We introduce the notion of a hypercycle in a hypergraph which is defined by a Τ χ Ν (0, l)-matrix A. The maximum number of independent hypercycles s(A) is connected with the rank r (A) of the matrix A by the equality r(A) + s(A) -T. We prove that for a regular random hypergraph with N vertices and T edges, whose each edge contains not more than r vertices, the total number of hypercycles S(A) = 2*( Λ ) -1 has the following threshold property as Ν, Τ -» oo, Ν/Τ -* α: there exists a constant a r such that MS(A) -> 0 for α < a r , and MS(A) -» oo for a > a r .
I. MAIN RESULTSWe consider a hypergraph G A which is defined by a Τ χ Ν matrix A = \\a tj \\ in GF (2). The set of vertices of the hypergraph G A is the set {1,..., TV} of the numbers of columns of the matrix, and the set of enumerated hyperedges is the set {61,..., &r}, where bt = {j: a tj = 1}, t = Ι,.,.,Τ. Thus, there exists a correspondence between a row a t = (a t i, -· -, CUN) and the hyperedge b t , t = 1,..., T. Note that the empty edge corresponds to a row consisting of zeros. The multiplicity of a vertex j in the set of hyperedges B = {6^,...,b tm } is the number of hyperedges in B which contain this vertex. A set of hyperedges B = {b tl ,..., b tm } is a hypercycle if every vertex of the hypergraph G A has an even multiplicity in B, in other words, if the sum by coordinates of rows a tl + ... + a tm in GF(2) is equal to the zero vector.If every row of the matrix A consists of exactly two units, then the hypergraph G A is an ordinary graph (parallel edges may be encountered), and a hypercycle is an ordinary cycle or a union of cycles.The set of numbers of hyperedges which form a hypercycle is called in [1] the critical set of rows of the matrix A.The union of two sets of hyperedges BI and B 2 is the set BI Δ Β 2 containing those, and only those, hyperedges which are contained in one of the sets BI and B 2 and are not contained in the both sets.Let ει,...,ε s take values 0 and 1. Hypercycles £?i,..., £ s are independent, if ειΒιΔε 2 Β 2 Δ...Δε 3 Β 3 = 0 iff ει = ... = ε β = 0. We denote by s(A) the maximum number of independent hypercycles in the hypergraph GA, and by r(A), the rank of the matrix A. In Section 2 it will be proved that r(A) + s(A) = T.(1)
