P→v-factorization of symmetric complete bipartite digraphs

Abstract

Let P→_v be the directed path on v vertices and K _m,n* be the symmetric complete bipartite digraph with two partite sets having m and n vertices. A P→_v-factorization of K_m,n* is a set of arcdisjoint P→_v-factors of K_m,n* which is a partition of the set of arcs of K _m,n*. When v is an even number, the spectrum problem for a P→_v-factorization of K_m,n* has been completely solved (see [4]). In this paper, it is shown that a necessary and sufficient condition for the existence of a P→_v-factorization of K _m,n* for v is an odd number.