The cyclicity of a hypergraph is an e ciently computable integer that extends the notion of the cyclomatic number of a graph The formula for the cyclicity is suggested by the join invariant of an acyclic hypergraph which is the multiset of all joining sets in any of its join trees Once we gure out how the multiplicity of a joining set depends on the structure of the acyclic hypergraph we de ne ...

Many applications of Szemerédi’s Regularity Lemma for graphs are based on the following counting result. If G is an s-partite graph with partition V (G) = ⋃s i=1 Vi, |Vi| = m for all i ∈ [s], and all pairs (Vi, Vj ), 1 i < j s, are -regular of density d, then G contains (1± f( ))d s 2 ms cliques Ks, provided < (d), where f( ) tends to 0 as tends to 0. Guided by the regularity lemma for 3-unifor...

A hypergraph is d-degenerate if every subhypergraph has a vertex of degree at most d. A greedy algorithm colours every such hypergraph with at most d+ 1 colours. We show that this bound is tight, by constructing an r-uniform d-degenerate hypergraph with chromatic number d + 1 for all r ≥ 2 and d ≥ 1. Moreover, the hypergraph is triangle-free, where a triangle in an r-uniform hypergraph consists...

We present a matrix multiplication based algorithm for counting the number of (induced) occurrences of a fixed r-uniform hypergraph in a larger hypergraph. In many cases, the running time is better than that of the näıve algorithm. We also present several useful applications of the algorithm, such as determining the dominant color among monochromatic simplices in a redblue edge-colored hypergra...

For every hypergraph on n vertices there is an associated subspace arrangement in R n called a hypergraph arrangement. We prove shellability for the intersection lattices of a large class of hypergraph arrangements. This class incorporates all the hypergraph arrangements which were previously shown to have shellable intersection lattices.

Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and Frankl-Rödl related to strong hypergraph regularity, the spectral approach of Friedman-Wigderson, and more recent results of...

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −1. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are ...