Complete 3-uniform Hypergraph Research Articles

Tight 9-Cycle Decompositions of λ-Fold Complete 3-Uniform Hypergraphs

For 2≤t≤m, let Zm denote the group of integers modulo m, and let TCm(t) denote the t-uniform hypergraph with vertex set Zm and hyperedge set {{i,i+1,i+2,…,i+t−1}:i∈Zm}. Any hypergraph isomorphic to TCm(t) is a t-uniform tight m-cycle. In this paper, we consider the existence of tight 9-cycle decompositions of λ-fold complete 3-uniform hypergraphs. According to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, we obtain the necessary and sufficient conditions for the existence of TC9(3)-decomposition of λKn(3). We show there exists a TC9(3)-decomposition of λKn(3) if and only if λn(n−1)(n−2)≡0(mod54), λ(n−1)(n−2)≡0(mod6) and n≥9.

On locally rainbow colourings

Given a graph H, let g(n,H) denote the smallest k for which the following holds. We can assign a k-colouring fv of the edge set of Kn to each vertex v in Kn with the property that for any copy T of H in Kn, there is some u∈V(T) such that every edge in T has a different colour in fu.The study of this function was initiated by Alon and Ben-Eliezer. They characterized the family of graphs H for which g(n,H) is bounded and asked whether it is true that for every other graph g(n,H) is polynomial. We show that this is not the case and characterize the family of connected graphs H for which g(n,H) grows polynomially. Answering another question of theirs, we also prove that for every ε>0, there is some r=r(ε) such that g(n,Kr)≥n1−ε for all sufficiently large n.Finally, we show that the above problem is connected to the Erdős–Gyárfás function in Ramsey Theory, and prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each fixed r the complete r-uniform hypergraph Kn(r) can be edge-coloured using a subpolynomial number of colours in such a way that at least r colours appear among any r+1 vertices.

Maximum Packing of λ-Fold Complete 3-Uniform Hypergraph with a Special Tetrahedron

Let Kv(3)=(V,E) be the complete 3-uniform hypergraph, where the vertex set is V={x1,x2,⋯,xv}, in which the edge set E is of all triples. Let ST denote the special tetrahedron with four edges, where each edge contains three vertices of degree 2. In this paper, we consider the decomposition and packing of a complete 3-uniform hypergraph of an λ-fold special tetrahedron. Firstly, the necessary conditions for the existence of the λ-fold ST-decomposition are discussed in four distinct cases. Secondly, according to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, it is proven that the above necessary conditions are sufficient. Finally, we prove that a maximum ST-packing of a complete 3-uniform hypergraph Kv(3) exists for all v≥6 and λ≥1.

Spectrum of 3-uniform 6- and 9-cycle systems over [formula omitted

We consider edge decompositions of Kv(3)−I, the complete 3-uniform hypergraph of order v minus a set of v/3 mutually disjoint edges (1-factor). We prove that a decomposition into tight 6-cycles exists if and only if v≡0,3,6 (mod 12) and v≥6; and a decomposition into tight 9-cycles exists for all v≥9 divisible by 3. These results are complementary to the theorems of Akin et al. [Discrete Math. 345 (2022)] and Bunge et al. [Australas. J. Combin. 80 (2021)] who settled the case of Kv(3).

Constrained Ramsey numbers for the loose path, cycle and star

Given a 3-uniform hypergraph H and a positive integer k, the k-colored Ramsey numberR(H;k) is the minimum integer n such that every k-edge-coloring of the complete 3-uniform hypergraph Kn3 contains a monochromatic copy of H. Given two 3-uniform hypergraphs H and G, the constrained Ramsey numberf(H,G) is defined as the minimum integer n, such that any edge-coloring of Kn3 with any number of colors contains either a monochromatic copy of H or a rainbow copy of G. Let P33, C33 and S33 be the 3-uniform loose path, loose cycle and loose star of size 3, respectively. We study the constrained Ramsey number for the cases H,G∈{P33,C33,S33}. For 3-graphs H,G∈{P33,C33,S33}, we reduce f(H,G) to the 2-colored Ramsey number of H, except when H=G=S33, in which case we prove that f(S33,S33)=R(S33;2)+1. We also prove that f(H,P33)=R(H;2) for infinitely many 3-uniform hypergraphs H.

