Some properties of irreducible coverings by cliques of complete multipartite graphs

In this paper some recursion formulas and asymptotic properties are derived for the numbers, denoted by N(p, q), of irreducible coverings by edges of the vertices of complete bipartite (labeled) graphs Kp,q. The problem of determining numbers N(p, q) has been raised by I. Tomescu (dans “Logique, Automatique, Informatique,” pp. 269–423, Ed. Acad. R.S.R., Bucharest, 1971). A result concerning the asymptotic behavior of the number of irreducible coverings by cliques of q-partite complete graphs is obtained and it is proved that limn→∞ I(n)1n2 = 3112, limn→∞ (log M(n))1n = 313, and limn→∞ C(n)1n(nln n) = 1e, where I(n) and M(n) are the maximal numbers of irreducible coverings, respectively, coverings by cliques of the vertices of an n-vertex graph, and C(n) is the maximal number of minimal colorings of an n-vertex graph. It is also shown that maximal number of irreducible coverings by n − 2 cliques of the vertices of an n-vertex graph (n ≥ 4) is equal to 2n−2 − 2 and this number of coverings is attained only for K2,n−2 and the value of limn→∞ I(n, n − k)1n is obtained, where I(n, n − k) denotes the maximal number of irreducible coverings of an n-vertex graph by n − k cliques.