Recent progress on the existence of perfect Mendelsohn designs

Abstract

Abstract A necessary condition for the existence of a (v,k,λ)-PMD is λv(v−1)≡0 ( mod k) . For k=3, 4 , and 5, some fairly conclusive results have already been established regarding the existence of (v,k,λ)-PMDs. For a very long time, the results for k=6 had been far from conclusive. In particular, for k=6 and λ=1, where the necessary condition for existence of a (v,6,1)-PMD is v≡0,1,3 or 4 ( mod 6) , only the cases of v≡0,1 ( mod 6) had been investigated with any measure of success. However, for v≡1 ( mod 6) , the problem of existence of a (v,6,1)-PMD is now completely settled; while for v≡0,3,4 ( mod 6) , the largest unknown cases are for v=198, 657, 148, respectively. The problem of existence of (v,7,λ)-PMDs has now been reduced to relatively few possible exceptions, where the most outstanding cases remain for λ=1. We shall present a brief survey of the current existence results for (v,k,λ)-PMDs. In addition, we shall mention some useful results relating to the existence of holey perfect Mendelsohn designs (HPMDs) and incomplete perfect Mendelsohn designs (IPMDs), with applications to packings and coverings.