Hypercycles in a random hypergraph

Abstract

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)

Systems of Random Equations over Finite Algebraic Structures

“…The studies initiated in [15] were continued in [49][50][51][52]. Although the latter analyze SREIS systems over GF(q) whose matrices A Nn of coefficients in each row have no more than r non-zero elements, the theorems proved here (similar to Theorem 1) yield the following result pertaining also to CCSRE systems over GF(q): relation (8) is true in the field GF(q).…”

“…The authors of [15] analyze the limit distribution of the rank of a random matrix A a Nn ij = || || over GF(2), with the rows being independent random vectors. Each row has r sequential unities, and each unity, irrespective of the others, takes any of n places with the probability 1/ n, and a ij = 1 0 ( ) if an odd (even) number of unities occupy the place j.…”

“…The main theoretical conclusion of [33] is: the limit distribution of the number n n of the solutions of system (14) Moreover, Kovalenko analyzes in [33] the geometrical structure of the set of linearly independent solutions and derives the limiting values of the mathematical expectation and variance of the number of nontrivial solutions of (14) (for deriving formulas (15) for the equiprobable matrix A Nn and for calculating lim n n ® ¥ M n when conditions (13) hold see also [54]). …”

“…Formulas (15) with the number 2 replaced by q specifies the form of the limit distribution n n for such a system.…”

“…The idea from [32,33] that Kovalenko used in [36] to prove the invariance, naturally delimit the coefficients of system (14), where the limiting distribution of the number of solutions as n ® ¥ has the form (15). This domain of invariance is given in the following terminology: d d , in the notation from [36].…”