On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

01/17/2020 ∙ by Gábor Damásdi, et al. ∙ Eötvös Loránd University KIT Microsoft University of Birmingham Berlin Institute of Technology (Technische Universität Berlin) Renyi Institute of Mathematics 0

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with 2kk steps with generalized rectangles there is a row or a column in the diagram that is used by at least k+1 rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim et al., we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height 2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension (1 + o(1))log_2log_2 n. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.

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1 Introduction

The covering number of a graph (host) with respect to a class  is the least  such that there are graphs with for such that their union covers the edges of and no other edges. We denote this number by . The study of covering numbers has a long tradition:

  • In 1891 Petersen [20] showed that the covering number of -regular graphs with respect to -regular graphs is .

  • In 1964 Nash-William [19] defined the arboricity of a graph as the covering number respect to forests and showed that it equals the lower bound given by the maximum local density.

  • The track number introduced by Gyárfás and West [11] and the thickness introduced by Aggarwal et al. [1] correspond to covering numbers with respect to interval graphs and planar graphs respectively.

Knauer and Ueckerdt [17] proposed the study of local covering numbers. This number is defined as the minimum number such that there is a cover of  with graphs from  (see above) such that every vertex of is contained in at most members of the cover. We denote the local covering number by . Fishburn and Hammer [9] introduced the bipartite degree which equals what we call the local covering number with respect to complete bipartite graphs. Motivated by questions regarding the local dimension of posets, Kim et al. [15] studied local covering number with respect to difference graphs and compare this to the local covering number with respect to complete bipartite graphs.

In this paper we continue the studies initiated in [15]. In Section 2 we discuss local coverings with difference graphs and complete bipartite graphs. With Theorem 1 we give a precise result regarding the local covering number of a difference graph (Young diagram) with respect to complete bipartite graphs (generalized rectangles). This answers a question raised in [15].

Section 3 relates the results to the local dimension of posets. In this section we also discuss aspects of the local dimension of Boolean lattices. Finally, in Section 4 we discuss covering numbers of directed graphs and their relation with order dimension and notions of Ferrers dimension.

2 Covering numbers

Following the notation in [17], local covering numbers are defined as follows. For a graph class and a graph , an -covering of is a set of graphs with . (In [17] this is called an injective -covering. But as all coverings considered here are injective, we omit this specification throughout.) An -covering of is -local if every vertex of is contained in at most of the graphs , and the local -covering number of , denoted by , is the smallest for which a -local -cover of exists.

A difference graph is a bipartite graph in which the vertices of one partite set can be ordered in such a way that for , i.e., the neighbourhoods of these vertices along this ordering are weakly nesting.

Difference graphs are closely related to Young diagrams. Let denote the set of positive integers. For we denote . A Young diagram with rows and columns is a subset such that whenever , then provided , as well as provided . A Young diagram444In the literature our Young diagrams are more frequently called Ferrers diagrams. We stick to Young diagram to be consistent with [15]. is visualized as a set of axis-aligned unit squares, called cells that are arranged consecutively in rows and columns, each row starting in the first column, and with every row (except the first) being at most as long as the row above.

A generalized rectangle (also called combinatorial rectangle) in a Young diagram is a set of the form with and and . Note that (unless ) not every set of the form with and satisfies . A generalized rectangle with being a set of consecutive numbers in and being a set of consecutive numbers in is an actual rectangle. A generalized rectangle uses the rows in  and the columns in . See Figure 2 for an illustrative example.

Difference graphs can be characterized as those bipartite graphs with bipartition , , which admit a bipartite adjacency matrix whose support is a Young diagram :

Moreover, a complete bipartite subgraph of corresponds to a generalized rectangle in . Rows and columns of correspond to vertices of in and , respectively.

Figure 1: Left: A Young diagram with rows and columns. Highlighted are the generalized rectangle (blue), and the actual rectangle (red). Right: A corresponding difference graph with corresponding red and blue .

In [15], Kim et al. introduced the concept of covering a Young diagram with generalized rectangles subject to minimizing the maximum number of rectangles intersecting any row or column. Their motivation was to investigate the relations between local difference cover numbers and local complete bipartite cover numbers.

Let denote the class of all difference graphs, and the class of all complete bipartite graphs. Clearly, we have for all graphs . Kim et al. [15] asked whether there is a sequence of graphs for which is constant while is unbounded. They prove that for all graphs on vertices,

by showing that whenever is a difference graph with one partite set of size . However, no lower bound on for is established in [15]. Specifically, Kim et al. ask for the exact value of for the difference graph with vertex set and for all . For the case that is a power of  they prove the upper bound .

