Metamorphosis problems for graph designs

Abstract

A lambda-fold G-design of order n is a pair (X, B), where X is a set of n vertices and B is a collection of edge disjoint copies of the simple graph G, called blocks, which partitions the edge set of lambda K-n, (the undirected complete graph with n vertices) with vertex set X. Let (X, B) be a G-design and H be a subgraph of G. For each block b is an element of B, partition b into copies of H and G\H and place the copy of H in B(H) and the edges belonging to the copy of G\H in D(G\H). Now if the edges belonging to D(G\H) can be arranged into a collection D(H) of copies of H, then (X, B(H) boolean OR D(H)) is a lambda-fold H-design of order n and is called a metamorphosis of the lambda-fold G-design (X, B) into a lambda-fold H-design and denoted by (G > H) - M lambda(n). In this paper, the existence of a (G > H) - M lambda(n) for graph designs will be presented, variations of this problem will be explained and recent developments will be surveyed.