removed, or the current incarnation of is -approximately and we stop applying the lemma. When we are finished applying Lemma 13, add the center vertices of the subgraphs to the appropriate sets . Put the leaves back into and denote the result as .
Place vertices from into the set , for , so that the resulting set, relabeled as , is of size or and .
Step 2: Finding a -factor in
Now we try to find a -factor among the remaining vertices in with the goal of matching them with the vertices. To do so, however, we will need to make the following adjustments:
Vertices in copies of where the center vertex is in some must be in a specified copy of in .
Recall that is the set of centers of -stars which were found in Step 1. If is the center of a with leaves in , then will be assigned to , where . This means that will be adjacent to vertices in in a in .
For , vertices will be assigned to or , respectively. This means that will be adjacent to either vertices in or vertices in in a in . We know this can be accomplished because if , then we may assume, without loss of generality, that is adjacent to at least vertices in .
Moreover, because all but a -proportion of the sets are typical, we have that . Recalling the definition of from Lemma 12, and there are at most copies of with the center vertex in a given .
Let us be given . Then there exists an and a positive integer such that the following occurs:
Let there be positive integers which are divisible by and with , for all and . Let be a tripartite graph such that for distinct indices , . For all , each vertex in is adjacent to at least vertices in . We will either find a -factor in the graph induced by with certain restrictions, or the graph induced by has a very specific structure.
The restrictions on the -factor in the graph induced by are as follows. For each pair , there are at most copies of which must be part of the factor. For each , there are at most vertices with the following property: can only be in copies of in the pair and is adjacent to at least vertices in .
If such a factor cannot be found, then, without loss of generality, the graph induced by can be partitioned such that , for and and for .
First, match vertices in that are assigned to with typical neighbors in and those vertices with typical neighbors in . As the name implies, a typical neighbor is a neighbor which is a typical vertex. This forms a copy of . Then, place the vertices that were moved in prior steps into copies of by matching the with vertices in the appropriate “” set. Remove all of these from , and apply Lemma 14 to what remains with and . If the appropriate -factor cannot be found, then we are in the case of Part 2. The diagram that defines that case is in Figure 3.
Step 3: Completing the -factor
If the -factor above is found, then we will recycle notation to define to be the vertices that remain from after removing copies of as above. It is easy to see that each will have size close to and divisible by . Further define , and so that each member of the -factor of lies in a pair , or , and so that each of the triples , and consist of sets of the same size. Note that this can be done arbitrarily.
We use Proposition 15, which allows us to complete a -factor into a -factor. The proof follows easily from König-Hall.
Let be a bipartite graph with , divides , and each vertex is adjacent to at least vertices in the other part. Then, we can find a -factor in .
Let be a tripartite graph with , divides , and each vertex is adjacent to at least vertices in each of the other parts. Furthermore, let there be a -factor in . Then, we can extend it into a -factor in .
3.2 Part 2: is approximately the graph represented by Figure 3
Let be the graph on vertices for with the following non-adjacencies: for for and . Notice that corresponds to the diagram in Figure 3.
In this part, our graph is -approximately . Note that the first column of consists of the pairwise sparse sets from the definition of -extremal, and the second and third columns are defined by the exceptional case of Lemma 14. We will group the vertices into sets of size between and so that each vertex in is adjacent to at least vertices in each set when . In other words, the vertices in are typical according to the rules established by Figure 3. The non-typical vertices in row will be collected in the set . From this point forward we have issues related to divisibility that we did not have before. Namely, we may need to modify and so that their sizes are divisible by .
Step 1: Ensuring small sets of proper size
Each set has a target size that we will denote . If , then for all . If , let for all such that 3 divides and otherwise. If , let for all such that 3 divides and otherwise. Note that if , we can remove one copy of from the triple , and if , we can remove two copies of from triples where is not divisible by 3.
Apply Lemma 12 to obtain disjoint stars with centers in and leaves in . Then move these star centers to where so that holds for all .
Step 2: Partitioning the sets
Before we partition the sets, we must examine the behavior of . If this is - approximately , then call the dense pairs and . Note that the sets and need not be uniquely-defined as long as they satisfy the given condition. If is not - approximately , do nothing.
For , , we say that and coincide if the intersection of their typical vertices is large and therefore the intersection of the typical vertices of and is small. We will determine the quantities that constitute “large” and “small” later. If and coincide with and , respectively, then is approximately . This case will be handled in Section 3.3. If and coincide with and , respectively, then is approximately . This case will be handled in Section 3.4. Otherwise, there may be no coincidence, or coincidence may occur in exactly one of and