The number of steps of a Young diagram is the number of different row lengths in , i.e., the cardinality of

The cells in are called the steps of . Young diagrams with elements, rows, columns, and steps, visualize partitions of into  unlabeled summands (row lengths) with summands of different values and largest summand being .

We say that is covered by a set of generalized rectangles if , i.e., is the union of all rectangles in . In this case we also say that is a cover of . If additionally the rectangles in are pairwise disjoint, we call a partition of . For example, Figure 2 shows a Young diagram with a partition into actual rectangles.

Theorem 1.

For any , any Young diagram can be covered by a set of generalized rectangles such that each row and each column of is used by at most rectangles in if and only if has strictly less than steps.

The Young diagram of the difference graph is , i.e., the (unique) Young diagram with rows, columns, and steps. Therefore Theorem 1 answers the questions raised by Kim et al..

2.1 Proof of Theorem 1

Throughout we shall simply use the term rectangle for generalized rectangles, and rely on the term actual rectangle when specifically meaning rectangles that are contiguous. For a Young diagram and , let us define a cover of to be -local if each row of is used by at most rectangles in and each column of is used by at most rectangles in . Recall that is the Young diagram with rows, columns, and steps. See Figure 2.

Figure 2: The Young diagram with steps and a -local partition of with actual rectangles.

We start with a lemma stating that instead of considering any Young diagram with steps, we may restrict our attention to just .

Lemma 2.

Let and be any Young diagram with steps. Then  admits an -local cover if and only if admits an -local cover with exactly rectangles.

Proof.

First assume that admits an -local cover . If consists of strictly more than rectangles, as every rectangle is contained in a for some step , by the pigeonhole principle there are , , such that for some step . However, in this case is also an -local cover of with one rectangle less, where denotes the rectangle whose row set and column set is the union of the row set and column set of and . Thus, by repeating this argument, we may assume that .

Figure 3: Transforming a cover of any Young diagram with steps into a cover of (left) and vice versa (right).

If , there is a row or a column that is not used by any step in . Apply the mapping with

respectively

Intuitively, we cut out row  (respectively column ), moving all rows below one step up (respectively all columns to the right one step left). This gives an -local cover of a smaller Young diagram with steps, and eventually leads to an -local cover of , as desired. See the left of Figure 3.

On the other hand, if admits an -local cover , this defines an -local cover of as follows. Index the rows used by the steps of by and the columns used by the steps of by and let . Defining

for gives an -local cover of . See the right of Figure 3.

Observe that the construction maps an actual rectangle of to an actual rectangle of . Also, if is a partition of , then is a partition of . This will be used in the proof of Item 1 of Theorem 3. ∎

Let us now turn to our main result. In fact, we shall prove the following strengthening of Theorem 1.

Theorem 3.

For any and any Young diagram with steps, the following hold.

  1. [label = ()]

  2. If , then there exists an -local partition of with actual rectangles.

  3. If , then there exists no -local cover of with generalized rectangles.

Proof.

First, let us prove Item 1. For shorthand notation, we define . It will be crucial for us that the numbers solve the recursion

(1)

This follows directly from Pascal’s rule for any with .

Due to Lemma 2 it suffices to show that for any and , there is an -local partition of with actual rectangles.

We define the -local partition by induction on and . For illustrations refer to Figure 4.

Figure 4: Left: The Young diagram with steps and a -local partition of into actual rectangles. Right: The Young diagram with steps, the rectangle with , and the Young diagrams and with and steps, respectively.

If , respectively , then is the set of rows of , respectively the set of columns of . If and , then by (1). Consider the actual rectangle for . Then splits into a right-shifted copy of and a down-shifted copy  of . Note that and .

By induction we have an -local cover of and an -local cover of , each consisting of pairwise disjoint actual rectangles. Define

this is a cover of consisting of pairwise disjoint actual rectangles. Rows to  are used by and at most rectangles in , and rows to are used by at most  rectangles in . Hence each row of is used by at most rectangles in . Similarly each column of is used by at most rectangles in . Thus is an -local partition of by actual rectangles, as desired.

For we obtain an -local partition of by restricting the rectangles of the cover of to the rows from to . This yields an -local partition of a down-shifted copy of .


Now, let us prove Item 2. Due to Lemma 2 it is sufficient to show that for the Young diagram with admits no -local cover. If with has an -local cover, then by restricting the rectangles of the cover to the rows from to we obtain an -local cover of a down-shifted copy of . Therefore, we only have to consider .

Let be a cover of . We shall prove that is not -local. Again, we proceed by induction on and , where illustrations are given in Figure 5.

