In this paper, some inequality relations between the Laplacian/signless Laplacian H-eigenvalues and the clique/coclique numbers of uniform hypergraphs are presented. For a connected uniform hypergraph, some tight lower bounds on the largest Laplacian H+-eigenvalue and signless Laplacian H-eigenvalue related to the clique/coclique numbers are given. And some upper and lower bounds on the clique/...

The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(...

Extending the notion of (random) k-out graphs, we consider when the k-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each r there is a k = k(r) such that the k-out r-uniform hypergraph on n vertices has a perfect fractional matching with high probability (i.e., with probability tending to 1 as n → ∞) and prove an analogous result for r-uniform r-...

For an r-uniform hypergraph H, let f(H) be the minimum number of complete r-partite r-uniform subhypergraphs of H whose edge sets partition the edge set of H. In this talk, I will discuss the value of f(H) for the random hypergraph H. Namely, I will prove that if (log n)/n ≤ p ≤ 1/2 and H ∈ H(n, p), then with high probability f(H) = (1− π(K(r−1) r ) + o(1)) ( n r−1 ) , where π(K(r−1) r ) is the...

For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K` must contain a complete `-partite graph with Ω(logn) vertices in each class. We give another proof of N...

In this paper we find, for n ≤ 16, the maximum number of edges in a 4-uniform hypergraph which does not have the complete 4-uniform hypergraph on five vertices,K4 5 , as a subgraph. Equivalently, we find all optimal (n, n−4, n−5) covering designs for n ≤ 16. Using these results we find a new upper bound for the Turán density ofK4 5 . π(K4 5 ) ≤ 1753 2380 = 0.73655 . . . . Finally wemake some no...

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-Parking functions. We observe a simp...

For 1 6 ` < k, an `-overlapping k-cycle is a k-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of k consecutive vertices and every two consecutive edges share exactly ` vertices. A k-uniform hypergraph H is `-Hamiltonian saturated if H does not contain an `-overlapping Hamiltonian k-cycle but every hypergraph obtained from H by adding one edge does contain such...

We use an algebraic method to prove a degree version of the celebrated Erdős-Ko-Rado theorem: given n > 2k, every intersecting k-uniform hypergraph H on n vertices contains a vertex that lies on at most ( n−2 k−2 ) edges. This result implies the Erdős-Ko-Rado Theorem as a corollary. It can also be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph m...

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H-eigenva...