On the maximum number of irreducible coverings of an n-vertex graph by n−3 cliques

Abstract

In this paper the structure of the irreducible coverings by n−3 cliques of the vertices of a graph of order n is described. As a consequence, the number of such coverings for complete multipartite graphs is deduced. Also, it is proved that for sufficiently large n the maximum number of irreducible coverings by n − 3 cliques of an n-vertex graph equals 3n−3−3·2n−3+3 and the extremal graph coincides (up to isomorphism) to K3,n−3. This asymptotically solves a problem raised in a previous paper by the author (J. Combinatorial Theory B28, 2(1980), 127-141). The second extremal graph is shown to be isomorphic to K3,n−3− e.