Figure 5: The Young diagram with steps, the rectangle with , and the Young diagrams and with and steps, respectively.

If , then each row is only used by a single rectangle in , otherwise,  would not be -local. Hence, each row of is a rectangle in . Thus column  of is used by rectangles, proving that is not -local.

The case is symmetric to the previous by exchanging rows and columns.

Now let and . We have . Consider the rectangle for . Then splits into a right-shifted copy of and a down-shifted copy of . Note that .

Let , respectively , be the subset of rectangles in using at least one of the rows in , respectively at least one of the columns in . Note that as each generalized rectangle is contained in .

Prune each rectangle in to the columns and each rectangle in  to the rows . This yields covers of and .

The Young diagram is a copy of and . Hence, by induction the pruned cover is not -local. If some column of is used by at least rectangles in , this column of is used by at least rectangles in , proving that is not -local, as desired. So we may assume that some row  of is used by at least rectangles in .

Symmetrically, is a copy of and . Hence, the pruned is a cover of , which by induction is not -local, and we may assume that some column of is used by at least rectangles in . Hence row in is used by at least rectangles in and column in is used by at least rectangles in . As and element is contained in some rectangle of , either row of is used by at least rectangles or column of is used by at least rectangles (or both), proving that is not -local. ∎

Finally, Theorem 1 follows from Theorem 3 by setting .

2.2 More about local covering numbers

Using Theorem 1 and , we see that

  • for every difference graph the exact value of is the smallest such that for the number of steps555For graphs, this is the number of different sizes of neighbourhoods in one partite set. of it holds that ,

  • the difference graphs , (corresponding to the Young diagrams , ), defined by Kim et al. satisfy

    (and, using more precise bounds on Stirling’s approximation, it can be shown that the term is at most for all ),

  • for this sequence of difference graphs is constant , while is unbounded, and

  • for all graphs on vertices,

It is also interesting to understand the worst case scenario in covering a bipartite graph by complete bipartite graphs or difference graphs. With a different proof, the following result was already shown in [15].

Theorem 4.

For any there exists a bipartite graph on vertices such that .

Proof.

Suppose is even. Consider a random bipartite graph with vertex classes and , where

and each edge is chosen with probability

. For any , the expected number of ’s in is . If we choose , then the expected number of ’s (and hence the probability that contains a ) is less than . The probability that is at least , so with nonzero probability and has no .

Now consider a cover of with difference graphs. We call a star an -star (resp. -star) if its centre is in (resp. ). No difference graph in the cover contains a and thus every difference graph in the cover can be decomposed into at most -stars and at most -stars. Without loss of generality, at least half the edges of are covered by -stars. As has vertices, among the edges covered by -stars there are at least incident to some vertex . Each difference graph in the cover contributes at most of the -stars containing . Therefore at least difference graphs of the cover contain . ∎

As Kim et al. [15] already observed, the upper bound follows from a theorem of Erdős and Pyber [7], which shows that a cover of corresponding size exists even with complete bipartite graphs.

Theorem 5 (Erdős and Pyber).

For any simple graph on vertices, .

Hansel [12] (see also Király, Nagy, Pálvölgyi and Visontai [16] and Bollobás and Scott [3]) proved that , which together with the obvious upper bound gives the following proposition:

Proposition 6.

For all , .

In fact, Hansel proved a somewhat stronger result, namely that, for every -covering of , the average number of complete bipartite graphs in which a vertex appears is at least .

An interesting case is , which is obtained by deleting the edges of a perfect matching from the complete bipartite graph . Note that is the union of two difference graphs. What is the best covering of this graph by complete bipartite graphs?

Proposition 7.

and therefore .

Proof.

Let us denote the vertices of by , and the vertices of by where . One can easily obtain a covering of from a covering of . For every complete bipartite graph in the covering with vertex classes and take the complete bipartite graph with vertex classes and and another one with vertex classes and . This will be a covering of where the vertices and are covered exactly as many times as in the covering of . This construction shows .

On the other hand, we can obtain a covering of from a covering of . For a complete bipartite graph in the covering with vertex classes and take the complete bipartite graph with vertex classes and . This will be a covering of where the vertex is covered exactly as many times as and are covered in total in the covering of . This construction shows . ∎

3 Local dimension of posets

The motivation for Kim et al. [15] to study local difference cover numbers comes from the local dimension of posets, a notion recently introduced by Ueckerdt [23].

For a partially ordered set (also called a poset) , define a realizer as a set of linear extensions such that if and are incomparable (denoted ), then in some and in some . The dimension of , denoted , is the minimum size of a realizer. The dimension of a poset is a widely studied parameter.

