Constructions For Large Sets And Overlarge Sets Of Disjoint Hybrid Triple Systems With Resolvablity

Let X be a finite set. A cycle triple on X is a set of three ordered pairs (x, y), (y,z) and (z,x) of X, denoted by (or , or ). A transitive triple on X is a set of three ordered pairs (x, y), (y, z) and (x, z) of X, denoted by (x, y, z), where x, y, z are distinct elements of X. A Hybrid triple system of order v HTS(v) is a pair (X,β), whereβis a collection of cycle and transitive triples on X such that every ordered pair of X belongs to one triple of B. An HTS(v) is resolvable (or almost resolvable), denoted by RHTS(v) (or ARHTS(v)), if its block set can be partitioned into parallel classes (or almost parallel classes).A large set of Hybrid triple systems of order v, denoted by LHTS(v), is a collection of {(X,β_i) : 1≤i≤4(v - 2)}, where every (X,β_i) is an HTS(v), and allβ_is form a partition of all cycle triples and transitive triples on X. An LRHTS(v) (or LARHTS(v)), is an LHTS(v), where every HTS(v) is resolvable (or almost resolvable).An overlarge set of Hybrid triple systems of order v, denoted by OLHTS(v), is a collection {(Y\{y},A_y~j) : y∈Y,j = 0,1,2,3}, where Y is a (v + 1)-set, each (Y\{y},j = 0,1,2,3,A_y~j) is a HTS(v) and all A_y~j s form a partition of all cycle and transitive triples on Y. An OLRHTS(v) (or OLARHTS(v)), is an OLHTS(v), where every HTS(v) is resolvable (or almost resolvable ). In this paper, we establish some directed and recursive constructions for LRHTS(v), LARHTS(v), OLRHTS(v), OLARHTS(v), and we obtain some new results.