Maximum Cliques of Hypergraphs and Polynomial Optimization

A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Buló and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local (global) minimizers of a homogeneous polynomial function of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Turán densities of complete non-uniform hypergraphs.     

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of \({\mathbb N}^{(r)}\) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For \(r=2\), Motzkin and Straus’ theorem confirms this conjecture. For \(r=3\), it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi’s conjecture holds for bigger ranges of m when \(r=3\). We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.     

Judicious bisection of hypergraphs

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously.In this paper,we prove that if G is a hypergraph with n vertices and m_i edges of size i for i=1,2,...,k,then G admits a bisection in which each vertex class spans at most(m_1)/2+1/4m_2+…+(1/(2~k)+m_k+o(m_1+…+m_k) edges,where G is dense enough or △(G)= o(n) but has no isolated vertex,which turns out to be a bisection version of a conjecture proposed by Bollobas and Scott.     

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least \(c \left( {\begin{array}{c}|V(H)|\\ r-1\end{array}}\right) \) has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomassé and then solved by Allen, Böttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turán density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobás–Erd?s graphs defined earlier by the authors.     

Judicious partitions of weighted hypergraphs

Let G be a weighted hypergraph with edges of size i for i = 1, 2. Let wi denote the total weight of edges of size i and α be the maximum weight of an edge of size 1. We study the following partitioning problem of Bollob′as and Scott: Does there exist a bipartition such that each class meets edges of total weight at least (w_1-α)/2+(2w_2)/3? We provide an optimal bound for balanced bipartition of weighted hypergraphs, partially establishing this conjecture. For dense graphs, we also give a result for partitions into more than two classes.In particular, it is shown that any graph G with m edges has a partition V_1,..., V_k such that each vertex set meets at least(1-(1-1/k)~2)m + o(m) edges, which answers a related question of Bollobás and Scott.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C _r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G ^c is (t ? 2 ? i)(t ? 2)t, then λ(G) ≤ λ(C _3,m ). As an implication, the conjecture of Frankl and Füredi is true for \(\left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).     

A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C _m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C _m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r ₁,r ₂,…,r _l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r ₁,r ₂,? ,r _l }-hypergraphs and {r ₁,r ₂,? ,r _l }-hypergraphs concerning the second question.     

The independent neighborhoods process

A triangle T^(r) in an r-uniform hypergraph is a set of r+1 edges such that r of them share a common (r-1)-set of vertices and the last edge contains the remaining vertex from each of the first r edges. Our main result is that the random greedy triangle-free process on n points terminates in an r-uniform hypergraph with independence number O((n log n)^1/r). As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number r(T^(r),K_s^(r)) has order of magnitude s^r/ log s. This answers questions posed in [4, 10] and generalizes the celebrated results of Ajtai–Komlós–Szemerédi [1] and Kim [9] to hypergraphs.     

Connection Between Polynomial Optimization and Maximum Cliques of Non-Uniform Hypergraphs

In Motzkin and Straus (Canad. J. Math 498 17, 533540 1965) provided a connection between the order of a maximum clique in a graph G and the Lagrangian function of G. In Rota Bulò and Pelillo (Optim. Lett. 500 3, 287295 2009) extended the Motzkin-Straus result to r-uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study similar optimization problems and obtain the connection to maximum cliques for {s, r}-hypergraphs and {p, s, r}-hypergraphs, which can be applied to obtain upper bounds on the Turán densities of the complete {s, r}-hypergraphs and {p, s, r}-hypergraphs.     

Learning a hidden uniform hypergraph

Motivated by applications in genome sequencing, Grebinski and Kucherov (Discret Appl Math 88:147–165, 1998) studied the graph learning problem which is to identify a hidden graph drawn from a given class of graphs with vertex set \(\{1,2,\ldots ,n\}\) by edge-detecting queries. Each query tells whether a set of vertices induces any edge of the hidden graph or not. For the class of all hypergraphs whose edges have size at most r, Chodoriwsky and Moura (Theor Comput Sci 592:1–8, 2015) provided an adaptive algorithm that learns the class in \(O(m^r\log n)\) queries if the hidden graph has m edges. In this paper, we provide an adaptive algorithm that learns the class of all r-uniform hypergraphs in \(mr\log n+(6e)^rm^{\frac{r+1}{2}}\) queries if the hidden graph has m edges.     

