Shadow Ratio of Hypergraphs with Bounded Degree

We study the size of the shadow of k-uniform hypergraphs with bounded degree. Lower bounds on the ratio of the size of the shadow and the size of the hypergraph are given as a function of the degree bound and k. We show that cliques are extremal for a long range of degree bounds, but not for every bound. We give a general, but not sharp lower bound on the shadow ratio and show, that sometimes we can get extremal hypergraphs by deleting disjoint maximal matchings from a clique .


Introduction
In general the shadow ratio can be as close to zero as we want for any fixed k, as the sequence of larger and larger cliques show: if H = [n] k , then |σ (H )| |H | = k n−k+1 → 0 as n → ∞. But many hypergraph classes with shadow ratio bounded away from zero are known. A few examples are l-intersecting hypergraphs [2], hypergraphs with bounded VC-dimension [1] and hypergraphs with a set of distinct representatives [5]. In all of these cases the largest possible cliques are extremal. What can we say about the family of k-uniform hypergraphs with (H ) ≤ d?
If k = 2, H is just a graph, and its shadow is the set of nonisolated vertices. Since every vertex is incident to at most d edges and every edge is incident to exactly 2 vertices, a double counting of the vertex-edge incidences shows us the following.

Proposition 1 If H is a 2-uniform hypergraph with degree bound d ≥ 1, then
And, as any d-regular graph confirms, the bound on the shadow ratio is tight. Extremal hypergraphs, for which the stated inequalities are tight, are of interest in general. As we will see, cliques are extremal in terms of shadow ratio not just among hypergraphs with the same degree bounds as theirs. But what happens, if the degree bound is so low, it excludes every nontrivial clique? This case is rather different. We start with that by stating a stronger result than the one about the shadow ratios.
H is a k-uniform hypergraph with degree bound d and |H | = qd + r with nonnegative integers r < d and q, then The tightness of the inequality is confirmed by the disjoint union of q hypergraphs of size d, each on k + 1 vertices and one hypergraph of size r , also on k + 1 vertices. The corresponding result about shadow ratios is an easy corollary.
What happens if d ≥ k? We first start with a general lower bound on the shadow ratio. It is not tight most of the times but gives a usable lower bound for every degree bound. If k is a positive integer and x ≥ k is a real number, we can define the binomial . Every positive integer can be expressed as such a binomial coefficient with a suitable x. The following Theorem will be proved by applying Lovász's shadow bound to the subhypergraph of hyperedges containing a fixed vertex.
This bound is tight only if x is an integer. In this case, an extremal graph is a clique on x + 1 vertices. What is interesting is that these cliques are extremal even among hypergraphs with much larger degree bound.

Theorem 3 If t ≥ k ≥ 3 are integers and H is a k-uniform hypergraph with degree bound
Note that the bound on d is of order t k − 1 + t k−2 and compare this with the max degree of the next clique, which is of order What happens after the degree bound in Theorem 3? We can show that the clique on t + 1 vertices is not extremal much longer.

Proposition 2
For every integer t ≥ k ≥ 3 there exists a k-uniform hypergraph H with degree bound d such that and The proof of Proposition 2 is just a construction of a hypergraph with max degree The paper is organised as follows. In Sect. 2 we cover the famous Kruskal-Katona Theorem and give some new bounds on the shadow ratio of bounded size hypergraphs. In Sect. 3 we prove Theorems 2, 3 and 4 by connecting the case of bounded degree hypergraphs to the case of bounded size hypergraphs. In Sect. 4 an elementary proof of Theorem 1 is given. Section 5 contains the description of a hypergraph family which proves Proposition 2. In the end, in Sect. 6 we pose some open questions.