A partial linear extension of is a linear extension of an induced subposet of . A local realizer of is a non-empty set of partial linear extensions such that (1) if in , then in some , and (2) if and are incomparable (denoted ), then in some and in some . The local dimension of , denoted , is then the smallest for which there exists a local realizer of with each appearing in at most  partial linear extensions . Note that by definition for every poset .

For an arbitrary height-two poset , Kim et al. consider the bipartite graph with partite sets (the minimal elements of ) and whose edges correspond to the so-called critical pairs:

They prove that

which also gives good bounds for when has larger height, since we have

for the associated height-two poset known as the split of (see [2], Lemma 5.5). Using these results and the ones from the previous section, we can conclude the following for the local dimension of any poset.

Corollary 8.

For any poset on elements with split we have

3.1 Local dimension of the Boolean lattice

Let denote the Boolean lattice of subsets of the element set (note that this lattice has elements, one for each subset of ). Since the dimension of is we immediately have .

For any integer let denote the family of all the subsets of of size . This we call layer or the ’th layer of and let be the subposet of induced by layers and . We denote .

The study of the dimension has a long history. Kierstead [14] is a valuable survey on the topic.

Kim et al. [15] give a lower bound for which implies that . For the height poset they in fact give bounds on the local covering numbers of the corresponding bipartite graphs.

A similar lower bound on can be obtained as follows: Let and consider a local realizer such that each subset of appears in at most  of the partial linear extensions. Altogether there are at most appearances of singletons and at most partial linear extensions containing a singleton. The singletons cut these partial linear extensions altogether into at most consecutive parts. Given a non-singleton fixed set of , for any given such part we have two options, is either present in this part or not. Moreover, is present in at most partial linear extensions, thus in at most such parts. Two sets cannot be present in exactly the same parts by the definition of a local realizer. Thus, the number of sets (which is equal to ) is at most the number of subsets of size at most on elements. Hence . From the inequality it follows that .

Problem 1.

Determine the asymptotics of .

A possible approach towards resolving the problem would be to study the local dimension of appropriate pairs of levels.

We continue with what we can say regarding for some specific values of and .

3.2 The subposet of the two middle levels

Let

be odd and consider the poset

induced by the two middle levels of the Boolean lattice. We are interested in . More specifically, let be the adjacency matrix of the bipartite graph defined by the critical pairs (see Section 3) of levels and . We want to find good local covers of .

First we give a recursive formula for (notice that it has rows and colums). is the matrix with a single entry (its single row corresponds to , its single column to and these are not connected in the corresponding bipartite graph as , that is, they do not form a critical pair. Then,

Here J denotes an all- (not necessarily square) matrix of appropriate size and is the complement of an identity (square) matrix of appropriate size. This recursion can be easily verified by considering two elements and ordering the rows and columns by first taking the sets containing  but not containing , then taking the sets containing but not containing , then the sets containing and and finally the sets containing none of the two.

Problem 2.

Determine the best local cover of by Young diagrams.

3.3 The subposet of the first two levels

In this subsection, we look at the poset and the graph of critical pairs in . Throughout this section, we identify the first layer with in the obvious way.

It is known that the dimension of grows asymptotically as . Spencer proved the upper bound in [22], and Füredi, Hajnal, Rödl, and Trotter proved the corresponding lower bound in [10]. The maximum such that is sometimes denoted , see OEIS666On-Line Encyclopedia of Integer Sequences; https://oeis.org Sequence A001206 and Hoşten and Morris [13].

Theorem 9.

As , .

Proof.

The upper bound follows from Spencer’s upper bound for . We prove the lower bound

Let be an -covering of . Recall that, for each , the singletons in are weakly ordered by reverse inclusion of their neighbourhoods. We define a sequence of difference graphs and a sequence of subsets as follows. Let be a fixed positive real number. First, choose such that contains at least singeletons, if there is such a graph in . If there isn’t, then each pair is contained in at least elements of . Otherwise, let be the set of singletons in . Now suppose and have already been chosen. We choose a graph such that , if such a graph exists. If so, then, by the Erdős-Szekeres theorem, there is a subset such that and the elements of appear in the same or opposite order in and . Continue in this way until either or and there is no graph in that contains elements of . In the former case, each element of appears in at least elements of , and , so . In the latter case, let and be the first and last elements of in the order induced by and look at the set of chosen difference graphs that contain the pair . Because the ordering on induced by begins with either or for every , none of these graphs can contain any edges from to . Every other difference graph in contains less than edges from to , so there must be at least such difference graphs containing . Now, if we take , then

Using the affine approximation