The Constructions Of Covering Arrays On Hypergraphs And Resolvable Directed Group Divisible Designs

In this paper,we study the existence and the constructions of two kinds of com-binatorial configurations including mixed covering arrays on hypergraphs and resolvable directed group divisible designs with block size 3.The first research content of this paper is mixed covering arrays on hypergraphs.With the further study of orthogonal arrays,covering arrays and mixed covering arrays have been proposed.They have been applied in widely applications in many areas,such as software testing,coding and pharmaceutical industry,etc.Recently,mixed covering arrays have been extended to hypergraphs by many scholars in order to enhance the flexibility of applications.The existence problem of optimal mixed covering arrays on hypergraphs is becoming one of the most important topics in the field of combinato-rial designs.In this paper,we focus on the constructions of balanced mixed covering arrays on hypergraphs.We first present four hypergraphs“movements”.Furthermore,we determine the existence of two kinds of optimal balanced mixed covering arrays on hypergraphs.The second research content of this paper is resolvable directed group divisible design(RDGDD).Group divisible design(GDD)plays an extremely important role in combi-natorial design theory.It also has many applications in informatics.Recently,GDD has been extended to directed group divisible design.This paper mainly studies the existence problem of resolvable directed group divisible design with uniform group type and block size 3.We first prove that the necessary conditions for the existence of a 3-Directed f rame of type g~nare su cient.Then,we also investigate directed holey frame(DHF)and give the necessary and su cient conditions for the existence of a 3-DHF of type(t,m~n).Furthermore,we determine the existence of 3-RDGDD of type g~n.