The full metamorphosis of l-fold K4-designs into l-fold 3-star systems

Abstract

Let (X, B) be a λ-fold λ-design. If a triangle is removed from each block of B the resulting collection of 3-stars; 5, is a partial λ-fold 3-star system; (X, S). If the edges belonging to the deleted triangles can be arranged into a collection of 3-stars T} then (X,S UT) is a A-fold 3-star system, called a metamorphosis of the A-fold K-design (X, B) into a A-fold 3-star system. Label the elements of each block 6 with b1,b2,b3 and 64 (in any manner). For each i = 1,2,3,4 define a set of triangles Ti and a set of stars S as follows: For each block b= [b1,b2,b3,b4) belonging to B) partition 6 into a star centered at 6 and the triangle b\ b, then place the star in Si and the triangle in T. (X, Si) forms a partial A-fold 3-star system. Now if the edges belonging to the triangles in T< can be arranged into a collection of stars T then (X, Si U T/) is a A-fold 3-star system and we say that Mi =(X, Si U T) is the ith metamorphosis of (X, B).