Constructing sparse Davenport-Schinzel sequences by hypergraph edge coloring

10/16/2018
by   Jesse Geneson, et al.
0

A sequence is called r-sparse if every contiguous subsequence of length r has no repeated letters. A DS(n, s)-sequence is a 2-sparse sequence with n distinct letters that avoids alternations of length s+2. Pettie and Wellman (2018) asked whether there exist r-sparse DS(n, s)-sequences of length Ω(s n^2) for s ≥ n and r > 2, which would generalize a result of Roselle and Stanton (1971) for the case r = 2. We construct r-sparse DS(n, s)-sequences of length Ω(s n^2) for s ≥ n and r > 2. Our construction uses linear hypergraph edge-coloring bounds. We also use the construction to generalize a result of Pettie and Wellman by proving that if s = Ω(n^1/t (t-1)!), then there are r-sparse DS(n, s)-sequences of length Ω(n^2 s / (t-1)!) for all r ≥ 2. In addition, we find related results about the lengths of sequences avoiding (r, s)-formations.

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1 Introduction

A Davenport-Schinzel sequence of order is a sequence with no adjacent same letters that avoids alternations of length [3]. A -sequence is a Davenport-Schinzel sequence of order with distinct letters. The function is defined as the maximum possible length of a -sequence. Davenport-Schinzel sequences have a variety of applications and connections to other problems, including upper bounds on the maximum complexity of lower envelopes of sets of polynomials of bounded degree [3], the maximum complexity of faces in arrangements of arcs [20], the maximum number of edges in certain -quasiplanar graphs [5, 7], extremal functions of forbidden matrices [11, 15], and extremal functions of tuples stabbing interval chains [6].

Most research on Davenport-Schinzel sequences has focused on when is fixed. It is easy to see that and [3]. Nivasch [14] and Pettie [16] proved that , while Agarwal, Sharir, and Shor [1] proved that . Pettie [16] and Nivasch [14] proved that and for all , where .

A more general upper bound from Davenport and Schinzel [3, 12] shows that , even when is not fixed. Roselle and Stanton [19] constructed a family of sequences to prove that if , then . For the case of , the coefficient of in their lower bound is , and it is an open problem [18] to determine what is the actual coefficient between and . Pettie and Wellman [18] proved several bounds for when is not fixed but smaller than linear in , including that if , then is between and .

Call a sequence -sparse if every contiguous subsequence of length has all letters distinct. Let be the maximum possible length of an -sparse -sequence. Klazar proved for fixed that for all [13], but the proof does not work when is not fixed. Pettie and Wellman [18] asked whether Roselle and Stanton’s bound can be generalized to -sparse -sequences.

In this note, we construct -sparse -sequences of length for , where the constant in the bound depends on . Our construction uses Kahn’s asymptotic bound on linear hypergraph edge-coloring [10]. As a corollary, we obtain that if , then there are -sparse -sequences of length for all .

We also prove related results about -formations. An -formation is a concatenation of permutations of distinct letters. The function is defined as the maximum possible length of an -sparse sequence with distinct letters that avoids all -formations. Similarly we define the function to be the maximum possible length of a -sparse sequence with distinct letters that avoids all -formations.

Nivasch [14] and Pettie [17] found tight bounds on for all fixed , which are mostly on the same order as the bounds for . Upper bounds on -formations have been used to find tight bounds on the extremal functions of several families of forbidden sequences [13, 8], including a family of sequences used to bound the maximum number of edges in -quasiplanar graphs in which every pair of edges intersect at most a constant number of times [7, 5].

We show that for all using the same family of sequences that we constructed for . The upper bound is from [12, 14]. We also prove that for all , where the constant in the bound again depends on . Using the previous bound as the initial case, we generalize the construction of -sparse -sequences of length to prove that for all and , where the constant in the bound depends on and .

2 Iterated hypergraph edge coloring

A hypergraph is called linear if every pair of distinct edges have intersection size at most . A proper edge-coloring of a hypergraph is a coloring of the edges of so that no intersecting distinct edges have the same color.

Erdős, Faber, and Lovász conjectured that any linear hypergraph on vertices has a proper edge-coloring with at most colors. The conjecture was originally stated for vertex colorings of graphs [4], but Hindman observed that both versions were equivalent and proved the conjecture for [9]. Chang and Lawler [2] proved an upper bound of , before Kahn proved the asymptotic upper bound of [10].

Our construction of -sparse -sequences of length is inductive. The initial case for is a family of sequences that looks similar to the sequences in [19]. For the inductive step, we turn each sequence into a linear hypergraph, color its edges with the minimal possible number of colors, and then add those colors into the sequence as letters to increase the sparsity by . The next lemma is part of what makes the induction work.

