Unbalanced Star-Factorizations of Complete Bipartite Graphs II

摘要

There are simple arithmetic conditions necessary for the complete bipartite graph K m,n to have a complete factorization by subgraphs which are made up of disjoint copies of K p,q . It is conjectured that these conditions are also sufficient. In any factor the copies of K p,q have two orientations depending which side of the bipartition the p-set lies. The balance ratio is the relative proportion, x:y of these where gcd(x,y)=1. In this paper, we continue the study of the unbalanced case (y > x) where p = 1, to show that the conjecture is true whenever y is sufficiently large. We also prove the conjecture for K 1,4-factorizations.