Shadow Ratio of Hypergraphs with Bounded Size
For uniform hypergraphs of fixed size, the famous Kruskal-Katona Shadow Theorem determines a tight lower bound on the size of the shadow. To state it, we will need the following

Lemma 1 ([3]) For every positive integer m and k there uniquely exists a k
We call this sum the k-binomial representation of m.
The k-binomial representation of positive integers plays a crucial role in the Shadow Theorem.
The k-order shadow function determines the exact lower bound on the size of the shadow of k-uniform hypergraphs of size m. This statement is the Kruskal-Katona Shadow Theorem [3,4].
The Shadow Theorem determines F k (m) m for every value of k and m, but these values are hard to handle, even if we just need a statement about hypergraphs of bounded size instead of fixed size. One possible solution is to use a weaker version of the Shadow Theorem due to Lovász [6], which gives a not tight, but easily applicable lower bound on the shadow. For any positive integer k and real number x, let

Equality holds if and only if x is an integer.
This lower bound is not always tight, but one can easily deduce a result about the shadow ratio of hypergraphs of bounded size from it.

Corollary 2 If k is a positive integer, x ≥ k is a real number and m ≤
x k is a positive integer, then This corollary is also tight if and only if x is an integer. And it gives a strictly monotone decreasing bound of F k (m) m in m. But the values of this shadow ratio function are far from being strictly decreasing. We show two sharpenings of Corollary 2, the first is about values of m which are slightly bigger than a k , the second is about values that are slightly smaller.
Proof We prove the implication by induction on l.
For l = 0 it is just Corollary 2. Let us now suppose that (1) is true for some l and also that l Proof Since the above sum equals to m, the only thing we need to show is a > a − a+1 In a similar manner one can prove other inequalities for F k (m) m . We only need one more.

Lemma 3 Let a ≥ k be positive integers. If
where the last inequality follows from the following claim.
Claim (1) If x 1 , x 2 , y 1 , y 2 ≥ 0, then For the m < d 0 case, we will need two more simple inequalities.

Another simple case is when
by Corollary 2, and we have just seen that We prove inequality (2) by induction on t − s. If t − s = 1, then either where the first inequality follows by Claim (2). The induction step from t − 1 to t goes in a similar way. Suppose the k-binomial by Claim (2) and the induction hypotheses. On the other hand, if (2) and (3). Let u be the smallest integer between s and t − 1 such that by the induction hypotheses. This finishes the proof of the induction step, the whole m < d 0 case and thus the proof of Lemma 3.

Shadow Ratio of Hypergraphs with Higher Maximum Degree
We prove Theorem 2, 3 and 4 by connecting the case of bounded degree hypergraphs to the case of bounded size hypergraphs.
If v is a vertex of H , then let H is k-uniform, H −v is a (k − 1)-uniform hypergraph, and its shadow ratio is closely related to the shadow ratio of H , as stated in the following Lemma 4 If there exists an α ∈ R such that for a k-uniform hypergraph H the inequality

Proof of Lemma 4
By double counting the vertices of H and the hyperedges containing them, we get k|H | = v |H v |. The same argument for the shadow shows Here σ (H ) v is the set of those elements of the shadow, which contain v. The equality |H −v | = |H v | holds for every v, as the bijection H → H ∪ {v} shows. Similarly, we can get |σ (H −v )| = |σ (H ) v | for every v. The following line of inequalities prove the lemma: Note that |H −v | = |H v | ≤ (H ). We can combine Lemma 4 with the results about the shadow ratio of bounded size hypergraphs from Sect. 2 to get the theorems about bounded degree hypergraphs.

Proof of Theorem 2 According to the degree bound, for every vertex v of H the inequality
We can apply Corollary 2, which guarantees Proof of Theorem 1 First, we show, that if a k-uniform hypergraph H has degree bound d < k, then its hyperedges can be partitioned into subhypergraphs of size at most d with the subhypergraphs having disjoint shadows.