Lemma 2.1.

Let be a linear hypergraph with vertices and let be a proper-edge coloring of , where denotes the set of colors. If is the hypergraph obtained from with vertex set equal to and with edge set consisting of every edge of the form for , then is also linear.

Proof.

Let and be any two distinct edges of . Then and for some edges . If , then since is a proper edge-coloring of . Otherwise if , then since is linear. ∎

In the next theorem for the case of , we could use Vizing’s Theorem instead of Kahn’s upper bound for the proper edge-coloring of , since is just a graph when . This would eliminate the from the proof.

Theorem 2.2.

Fix integer and real number . Then for all , where the constant in the bound depends on and .

Proof.

Since the upper bound was already prove in [3, 12], it suffices to prove that for all , where the constant in the bound depends on and .

Roselle and Stanton already proved the result for [19]. Our construction is inductive. Rather than using Roselle and Stanton’s construction for the initial case of , we use a slightly simpler construction with a worse constant in the bound.

Let be two parameters that will be chosen at the end of the proof in terms of , , , and . Define to be the sequence obtained by starting with the empty sequence and then addending copies of the subsequence for each and , for in lexicographic order. For each with and , we call the consecutive copies of the subsequence in a block. We call the set of adjacent blocks with the block-row . There are a total of block-rows, and each block-row has fewer than blocks.

First observe that has length . Moreover for any pair of distinct letters in , the length of an alternation on the letters is less than since the block-row containing the block on letters contains an alternation on of length at most , and all other block-rows contain alternations on of length at most . Note also that is -sparse, but not -sparse, and has distinct letters.

For every , we will construct so that has length and any pair of distinct letters in have alternation length less than . In addition, will be -sparse but not -sparse, and will have at most distinct letters.

Like , the sequences for also have blocks, where each block consists of a sequence of distinct letters repeated times. In order to construct from , we treat each block in as an edge in a -uniform hypergraph.

Specifically, is the -uniform hypergraph with vertex set equal to the letters of and edge set with if and only if there is a block in on the letters . Note that in the case that , is a graph by construction, so is also a linear hypergraph for .

Suppose for inductive hypothesis that is a linear hypergraph. Then by the theorem of Kahn [10], there exists a proper edge-coloring of with at most colors.

For each edge , insert the color after each of the occurrences in of the letters in . The resulting sequence is -sparse but not -sparse. It has length and at most distinct letters.

As for alternations, we note that any pair of letters in that were also in have alternation length less than by inductive hypothesis. If and are both letters that are new to , then and make an alternation of length less than . If was in but was not, then there are two cases.

If appears in no block with , then and make an alternation of length less than . If and appear in a single block together, then and make an alternation of length less than . Note that and cannot occur in two blocks together since is a proper edge-coloring.

We have one last step for the induction. Let be the -uniform hypergraph with vertex set equal to the letters of and edge set with if and only if there is a block in on the letters . was a linear hypergraph, so is also a linear hypergraph by Lemma 2.1. This completes the induction.

The last part of the proof is choosing and in terms of and . Since is an -sparse sequence avoiding alternations of length with at most distinct letters and length , choosing e.g. and suffices to give the bound of for all , where the constant in the bound depends on and .

Note also that even if has fewer than distinct letters, we can add a sequence of new distinct letters at the end of to increase the number of distinct letters to without increasing the maximum alternation length. ∎

One of the constructions in Pettie and Wellman’s paper [18] is an inductive construction that uses Roselle and Stanton’s construction as its initial case. The construction in Theorem 2.2 can be substituted for Roselle and Stanton’s construction in Pettie and Wellman’s proof to generalize the result in [18].

Corollary 2.3.

If , is between and for all .

Proof.

Pettie and Wellman proved the case in Theorem 4.1 of their paper [18]. The initial case of their construction uses the Roselle-Stanton construction for a prime power and . For , their construction and the analysis in their proof also work if we replace the Roselle-Stanton construction in their initial case with our construction in Theorem 2.2 using as the bound for . ∎

3 Long -formations

We obtain the next result just from the construction in the last section and previously known upper bounds.

Theorem 3.1.

Fix integer and real number . for all , where the constant in the lower bound depends on and .

Proof.

The upper bound follows from Klazar’s bound for all [12, 14]. The lower bound follows from our construction in Theorem 2.2. ∎

Klazar actually proved the more general result that for all [12, 14]. In the next theorem, we show for sufficiently large that this bound is tight up to a factor that depends only on (and not on or ).

Theorem 3.2.

