Equivalence of lower bounds on the number of perfect pairs

TL;DR

This paper demonstrates that two previously established lower bounds on the number of perfect pairs in certain graph factorizations are fundamentally equivalent, unifying their theoretical implications.

Contribution

It proves the equivalence of two different lower bounds on perfect pairs, showing they lead to the same fundamental limit in graph factorizations.

Findings

Both bounds are mathematically equivalent.

Unification of bounds simplifies understanding of perfect pairs.

Results apply to odd complete graphs and their factorizations.

Abstract

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.