On the metamorphosis of a -design into a -design

A G -design of order v is an edge disjoint decomposition of K v 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 2 n ( n − 1 ) x + 1 into a ( G − e ) -design of the same order for all integers x .