Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
We investigate a covering problem in -uniform hypergraphs (-graphs): given a -graph , what is , the least integer such that if is an -vertex -graph with minimum vertex degree then every vertex of is contained in a copy of in ?
We asymptotically determine when is the generalised triangle , and we give close to optimal bounds in the case where is the tetrahedron (the complete -graph on vertices).
This latter problem turns out to be a special instance of the following problem for graphs: given an -vertex graph with edges, what is the largest such that some vertex in must be contained in triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.
Let be a graph with at least one edge. What is the maximum number of edges an -vertex graph can have if it does not contain a copy of as a subgraph? This is a classical question in extremal graph theory. If is a complete graph, then the exact answer is given by Turán’s theorem , one of the cornerstones of extremal graph theory. For other graphs , the value of is determined up to a error term by the celebrated Erdős–Stone theorem .
Ever since Turán’s foundational result, there has been significant interest in obtaining similar “Turán–type” results for -uniform hypergraphs (-graphs), with . The extremal theory of hypergraphs has however turned out to be much harder, and even the fundamental question of determining the maximum number of edges in a -graph with no copy of the tetrahedron (the complete -graph on vertices) remains open — it is the subject of a 70-years old conjecture of Turán, and of an Erdős $ 1000 prize***In fact, to earn this particular Erdős prize, it is sufficient to determine the limit for any integers .. Most of the research efforts have focussed on the case of -graphs, where a small number of exact and asymptotic results are now known — see [3, 4, 12, 20, 24], as well as the surveys by Füredi , Sidorenko , and Keevash .
It is well-known that the Turán problem for an -graph is essentially equivalent to identifying the minimum vertex-degree required to guarantee the existence of a copy of . More recently [11, 44, 51], there has been interest in variants where one considers what minimum -degree condition is required to guarantee the existence of a copy of . Given an -set with , its neighbourhood in is the collection
of -sets whose union with makes an edge of . The neighbourhood of defines an -graph
which is called the link graph of . The degree of in is the size of its neighbourhood. The minimum -degree of is the minimum of over all -subsets . In particular, the case has received particular attention; is known as the minimum codegree of , and a minimum codegree condition is the strongest single degree condition one can impose on an -graph. Determining what minimum codegree forces the existence of a copy of a fixed -graph is known as the codegree density problem . A few results on the codegree density for various small -graphs are known, see [18, 19, 36, 49].
In a different direction, there has been significant recent research activity devoted to generalising another foundational result in extremal graph theory. Let be a graph whose order divides . What minimum degree condition is required to guarantee that a graph on vertices contains an -tiling — a collection of vertex-disjoint copies of ? In the case of complete graphs, this was answered by the celebrated Hajnal–Szemerédi theorem , which (under the guise of equitable colourings) has applications to scheduling problems. For a general graph , the Kühn–Osthus theorem  determines the minimum degree-threshold for -tilings up to a constant additive error.
There has been a growing interest in determining analogous tiling thresholds in -graphs for , see the surveys by Rödl and Ruciński , and Zhao  devoted to the subject. In an effort to generalise Dirac’s theorem on Hamilton cycles to hypergraphs, Rödl, Ruciński and Szemerédi  determined the minimum codegree threshold for the existence of a perfect matching in -graphs for . The paper also introduced the hugely influential absorption method, which has been used as a key ingredient in many of the results in the area obtained since. Beyond perfect matchings, codegree tiling thresholds have by now been determined for a number of small -graphs, including [35, 45], [30, 43], and ( with two edges removed) [10, 39]. In addition, the codegree tiling thresholds for -partite -graphs have been studied recently [9, 25, 26, 29, 52]
Turning to minimum vertex-degree tiling thresholds, fewer results are known. The vertex-degree thresholds for perfect matchings were determined for -graphs by Han, Person, and Schacht  (asymptotically) and by Kühn, Osthus and Treglown  and Khan  (exactly). Han and Zhao  determined the vertex-degree tiling threshold for , while Han, Zang, andZhao  asymptotically determined the vertex-degree tiling threshold for all complete -partite -graphs.
As a key part of their argument, Han, Zang, and Zhao considered a certain -graph covering problem and showed it was distinct from the corresponding Turán-type existence problem. This stands in contrast with the situation for ordinary graphs, where existence and covering thresholds essentially coincide. Given an -graph , Falgas–Ravry and Zhao  introduced the notion of an -covering, which is intermediate between that of the existence of a single copy of and the existence of an -tiling.
We say that an -graph has an -covering if every vertex in is contained in a copy of in . Equivalently an -covering of is a collection of copies whose union covers all of . For every positive integer , the -degree -covering threshold is the function
We further let the -degree -covering density to be the limit†††This limit can be shown to exist — see [21, Footnote 1].
