A Recursive Construction For Regular Difference Triangle Sets

Abstract

A difference triangle set (D$\Delta$S) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct D$\Delta$Ss, we present a new type of combinatorial design, monotonic directed $(v,k,\lambda)$-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (D$\Delta$Ss). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) D$\Delta$Ss if the existence of a single regular D$\Delta$S is known.