On the metamorphosis of a G-design into a (G−e)-design

A G-design of order v is an edge disjoint decomposition of Kv into copies of the graph G. A metamorphosis of a G-design of order v into a (G−e)-design of order v is obtained by retaining the graph G−e from each block of G in the design, and rearranging the remaining edges to form further copies of G−e. Here, we prove that if a graph G with n edges admits an α-labelling and the graph G−e admits a ρ+-labelling, then there is a metamorphosis of a G-design of order 2n(n−1)x+1 into a (G−e)-design of the same order for all integers x.