The author of the present paper investigates systems of Boolean equations. As we know the description of the solution of the Boolean equation \(f(x_1, \dots, x_n) = 1\) \(( = 0) \) is equal to the description of the subgraph \(A_f = \{\widetilde \alpha \in \{0,1\}^n : f(x_1, \dots, x_n) = 1\}\) of the Boolean cube \(\{0,1\}^n\) (or the subgraph \(B_f\), \(B_f = \{0,1\}^n \backslash A_f)\). The paper consists of three sections. In §1 the author deals with the problem of the characterization of all so-called Boolean graphs (graphs embedded into the Boolean cube). Thus this section is a review of the results from the paper of the autor and D. S. Shevelev [Diskretn. Mat. 3, No. 4, 52-61 (1991; Zbl 0751.05084) and the papers of K. V. S. Bhat [Inform. Process. Lett. 1, No. 1, 81-92 (1980)], M. Mulder [Discrete Math. 28, 179-188 (1979; Zbl 0418.05034)], S. Foldes [Discrete Math. 17, 155-159 (1977; Zbl 0354.05045)].
In §2 the author studies covers of Boolean graphs by Boolean graphs from the following three sets: the set of all Boolean graphs of threshold functions; the set of all Boolean graphs of threshold functions of \(\leq n\) variables; the set of all Boolean graphs of elementary conjunctions; the set of all Boolean graphs of elementary conjunctions of \(\leq n\) variables (or the set of all Boolean sub cubes of the Boolean \(n\)-cube in this case). The cover of some Boolean graph by Boolean graphs from one of the previous families is equal to the representation of the Boolean functions by the disjunction of functions from these families. Thus every Boolean function \(f(x_1, \dots, x_n)\) and each previous set \(\Gamma\) of Boolean graphs define the pair \((k_1, k_2)\) of integers, where \(k_1\) is the minimal number of Boolean graphs from \(\Gamma\) in all possible covers of \(A_f\) and \(k_2\) is the similar number for \(B_f\). The author proves in §3 some interesting theorems about properties of these pairs for all three families of Boolean graphs and for the set of all connected Boolean graphs.