Definition 2
The Johnson graph G H of a k-uniform hypergraph H is a graph whose vertices correspond to hyperedges of H and whose edges correspond to pairs of hyperedges H 1 , The maximal connected subhypergraphs of H are called the connected components of H .
If H and H are two different connected components of H , then their shadows are disjoint. First, we consider only connected hypergraphs. Their size is at most d, as stated in the following

Lemma 5 If d < k and H is a connected k-uniform hypergraph with degree bound d, then |H | ≤ d.
Proof Let H = {H 1 , . . . , H t } and let the hyperedges be ordered in such a way, that for every H i there is an H j with j < i and |H i ∩ H j | = k − 1. Such an ordering exists as can be seen by a breadth-first search in the Johnson graph G H . We prove the inequality Equality is achieved, if H has exactly k + 1 vertices. Finally, if H is a k-uniform hypergraph with degree bound d < k and arbitrary size t, then by Lemma 5 it is a disjoint union of subhypergraphs H 1 , . . . , H q+1 each of size at most d and with disjoint shadows. Note that the vertex sets of the subhypergraphs may be intersecting, but their shadows are disjoint. Let the sizes of the subhypergraphs be t 1 , . . . , t q+1 . We claim that if there are i = j with 0 < t i ≤ t j < d, then H is not extremal. Indeed, if this is the case, then |σ ( . But if we replace this two subhypergraphs with H i and H j , subhypergraphs on separate k + 1 vertices and size t i − 1 and t j + 1, then As a conclusion, extremal k-uniform hypergraphs with degree bound d < k and size qd + r are those, which can be decomposed into q + 1 connected hypergraphs, each on k + 1 vertices and q of them having size d, one of them size r .

Nonregular Initial Segments with Low Shadow Ratio
The Katona-Kruskal Shadow Theorem can be stated in the following way: a k-uniform hypergraph with m hyperedges has shadow size at least that of the size of the shadow of the k-uniform initial segment of size m. Initial segments are in a way a generalization of cliques and can be defined with the help of colexicographic ordering, where if given two sets A, B ⊂ [n] we say that A > B if and only if the largest element of the symmetric difference of A and B are in A. A k-uniform initial segment of size m is then just the first m subsets of size k in the colexicographic ordering. These constructions are easier to describe when the k-binomial representation of m contains only a few terms, but all the initial segments have a pretty low shadow ratio and some of them might even be extremal for a given degree bound. The computation and comparison of the shadow ratios of initial segments are somewhat obscure, so we only do it for one more class of initial segments, to prove Proposition 2.

Proof of Proposition 2
We need a hypergraph with degree bound t k − 1 + t + 1 − t+2 k k − 2 and lower shadow ratio than the shadow ratio of is such a hypergraph, as will be shown in the following. Note that this hypergraph is just H  + 4). All the extremal hypergraphs we have seen so far could have been obtained by deleting some (sometimes zero) disjoint maximal matchings from an initial segment.

Question 1
Is it true, that for any k and d, among k-uniform hypergraphs with degree bound d, there is an extremal hypergraph with minimal shadow ratio which can be obtained by deleting disjoint maximal matchings from an initial segment?
In all the solved cases, the extremal hypergraph were regular, because Lemma 4 can not guarantee extremality if the hypergraph is nonhomogene in term of the shadow ratios of the local subhypergraphs around the vertices.
A harder question is: how can we bound the size of the shadow as a function of k, d and |H |? Can we find functions F d k for which |σ (H )| ≥ F d k (|H |) holds sharply, that is the inequality holds for any k uniform hypergraph with degree bound d, and for any size m there is a hypergraph H with |H | = m and |σ (H )| = F d k (m)? We have seen in Theorem 1, that if d < k and m = qd + r with r < d and q integers, What can we say about the case k ≤ d?
Question 2 Determine functions F d k for k ≤ d such that |σ (H )| ≥ F d k (|H |) sharply holds for k-uniform hypergraphs with degree bound d.
If d is large enough compared to k and m, then F d k (m) equals to F k (m), the shadow function from the Kruskal-Katona Theorem. Hence an answer to this question would generalize Kruskal-Katona. in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.