Excess graphs and bicoverings

A bstract The classical bicovering problem seeks to cover all pairs from a v-set by a family F of k-sets so that every pair occurs at least twice and the cardinality of F is minimaL A weight function is introduced for blocks in such a design, and its use in constructing bicoverings is illustrated. A covering is a collection of k-sets (blocks) chosen from elements of a v-set so that each pair from the v elements occurs at least once. The cardinality of a minimal covering is written N 1 (2,k,v), or simply N(2,k,v). A 'A-covering is a covering where each pair appears at least ' A times; if ' A = 2, we have a bicovering. The cardinality of a minimal bicovering is denoted by N A (2,k,v). Henceforth, we shall use the term bicovering to denote a minimal bicovering. 2. The Weight Function For any 'A-covering consisting ofb blocks, take a block B. Let Xi be the number of blocks meeting B in exactly i elements counting the number of other occurrences of elements from B and the other occurrences of element pairs from B gives: