Shortening Array Codes and the Perfect $1$-Factorization Conjecture

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K
_n
, for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of
_Kp+1
and K
<sub>2

p</sub>
, where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1 -factorization of K
_n
is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K
_p+1
family of perfect 1 -factorization from the K
_2p
family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related.