On tight blocking set in minimum coverings

Let ( X , B ) be a ( λ K n , G ) -covering with excess E and a blocking set T. Let Γ 1 , Γ 2 , … , Γ s be all connected components of E with at least two vertices (note that s = 0 if E = ∅ ). The blocking set T is called tight if further V ( Γ i ) ∩ T ≠ ∅ and V ( Γ i ) ∩ ( X \ T ) ≠ ∅ for 1 ⩽ i ⩽ s . In this paper we give a complete solution for the existence of a minimum ( λ K n , G ) -covering admitting a blocking set (BS), or a tight blocking set (TBS) for any λ and when G = K 3 and G = K 3 + e .