The full metamorphosis of lambda-fold K-4-designs into lambda-fold 3-star systems

Abstract

Let (X, B) be a lambda-fold K-4-design. If a triangle is removed from each block of B the resulting collection of 3-stars; S, is a partial lambda-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 boolean OR T*) is a lambda-fold 3-star system, called a metamorphosis of the lambda-fold K-4-design (X, B) into a lambda-fold 3-star system. Label the elements of each block b with b(1), b(2), b(3) and b(4) (in any manner). For each i = 1,2, 3, 4 define a set of triangles T-i and a set of stars S-i as follows: for each block b = [b(1), b(2), b(3), b(4)] belonging to B, partition b into a star centered at b(i) and the triangle b\b(i), then place the star in S-i and the triangle in T-i. (X, S-i) forms a partial lambda-fold 3-star system. Now if the edges belonging to the triangles in T-i can be arranged into a collection of stars T-i* then (X, S-i boolean OR T-i*) is a lambda-fold 3-star system and we say that M-i = (X, S-i boolean OR T-i*) is the ith metamorphosis of (X, B). The full metamorphosis of (X, B) is the set of four metamorphoses {M-1, M-2, M-3, M-4}. The purpose of this work is to give a complete solution of the following problem: For which n and lambda does there exist a lambda-fold K-4-design having a full metamorphosis into lambda-fold 3-star systems?