Equivalence of lower bounds on the number of perfect pairs
  Let c(F) be the number of perfect pairs of F and c(G) be the maximum of c(F) over all (near-) one-factorizations F of G. Wagner showed that for odd n, c(K_{n}) \geq n*phi(n)/2 and for m and n which are odd and co-prime to each other, c(K_{mn}) \geq 2*c(K_{m})*c(K_{n}). In this note, we establish that both these results are equivalent in the sense that they both give rise to the same lower bound. 
V. Ch. Venkaiah, K. Ramanjaneyulu, Neelima Jampala, J. Rajendra Prasad
cs.DM math.CO