Let denote the complete -graph on vertices and denote the -graph obtained by removing one edge from . A tight -uniform -cycle is an -graph with a cyclic ordering of its vertices such that every consecutive vertices under this ordering form an edge. Falgas–Ravry and Zhao  determined , where is , , , and . Han, Lo, and Sanhueza-Matamala  determined for all and .
In this paper we investigate and for various 3-graphs . We first consider . Let be the function
Observe that for fixed , is a decreasing function of over the interval . On the other hand is an increasing function of , so there exists a unique such that , namely
For all odd integer , . In particular, .
The upper and lower bounds on in Theorem 1.1 are apart by less than . However it seems much more work will be needed to determine exactly. As a first step in this direction, we prove the following stability theorem characterising near-extremal configurations. Let .
For every , there exists and such that the following holds: for every , if is a -graph on vertices with minimum vertex degree at least and is not covered by a copy of in , then the link graph can be made bipartite by removing at most edges.
Next we consider .
The upper bound was derived from the flag algebra method. We believe that the lower bound is tight. As we show in Section 2.3, the problem of determining is equivalent to (a special case of) a problem about triangle-degrees in graphs.
Given a graph , the triangle-degree of a vertex is the number of triangles that contains . The well-studied Rademacher–Turán problem concerns the smallest average triangle-degree among all graphs with a given edge density (the edge density is defined as ). This problem attracted significant attention (see [5, 14, 22, 46, 47]) until it was resolved asymptotically by Razborov  using the framework of his newly-developed theory of flag algebras. Different proofs expressed in the language of weighted graphs were later found by Nikiforov  and by Reiher  (who generalised Razborov’s result to cliques of order and of arbitrary order , respectively).
Let denote the maximum triangle-degree in . (This is related to but different from the well-studied book number, which is the maximum number of triangles containing a fixed edge of , see the discussion in Section 4 for details.) For , we define
which is the asymptotically smallest maximum scaled triangle-degree in a graph with edge density . We derive the following upper bounds for and conjecture that they are tight. If Conjecture 1.5 holds, then (see Proposition 3.1).
Suppose , for some . Then
As we will see, the constructions underpinning Theorem 1.4 are very different from the extremal ones for the Rademacher–Turán problem.
The upper bounds on given in Theorem 1.4 are tight for every .
gave a tripartite analogue of Razborov’s triangle-density result. In a similar spirit, we prove Conjecture 1.5 holds for tripartite graphs. Note that a tripartite graph on vertices can have between and edges.
Let be a tripartite graph on vertices. Then
Structure of the paper
In Section 2 we prove Theorems 1.1–1.3 along with bounds for and for . In Section 3 we prove Theorems 1.4 and 1.6, and give flag algebra bounds on . We end the paper in Section 4 with a discussion of book numbers in graphs and a comparison of known results and conjectures on minimal triangle density, triangle-degree and book-number as functions of edge density.
We use standard graph and hypergraph theory notation throughout the paper. In addition, we use to denote the set and to denote the collection of all -subsets of a set . Where there is no risk of confusion, we identify hypergraphs with their edge-sets.
2 Covering in -graphs
2.1 Proof of Theorem 1.1
Recall that is the (unique up to isomorphism) -graph on -vertices spanning edges, also known as the generalised triangle. In this subsection, we prove Theorem 1.1.
Proof of Theorem 1.1.
Lower bound: let be odd, and let . We construct a -graph on vertices as follows. Set aside a vertex , and let be a bipartition of into two sets of equal size. Let be an arbitrary -regular bipartite graph with partition . Now let be the -graph whose -edges are the union of the triples together with all the triples of vertices from inducing at most one edge in .
Clearly, for every triple of vertices , induces at most two edges of and is not contained in any copy of . Thus . This latter quantity is easily calculated: the degree of in is . For any , there are exactly pairs such that both and lie in , and exactly pairs such that both and lie in ; such pairs are the only pairs from that do not form an edge of with . In addition, there are exactly edges of containing the pair . Thus the degree of in is
By symmetry, the degree of any vertex in is also . Thus because . Since has no -covering, it follows that .
Upper bound: suppose is a -graph on vertices with and no copy of covering a vertex (here is not necessarily odd). We shall show that . Note that the link graph of is triangle-free. Furthermore, for any triple spanning two edges in . Let denote the collection of pairs such that induces two edges in . We know that for every . Observe that consists of all pairs , where either or and exactly one of , is in .
Counting non-edges of over all , we thus have
where in the last line we used Jensen’s inequality and our minimum degree assumption . By averaging, there exists a vertex with
Applying our minimum degree assumption yields and hence . Thus as claimed.
2.2 Proof of Theorem 1.2
Our proof shall make use of a consequence of Karamata’s inequality. Let and be real numbers. We say that majorises if for all , with equality attained in the case . Karamata’s inequality states that if majorises and is a convex function then .
