Quadratic excesses or paddings of covers with triples of λKn

A cover (with triples) of λKn is an ordered pair (V,B) where V is an n-element set and B is a set of 3-element subsets of V called blocks such that each 2-element subset of V appears in at least λ blocks. Define E(B) = {{x, y}, {x, z}, {y, z} | {x, y, z} ∈ B}. The excess (sometimes called the padding) of a cover is defined to be E(B)−E(λKn), although it is often convenient to think of the excess as the graph on n vertices with edge set induced by E(B)−E(λKn). A quadratic graph is a graph in which every vertex has degree 2 or 0. In 1987, Colbourn and Rosa characterized which quadratic graphs are the excess of some cover of Kn. In this paper, these results are extended by characterizing which quadratic graphs are the excess of some cover of λKn for λ ≥ 2.