Total domination in K 5- and K 6-covered graphs

A graph G is K r-covered if each vertex of G is contained in a K r-clique. Let γ t (G) denote the total domination number of G. It has been conjectured that every K r-covered graph of order n with no K r-component satisfies γ t (G) ≤ 2n/(r+1). We prove that this conjecture is true for r=5 and 6.