Successive vertex orderings of fully regular graphs

A graph G=(V,E) is called fully regular if for every independent set I⊂V, the number of vertices in V∖I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph.As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph.As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.

A jump to the Narayana number for hereditary properties of ordered 3-uniform hypergraphs

For k,l≥2 we consider ideals of edge l-colored complete k-uniform hypergraphs (n,χ) with vertex sets [n]={1,2,…n} for n∈N. An ideal is a set of such colored hypergraphs that is closed under the induced ordered sub-hypergraph relation. We obtain analogues of two results of Klazar (2010) who considered graphs, namely we prove two jumps for growth functions of such ideals of colored hypergraphs. The first jump is for any k,l≥2 and says that the growth function is either eventually constant or at least n−k+2. The second jump is only for k=3 and l=2 and says that the growth function of an ideal of edge two-colored complete 3-uniform hypergraphs grows either at most polynomially, or for n≥23 at least as Gn where Gn is the sequence defined by G1=G2=1, G3=2 and Gn=Gn−1+Gn−3 for n≥4. The lower bounds in both jumps are tight.

Helly-Type Theorems for the Ordering of the Vertices of a Hypergraph

Let H be a complete r-uniform hypergraph such that two vertices are marked in each edge as its ‘boundary’ vertices. A linear ordering of the vertex set of H is called an agreeing linear order, provided all vertices of each edge of H lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of H with at most 2r − 2 vertices, then there is an agreeing linear order on the vertex set of H. We also show that the constant 2r − 2 cannot be reduced in the theorem. The case r = 3 of the theorem has particular interest in the axiomatic theory of betweenness. Similar results are obtained for further r-uniform hypergraphs (r ≥ 3), where one or two vertices are marked in each edge, and the linear orders need to satisfy various rules of agreement. In one of the cases we prove that no such Helly-type statement holds.

Monochromatic Paths in 2-Edge-Coloured Graphs and Hypergraphs


 
 
 We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete k-partite k-uniform hypergraph. Finally, we show that any 2-edge coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.
 
 

On extremal spectral radius of blow-up uniform hypergraphs

Let G be an r-uniform hypergraph of order t, and let ρ(G) be the spectral radius of A(G), where A(G) is the adjacency tensor of G. A blow-up of G respected to a positive integer vector (n1,n2,…,nt), denoted by G∘(n1,n2,…,nt), is an r-uniform hypergraph obtained from G by replacing each vertex j of G with a class of vertices Vj of size nj≥1 and if {j1,j2,…,jr}∈E(G), then {vi1,vi2,…,vir}∈E(H) for every vi1∈Vj1,vi2∈Vj2,…,vir∈Vjr. Let Bn(G) be the set of all the blow-ups of G such that each ni≥1 and ∑i=1tni=n. Let Ktr be the complete r-uniform hypergraph of order t, and let SH(m,q,r) be the r-uniform sunflower hypergraph with m petals and a kernel of size r−q on t vertices. For any H∈Bn(Ktr), we prove thatρ(Ktr∘(n−t+1,1,1,…,1))≤ρ(H)≤ρ(Ttr(n)), with the left equality holds if and only if H≅Ktr∘(n−t+1,1,1,…,1), and the right equality holds if and only if H≅Ttr(n), where Ttr(n) is the complete t-partite r-uniform hypergraph of order n, with parts of size ⌊n/t⌋ or ⌈n/t⌉. For any H∈Bn(SH(m,q,r)), we determine the exact value of the spectral radius of H and characterize the hypergraphs with maximum spectral radius and minimum spectral radius in Bn(SH(m,q,r)), respectively.

Stability theorems for some Kruskal–Katona type results

The classical Kruskal–Katona theorem gives a tight upper bound for the size of an r-uniform hypergraph H as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of H is close to the maximum with respect to the size of its shadow, then H is structurally close to a complete r-uniform hypergraph. We prove similar stability results for two classes of hypergraphs whose extremal properties have been investigated by many researchers: the cancellative hypergraphs and hypergraphs without expansion of cliques.

