Shortening Array Codes and the Perfect 1-Factorization Conjecture

The existence of a perfect 1-factorization of the complete graph K n, for arbitrary n, is a 40-year old open problem in graph theory. Two infinite families of perfect 1-factorizations are known for K_(2p) and K_(p+1), where p is a prime. It was shown in L. Xu et al. (1999) that finding a perfect 1-factorization of K_n can be reduced to a problem in coding, i.e. to constructing an MDS, lowest density array code of length n. 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-factorizations from the K_(2p) family, by applying the reduction mentioned above. Namely, techniques from coding theory are used to prove a new result in graph theory.