We generalize the Harary-Sachs theorem to k-uniform hypergraphs: codegree-d coefficient of characteristic polynomial a uniform hypergraph H can be expressed as weighted sum subgraph counts over certain multi-hypergraphs with d edges. include detailed description aforementioned and formula for their corresponding weights.

A h-uniform hypergraph H = (V,E) is called (l, k)-orientable if there exists an assignment of each hyperedge e ∈ E to exactly l of its vertices v ∈ e such that no vertex is assigned more than k hyperedges. Let Hn,m,h be a hypergraph, drawn uniformly at random from the set of all h-uniform hypergraphs with n vertices and m edges. In this paper, we determine the threshold of the existence of a (l...

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...

We introduce the notion of a of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both and are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by X(H). An algorithm for c...

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a canonical regularization, the phases of the components of t...

In this paper, we show that the eigenvectors associated with the zero eigenvalues of the Laplacian and signless Lapacian tensors of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector associated with the zero eigenvalue of the Laplacian or signless Lapacian tensor have the same modulus. Moreover, under a cano...

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n, r). Erdős and Lovász conjectured thatm(n, 2) = θ (n2n). The best known lower boundm(n, 2) = Ω (√ n/ logn2n ) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on analysis of random greedy coloring algorithm investigated by Plu...

The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m(n, r) denote the minimum possible number of edges in an n-uniform non-r-colorable hypergraph. We show that for r > n, c1 n lnn m(n, r) rn C1 n lnn, where c1, C1 > 0 are some absolute constants.

As is well known, Lovfisz Local Lemma implies that every d-uniform d-regular hyper-graph is 2-colorable, provided d > 9. We present a different proof of a slightly stronger result; every d-uniform d-regular hypergraph is 2-colorable, provided d > 8.

A hypergraph H is simply a collection of subsets of a finite set, which we will always denote by V. Elements of V are called vertices and elements of H edges. A hypergraph is k-uniform (k-bounded) if each of its edges has size k (at most k). The degree in H of a vertex x is the number of edges containing x, and is denoted by d(x). Similarly, d(x,y) denotes the number of edges containing both of...