Kirkman Holey Packings And Coverings

Packings and coverings are important subjects in the combinatorial design theory and have been widely used in many areas.In this dissertation,we investigate the properties,structure and existence of optimal resolvable packings and coverings.In chapter 2,we introduce two essential auxiliary designs,RMGDD and RHGDD, and study their recursive constructions systematically.Especially,the methods used by R.S.Rees to construct RGDD are generalized to construct RHGDD.Besides,we give a powerful composition construction for RHGDD.By the constructions established above,we completely solve the existence problems of 3-RMGDD and 3-RHGDD in chapter 3.In chapter 4,we analyze the structure and study the construction of KHPD and KHCD.In special,we make use of FDGDD and IRHGDD to construct these two kinds of designs.When gu(?)0(mod 3) and g(u-1)(?)0(mod 2),a KHPD(g~u) or a KHCD(g~u) is just a 3-RGDD of type g~u,whose existence has been completely determined by R.S. Rees(1993).In Chapter 5,we almost solved the existence problems of KHPD(g~u)'s and KHCD(g~u)'s when gu(?)0(mod 3) and g(u-1)(?)1(mod 2).So,when gu(?) 0(mod 3), the existence of KHPD(g~u)'s and KHCD(g~u)'s are almost established.Furthermore,we consider the existence of KHCD(g~u)'s when gu(?) 0(mod 3) in chapter 6.It is proved that for any even g and positive integer u≥6,there exists a KHCD(g~u) except for a class of possible exceptions.In chapter 7,several further study problems are presented.