Generalisations of Hypomorphisms and Reconstruction of Hypergraphs

We define certain generalisations of hypergraph hypomorphisms, which we call k-morphisms, \((k,n-k)\)-hypomorphisms, partial \((k,n-k)\)-hypomorphisms. They are special bijections between collections of k-subsets of vertex sets of hypergraphs. We show that these mappings lead to alternative representations of the automorphism groups of r-uniform hypergraphs and vertex stabilisers of graphs. We also use them to show that almost every r-uniform hypergraph is reconstructible and \((k,n-k)\)-reconstructible. As a consequence we also obtain the result that almost every r-uniform hypergraph is asymmetric.     

Graphs with small total rainbow connection number

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that \({\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}}{n - 1} \\2\end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}}n \\2\end{array}} \right) - 1\), and \({\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}}{n - 2} \\2\end{array}} \right) + 2\). Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n ? s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.     

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.     

On Jumping Densities of Hypergraphs

A number \({\alpha\in [0, 1)}\) is a jump for an integer r ≥ 2 if there exists a constant c > 0 such that for any family \({{\mathcal F}}\) of r-uniform graphs, if the Turán density of \({{\mathcal F}}\) is greater than α, then the Turán density of \({{\mathcal F}}\) is at least α + c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0, 1) is a jump for r = 2. Erd?s also showed that every number in [0, r!/r ^r ) is a jump for r ≥ 3. However, not every number in [0, 1) is a jump for r ≥ 3. In fact, Frankl and Rödl showed the existence of non-jumps for r ≥ 3. By a similar approach, more non-jumps were found for some r ≥ 3 recently. But there are still a lot of unknowns regarding jumps for hypergraphs. In this note, we show that if \({c\cdot{\frac{r!}{r^r}}}\) is a non-jump for r ≥ 3, then for every p ≥ r, \({c\cdot{\frac{p!}{p^p}}}\) is a non-jump for p.     

A Comment on Ryser’s Conjecture for Intersecting Hypergraphs

Let \(\tau({\mathcal{H}})\) be the cover number and \(\nu({\mathcal{H}})\) be the matching number of a hypergraph \({\mathcal{H}}\). Ryser conjectured that every r-partite hypergraph \({\mathcal{H}}\) satisfies the inequality \(\tau({\mathcal{H}}) \leq (r-1) \nu ({\mathcal{H}})\). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with \(\nu({\mathcal{H}}) = 1\), Ryser’s conjecture reduces to \(\tau({\mathcal{H}}) \leq r-1\). Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with \(\tau({\mathcal{H}}) = r-1\), demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely \(\tau({\mathcal{H}}) \ge r-1\)? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r ? 1 must have at least \((3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))\) edges, and conjecture that there exist constructions with Θ(r) edges.     

Turán-type bounds for distance graphs

A lower bound is obtained for the number of edges in a distance graph G in an infinitesimal plane layer ?² × [0, ε] ^d , which relates the number of edges e(G), the number of vertices ν(G), and the independence number α(G). It is proved that \(e\left( G \right) \geqslant \frac{{19\nu \left( G \right) - 50\alpha \left( G \right)}}{3}\). This result generalizes a previous bound for distance graphs in the plane. It substantially improves Turán’s bound in the case where \(\frac{1}{5} \leqslant \frac{{\alpha \left( G \right)}}{{\nu \left( G \right)}} \leqslant \frac{2}{7}\).     

Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ <Emphasis Type="Italic">α</Emphasis> &lt; 0

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randi? index \(R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} \), where d_G(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randi? index for ?1 ≤ α < 0, respectively.     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

Critical Independent Sets and König–Egerváry Graphs

A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König–Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = | V(G) | ?2μ(G) is the deficiency of G. The number \({d(G)=\max\{\left\vert S\right\vert -\left\vert N(S)\right\vert :S\in\mathrm{Ind}(G)\}}\) is the critical difference of G. An independent set A is critical if \({\left\vert A\right\vert -\left\vert N(A)\right\vert =d(G)}\) , where N(S) is the neighborhood of S, and α _c (G) denotes the maximum size of a critical independent set. Larson (Eur J Comb 32:294–300, 2011) demonstrated that G is a König–Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α _c (G) = α(G). In this paper we prove that: (i) \({d(G)=\left \vert \mathrm{core}(G) \right \vert -\left \vert N (\mathrm{core}(G))\right\vert =\alpha(G)-\mu(G)=def \left(G\right)}\) holds for every König–Egerváry graph G; (ii) G is König–Egerváry graph if and only if each maximum independent set of G is critical.