Suppose is a convex function. Let be real numbers such that , and let . Set . Then
Since , our assumption on tells us that . If , then the claimed inequality is just Jensen’s inequality. So assume is nonempty and set for some .
Let be given by
Observe that . Setting
It follows readily from this that the -tuple majorises . Applying Karamata’s inequality to the convex function we obtain
Another ingredient in the proof of Theorem 1.2 is a classical result of Andrásfai, Erdős and Sós.
Theorem 2.2 (Andrásfai, Erdős, Sós ).
Let be a triangle-free graph on vertices with minimum degree . Then is bipartite.
With these two preparatory results in hand, the proof of Theorem 1.2 is straightforward: we first use Lemma 2.1 to show that the overwhelming majority of vertices in the link graph have degree much larger than , whereupon we deduce from the Andrásfai–Erdős–Sós theorem that is almost bipartite.
Proof of Theorem 1.2.
Recall . Fix . Without loss of generality, assume that . Pick and such that
Let be a -graph with , . Suppose is a vertex in not covered by any copy of . Without loss of generality, assume . By the vertex-degree assumption, , for some . Let be the collection of vertices in whose codegree with is smaller than average by a multiplicative factor of . Set .
Since is not covered by a copy of in , the following hold:
for every triple of vertices inducing two edges in , the -edge is missing from .
Property (i) implies that for every , the neighbourhood is an independent set in , while property (ii) implies that for every and every , the -edges and are both missing from . In particular for every , we have
Summing this inequality over all and using the fact , we get
Inserting this inequality back into (2.3), dividing through by and using yields
By the second part of (2.2) and the assumption that is sufficiently large, we have
and . Remove from all vertices from . By the definitions of , the resulting triangle-free graph has at most vertices and minimum degree at least
By Theorem 2.2, is bipartite. Since we removed only at most vertices from to obtain , it follows that can be made bipartite by removing at most edges, as claimed. This concludes the proof of Theorem 1.2.
2.3 Proof of Theorem 1.3
Given an -graph , write for the number of copies of in that cover .
There exists an -graph on vertices with minimum vertex-degree and no -covering if and only if there exists an -graph on vertices with at least edges such that for every vertex , .
In one direction, let be an -graph on vertices with minimum degree . Suppose is not covered by any in . By the minimum degree condition on , the -uniform link graph contains at least edges. Also, every copy of in the -graph must be a non-edge in the -graph , else together with it would make a copy of in covering . The minimum degree condition in then implies that for every vertex in the -vertex -graph ,
implying as desired.
In the other direction, let be an -graph on vertices with at least edges such that for all . We add a new vertex to and define an -graph on by setting the link graph of be equal to , and adding in as edges all -sets from which do not induce a copy of in . This yields an -graph on vertices in which is not covered by a copy of , , and for every ,
so as desired. ∎
For any , the -degree covering density is the least such that if is an -graph on vertices with at least edges, then there is a vertex contained in copies of in .
Proof of Theorem 1.3.
Lower bound: suppose and partition into three sets of size . Further partition each into two sets and of size . Now let be the -graph on obtained by putting in all edges of the form , with and adding for each an arbitrary -regular bipartite graph with partition . An easy calculation shows is both regular and triangle-degree regular, with every vertex satisfying and . We have thus . It follows from Proposition 2.3 that there exists a -graph on vertices with minimum degree and no -covering, establishing the desired lower bound on .
Lower bound: we construct a -graph on as follows. Set aside , and partition the remaining vertices into an -set and a -set . Let be the -graph on obtained by setting the link graph of to be the union of a clique on and a clique on , and adding all triples of the form or . Every path of length in the link graph of gives rise to an independent set in , hence there is no copy of the strong -cycle covering in . The degree of in is , and the degree in the rest of the graph are all at least
Thus , as desired.
Upper bound: Mubayi and Rödl [50, Theorem 1.9] proved that . An easy modification of their proof shows that is in fact an upper bound for the covering threshold. Indeed, let be a graph on vertices with , for some . Let be an arbitrary vertex in . By averaging, there exists such that . Form the multigraph as in [50, Proof of Theorem 1.9, p 151]. Then [50, Claim, p 151] shows that if there is no copy of covering the pair , then satisfies the conditions of [50, Lemma 6.2, p 149], and one can conclude as Mubayi and Rödl do that one of and has degree at most in , contradicting our minimum degree assumption.
For all , .
Let and be sufficiently large. Suppose that is a -graph on vertices with for some satisfying . Let be an arbitrary vertex. By averaging, there exists a vertex and an -set such that . Observe that
and an analogous bound holds for . Thus
On the other hand, for any , we have
Let be the -graph obtained by taking and adding a new vertex whose link graph consists precisely of those pairs