Fix integer and real number . Then for all , where the constant in the lower bound depends on and .

Proof.

It suffices to prove that for all , where the constant in the bound depends on and . We will use a family of sequences similar to the one used in Theorem 2.2.

Let be two parameters that will be chosen at the end of the proof in terms of , , , and . Define to be the sequence obtained by starting with the empty sequence and then addending copies of the subsequence for each and and , for in lexicographic order.

We call the consecutive copies of the subsequence in a block. We call the set of adjacent blocks with the block-row . Note that each block-row contains fewer than blocks, and there are a total of block-rows.

First observe that has length , and that is -sparse. Next we explain why the formations on letters have length less than . Let be arbitrary distinct letters in .

Note that we can find a longest formation on the letters by searching greedily from left to right in . Suppose that we go through the block-rows from beginning to end, and we mark block-rows greedily on the letters wherever the formation length increases by (in other words, on the last letter of each permutation of the formation that we find greedily).

Then every block-row not equal to increases the length of the formation on letters by at most . Block-row increases the length of the formation on letters by at most , so all -tuples of letters in have formation length less than .

Since is an -sparse sequence avoiding formations of length with distinct letters and length , choosing e.g. and suffices to give the bound of for all , where the constant in the bound depends on and . ∎

The lemma below generalizes the first part of Chang and Lawler’s argument for their upper bound of on proper edge-coloring for linear hypergraphs. We use this lemma in place of Kahn’s theorem in the main result of this section, which parallels the proof of Theorem 2.2.

Lemma 3.3.

Suppose that is a -uniform hypergraph in which every pair of edges have intersection size at most for . Then it is possible to color the edges of with colors so that no pair of edges with intersection size receive the same color.

Proof.

We color the edges of in an arbitrary order. Assume that we next color an edge . Since every pair of edges in have intersection size at most , there are at most edges already assigned colors that meet at each of the size- subsets of vertices that are contained in . Thus there will be an unused color for if , which holds for all , so we color with any unused color. ∎

The construction for the theorem below generalizes the construction in the proof of Theorem 2.2. Also, the intiial case of the construction uses the construction in Theorem 3.2.

Theorem 3.4.

Fix integers and real number . Then for all , where the constant in the lower bound depends on , , and .

Proof.

The upper bound was already proved in [12, 14], so it suffices to prove that for all , where the constant in the lower bound depends on , , and .

In the last section, we proved that the theorem is true for (Theorem 2.2) and also for (Theorem 3.2). As in Theorem 2.2, our construction for this theorem is inductive. For the initial case of , we set , where is the same sequence defined in Theorem 3.2.

For every , we will construct so that has length and any -tuple of distinct letters in has formation length less than . In addition, will be -sparse but not -sparse, and will have at most distinct letters.

Like , the sequences for also have blocks, where each block consists of a sequence of distinct letters repeated times. In order to construct from , we treat each block in as an edge in a -uniform hypergraph.

Similarly to Theorem 2.2, is the -uniform hypergraph with vertex set equal to the letters of and edge set with if and only if there is a block in on the letters .

Suppose for inductive hypothesis that is a hypergraph in which every pair of edges have intersection size at most . Then by Lemma 3.3, it is possible to color the edges of with some coloring using colors so that no pair of edges with intersection size receive the same color.

For each edge , insert the color after each of the occurrences in of the letters in . The resulting sequence is -sparse but not -sparse. It has length and at most distinct letters.

For formations, we consider arbirary distinct letters in . Note that if all of the letters were also in , then they have formation length less than by inductive hypothesis. If two of the letters are both new to , then the maximum possible formation length on is at most . If there is a single letter that was not in , then there are two cases.

If there is some that appears in no block with , then make a formation of length at most . If all of the letters appear in a single block together, then their maximum formation length is less than , since each block-row not containing that block contributes at most to the formation length. Note that the letters cannot all occur together in two blocks by the definition of the coloring .

We have one last step for the induction. Let be the -uniform hypergraph with vertex set equal to the letters of and edge set with if and only if there is a block in on the letters . was a hypergraph in which every pair of edges have intersection size at most , so is also a hypergraph in which every pair of edges have intersection size at most by the definition of the coloring . This completes the induction.

The last part of the proof is choosing and in terms of and . Since is a -sparse sequence avoiding formations of length with distinct letters and length , choosing e.g. and suffices to give the bound of for all , where the constant in the lower bound depends on , , and .

As in Theorem 2.2, note also that even if has fewer than distinct letters, we can add a sequence of new distinct letters at the end of to increase the number of distinct letters to without increasing the maximum formation length. ∎

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