Oberwolfach rectangular table negotiation problem

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H ( k , 3 ) be a bipartite graph with bipartition X = { x 1 , x 2 , … , x k } , Y = { y 1 , y 2 , … , y k } and edges x 1 y 1 , x 1 y 2 , x k y k − 1 , x k y k , and x i y i − 1 , x i y i , x i y i + 1 for i = 2 , 3 , … , k − 1 . We completely characterize all complete bipartite graphs K n , n that can be factorized into factors isomorphic to G = m H ( k , 3 ) , where k is odd and m H ( k , 3 ) is the graph consisting of m disjoint copies of H ( k , 3 ) .