Betweenness structures of small linear co-size

One way to study the combinatorics of finite metric spaces is to study the betweenness relation associated with the metric space. In the hypergraph metrization problem, one has to find and characterize metric betweennesses with collinear triples (or alternatively, non-degenerate triangles) that coincide with the edges of a given 3-uniform hypergraph. Metrizability of different kinds of hypergraphs was investigated in the last decades. Chen showed that Steiner triple systems are not metric, while Richmond and Richmond characterized linear betweennesses, i.e. metric betweennesses that realize the complete 3-uniform hypergraph. The latter result was also generalized to pseudometric (almost-metric) betweennesses by Beaudou et al. In this paper, we further extend this theory by characterizing the largest non-linear almost-metric betweennesses that satisfy certain hereditary properties, as well as the ones that contain a small linear number of non-degenerate triangles.

Some result on integrality of several matrix representation of complete r-uniform hypergraph

The eigenvalues of matrix representation hypergraph have the possibility that all of it can be an integer or otherwise. The Laplacian integral hypergraphs are those hypergraphs whose Laplacian spectrum consists entirely of integers likewise the definition of signless Laplacian integral and Seidel integral. This research focuses on the properties of the entry matrix and formulates the spectrum of the Laplacian matrix, Signless Laplacian matrix, and Seidel Matrix of complete r-uniform hypergraphs. Hence, it can determine the integrality of the representation matrix respectively. The result concludes that complete r-uniform hypergraphs are Laplacian integral, signless Laplacian integral, and Seidel integral.

On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set {a,b,c,d,e,f} and edge set {{a,b,c},{b,c,d},{c,d,e},{d,e,f},{e,f,a},{f,a,b}}. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if v≡1, 2, 10, 20, 28, or 29(mod36).

Anti-Ramsey Hypergraph Numbers

The anti-Ramsey number a r n ( H ) of an r-uniform hypergraph is the maximum number of colors that can be used to color the hyperedges of a complete r-uniform hypergraph on n vertices without producing a rainbow copy of H. In this paper, we determine anti-Ramsey numbers for paths of length 2, certain stars and complete hypergraphs, and the complete 3-uniform hypergraph of order 4 with a single hyperedge removed.

On maximum packings of λ-fold complete 3-uniform hypergraphs with triple-hyperstars of size 4

A symmetric triple-hyperstar is a connected, 3-uniform hypergraph where, for some edge { a , b , c } , vertices a, b, and c all have degree k > 1 and all other edges contain exactly 2 vertices of degree 1. Let H denote the symmetric triple-hyperstar with 4 edges and, for positive integers λ and v , let λ K v (3) denote the λ -fold complete 3-uniform hypergraph on v vertices. We find maximum packings of λ K v (3) with copies of H .

Anti-Ramsey number of matchings in a hypergraph

Given an edge-coloring of a hypergraph G, G is said to be rainbow if any two edges of G receive different colors. The anti-Ramsey numberAR(G,H) of a hypergraph H in a hypergraph G is defined to be the maximum integer k such that there exists a k-edge-coloring of G avoiding rainbow copies of H. The anti-Ramsey numbers of matchings have been extensively studied in several graph classes. But there is few results in hypermatchings and up to now, we only know the anti-Ramsey number of matchings, paths and cycles in complete uniform hypergraphs. In this paper, we determine the exact value of the anti-Ramsey number of a k-matching in a complete tripartite 3-uniform hypergraph. Interestingly, our results are parallel to the results in complete bipartite graphs.

The characteristic polynomial of the complete 3-uniform hypergraph

In this paper, we give the characteristic polynomial of the complete 3-uniform hypergraph. We show that the characteristic polynomial is the product of some cubic polynomials, so we can easily compute the spectra of complete 3-uniform hypergraphs. This computation includes a complete description of the eigenvalues' multiplicities. We also show the algebraic multiplicities of 0, 1 and (n−12) of a complete 3-uniform hypergraph with n vertices.

Exponential Lower Bound for Berge-Ramsey Problems

We give an exponential lower bound for the smallest N such that no matter how we c-color the edges of a complete r-uniform hypergraph on N vertices, we can always find a monochromatic Berge-K_n.

Multicovering hypergraphs

Let fr(n,p) represent the minimum number of complete r-partite r-graphs required to cover every edge of the complete r-uniform hypergraph on n vertices at least once and at most p times.Graham–Pollak theorem states that f2(n,1)=n−1. Upper and lower bounds were known for r=2 and a general p. In this note we give bounds for fr(n,p) for general r and p.