We propose the algebraic connectivity of an undirected 2m-uniform hypergraph under Einstein product. generalize to a directed and reveal relationship between vertex connectivity. present some results on eigenvalue multiplicity M-splitting tensors by method

Fix an integer k ≥ 3. A k-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant c depending only on k such that every simple k-uniform hypergraph H with maximum degree ∆ has chromatic number satisfying χ(H) < c ( ∆ log ∆ ) 1 k−1 . This implies a classical result of Ajtai-Komlós-Pintz-Spencer-Szemerédi and its strengthening due to Duke-Lefman...

A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every k, and s > s0(k), every k-uniform s-almost intersecting hypergraph has at most (s + 1) ( 2k−2 k−1 ) edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related result...

Network data has attracted tremendous attention in recent years, and most conventional networks focus on pairwise interactions between two vertices. However, real-life network may display more complex structures, multi-way among vertices arise naturally. In this article, we propose a novel method for detecting community structure general hypergraph networks, uniform or non-uniform. The proposed...

A perfect matching in a 3-uniform hypergraph on n = 3k vertices is a subset of n3 disjoint edges. We prove that if H is a 3-uniform hypergraph on n = 3k vertices such that every vertex belongs to at least ( n−1 2 ) − ( 2n/3 2 ) + 1 edges then H contains a perfect matching. We give a construction to show that this result is best possible.

We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper we ask about the smallest size of a k-uniform hamiltonian chain saturated hypergraph. We present a construction of a family of k-uniform hamiltonian chain saturated hypergraphs with O(nk−1/2) edges.

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. in [7], and continued in [8] and [16]. Here we obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergr...

A perfect matching in a 4-uniform hypergraph is a subset of b4 c disjoint edges. We prove that if H is a sufficiently large 4-uniform hypergraph on n = 4k vertices such that every vertex belongs to more than ( n−1 3 ) − ( 3n/4 3 ) edges then H contains a perfect matching. This bound is tight and settles a conjecture of Hán, Person and Schacht.

Let F0 be a fixed k-uniform hypergraph. The problem of finding the integer F0-packing number νF0 (H) of a k-uniform hypergraph H is an NP-hard problem. Finding the fractional F0-packing number ν∗ F0 (H) however can be done in polynomial time. In this paper we give a lower bound for the integer F0-packing number νF0 (H) in terms of ν∗ F0 (H) and show that νF0 (H) ≥ ν∗ F0 (H)− o(|V (H)| k).

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraphwith 2 Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. In this note, we show that we can reduce the arity of their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hyperg...