Optimal Cyclic Packings With Block Size Three

Let v, k, A be positive integers and v≥k. A (v,k,λ)-packing is a pair (V,B), where V is a u-set of elements (points) and B is a collection of k-subsets of V (blocks), such that every pair of points occurs in at most A blocks in B, denoted by CP(k,λ; v).An automorphism of a P(k,λ; v), (V, B), is a bijectionψ: V→V, such taht the induced mappingσ: B→B is also a bijection. A P(k,λ; v) is cyclic if it admits a cyclic automorphism group of order v that is spanned by a v-cycle. A cyclic P(k,λ; v) having only full block orbits is denoted by CP(k,λ; v).A cyclic P(k,λ; v) is said to be optimal if it has maximum possible number of base blocks, which is called cyclic packing number. In particular, the cyclic packing number of CP(k,λ;v) is denoted by CD(v,k,λ). If every pair of points occurs in exactly A blocks, the design is referred to as a cyclic balanced incomplete block design, denoted by CB(k,λ;v).Ernie F. Brickell and Victor K. Wei [7] have determined the cyclic packing num-bers CD(v,3,1). forλ> 1, M. J. Colbourn and C. J. Colbourn [20] have determined the existence of a CB(3,λ;v). In this paper, we research the existence of optimal CP(3,λ;v) withλ> 1, and determine the cyclic packing CD(v,3,λ), i.e.,...