Shortening Array Codes and the Perfect 1-Factorization Conjecture.

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, Kn, 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 K2p, where p is an arbitrary prime number (p > 2). It was shown in previous work that finding a perfect 1-factorization of Kn 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 Kp+1 family of perfect 1-factorization from the K2p family. Namely, techniques from coding theory are used to prove a new result in graph theory—that the two factorization families are related. [ABSTRACT FROM AUTHOR]